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Supplementary notes for some of the listed theorems are provided below. Any suggestions for additions or corrections can be emailed to me and will be most welcome.

Theorem no. 1: The Four Colour Theorem
  1. The original announcement (September 1976) by Appel and Haken of their proof is available on free access here courtesy of Project Euclid. The full publication followed a year later: Part I and Part II (there are also microfiche supplements).
  2. The weblink from this theorem page used to be to a nice overview at Robin Thomas's homepage. This no longer seems to be available but I find there is a copy here (February 2024).
  3. The obvious progression from the sophisticated computer-assisted proofs of 4CT to formalised, computer-generated proofs, is discussed here (or direct pdf download, 2.6MB). The formal proof of 4CT by Georges Gonthier is also announced here (official report here).
  4. Brendan McKay "A note on the history of the four-colour conjecture", Journal of Graph Theory, Vol. 72, No. 3, 2013, 361–363, has given the earliest publication date for the Four Colour Conjecture as 1854 (it was discussed in correspondence in 1852). A preprint is here.
  5. There is a reference by Isabel Maddison to a "slightly different form" of the map-colouring question due to Möbius and his amateur mathematician friend Adoplh Weiske, publicised by Möbius in 1840 ("Note on the history of the map-coloring problem", Bull. Amer. Math. Soc., Volume 3, Number 7, 1897, page 257; online here). On p. 146 of Alexander Soifer, The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, Springer, 2009, you can find the details: a country is to be divided into 5 regions each bordering every other. Essentially, prove that \(K_5\) is nonplanar (cannot be the graph of a map drawn in the plane), so perhaps more a precursor to the deeper Hadwiger's Conjecture than the four-colour conjecture.
  6. In 1976, concurrently with the proof of 4CT, Richard Steinberg, while a PhD student of Bill Tutte at Waterloo, conjectured that any graph having no cycles of length 4 or 5 should be 3-colourable. A counter-example was found in 2016.
  7. Chris Budd tests the hypotheses of the 4CT (in its original map-colouring formulation).
  8. A glimpse of algebraic connections to 4CT are given in this Secret Blogging Seminar post; see also this by Dror Bar-Natan.
  9. A good overview of 4CT for Quanta by David S. Richeson.

Theorem no. 2: The Fundamental Theorem of the Calculus
  1. Making the accumulation function \(\int_0^x f(t)dt\) of Part II of the theorem the starting point for explaining the whole Fundamental Theorem makes good pedagogical and aesthetic sense, as argued by McQuillan, D. and Olsen, D. M., "A Truly Beautiful Theorem: Demonstrating the Magnificence of the Fundamental Theorem of Calculus," Journal of Humanistic Mathematics, Volume 6 Issue 2, pages 148-160. Online here.
  2. Part 1 and 2 of this theorem are not converse. Indeed a counterexample disproving the converse of Part 2 is provided by the always-insightful
  3. This theorem is the choice of Amie Wilkinson in Episode 1 and Aris Winger in Episode 64 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 3: The Bruck–Ryser–Chowla Theorem on Finite Projective Planes
  1. Original sources for this theorem:
    1. Bruck, R. H. and Ryser, H. J., "The nonexistence of certain finite projective planes", Canadian Journal of Mathematics, Vol. 1, Issue 1, 1949, pp. 88–92; online.
    2. Chowla, S. and Ryser, H.J., "Combinatorial problems", Canadian Journal of Mathematics, 2: 1950, pp.93–99; online.
    3. Lam, C.W.H., Thiel, L. and Swiercz, S., "The non-existence of finite projective planes of order 10", Canadian Journal of Mathematics, Vol. 41, Issue 6, 1989, pp. 1117–1123; online.
  2. Related: note(4) to Theorem #72.
  3. How to draw finite projective planes in the Euclidean plane is somewhat a matter of taste or convenience. I have chosen in the theorem description to illustrate the order 2 plane with a 'broken' middle circle; it is often drawn with this circle completed. Neither is correct if you require that intersections of lines of the projective plane correspond to intersections of lines drawn in the Euclidean plane. The issue (thanks to Dr. Pravas K for drawing my attention to it) is discussed here.

Theorem no. 4: Euclid's Infinity of Primes
  1. Our description of Euclid's theorem follows conventional practice in casting the proof as 'by contradiction'. One may take issue with this: see Michael Hardy and Catherine Woodgold, "Prime Simplicity", The Mathematical Intelligencer, December 2009, Volume 31, Issue 4, pp 44–52; online (paywalled). A good overview of the issue is given here. A mathoverflow entry on misconceptions in mathematics also addresses the issue: scroll down to find the relevant response + debate.
  2. A magisterial survey of proofs of this theorem, with extensive cross indexing, is given by Romeo Meštrović in "Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.–2017) and another new proof"; online.
  3. Another account of the "infinitude" of proofs of this theorem can be found, encapsulated in an elegant contextual discussion, at Gödel's Lost Letter. They have a further posting on this.
  4. Among the proofs to be found via (3), Fürstenberg’s 1955 topological proof has been given a more gentle exposition by Tai-Danae Bradley; for that matter it has its own Wiki page!
  5. Among the proofs not found via (3) (it would have appeared too late, I think) is this lovely one-line proof by contradiction by Sam Northshield, with products taken over all primes \(p\), supposedly finite in number and with \(P\) denoting the product of all these primes: $$0<\prod_p \sin(\tau/2p)=\prod_p \sin(\tau/2p+\tau P/p)=0.$$ (As usual, \(\tau\) denotes circumference of unit circle. More details are given by John D Cook here and Cut-The-Knot here, and by Northshield himself here.
  6. Another intriguing approach is taken by Christian Elsholtz in "Fermat's Last Theorem Implies Euclid's Infinitude of Primes"; online, in which FLT and many other classic theorems, not all number theoretic, are shown to be false in a world where there are only finitely many primes. This and other recent work on 'infinity' proofs is described by Anna Kramer for Quanta magazine here.
  7. The arxiv preprint by Chris Caldwell and Yeng Xiong cited on this theorem page was published as "What is the Smallest Prime?" Journal of Integer Sequences, Vol. 15 (2012), Article 12.9.7; online. (I've kept to the arxiv citation on the theorem page because it's a much shorter URL. It also has facsimiles of historical documents which are typeset in the published article: although the resulting pdf is much smaller this seems a pity). Another discussion of the role of 1 as a non-prime is here by Evelyn Lamb.
  8. This theorem is the choice of Ken Ribet in Episode 22 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 5: The Chinese Remainder Theorem
  1. John D. Cook gives a neat description of how our congruence system \(x\equiv y_i\pmod{n_i}\) may be solved as $$y_1(N/n_1)^{\varphi(n_1)}+y_2(N/n_2)^{\varphi(n_2)}+\ldots + y_r(N/n_r)^{\varphi(n_r)},$$ where \(N=n_1n_2\cdots n_r\) and \(\varphi\) is Euler's totiant function. Thus our system with \(y_1=3, y_2=4,n_1=4, n_2=5\) is solved by \(3\times 5^2+4\times 4^4=1099\pmod{20}=19.\)
  2. A good online CRT solver is this by which gives all the working and accepts negative number inputs.
  3. Numberphile have a great video of Tadashi Tokieda applying CRT in card magic.
  4. A multivariable CRT (i.e. a system of linear modular equations) is given by Oliver Knill.
  5. Quanta magazine offers an excellent account of CRT and its place in the modern mathematical sciences by Lakshmi Chandrasekaran.

Theorem no. 6: The Fundamental Theorem of Algebra
  1. The current page describing this theorem replaces an older version using a quadratic polynomial, easier to assimilate perhaps but it seemed more revealing to have a cubic with both real and imaginary roots. The old version is archived here.
  2. Daniel Litt gives a nice 'minimal' proof of the theorem. Dating from 1941, a proof essay by Littlewood remains very accessible and enjoyable: J.E. Littlewood, "Every Polynomial has a Root", J. London Math. Soc., Vols. 1-16, Issue 2, 1941, pp. 95–98; online (paywall; reprint here, January 2024). On the subject of proofs of the theorem, I found this Quora answer enlightening.
  3. Daniel J. Velleman offers "The Fundamental Theorem of Algebra: A Visual Approach", The Mathematical Intelligencer, December 2015, Volume 37, Issue 4, pp 12–21, preprint found online here (January 2024).
  4. Paul Taylor has posted this English translation of "Gauss's second proof of the fundamental theorem of algebra"; as I discovered thanks to John D. Cook who, on the same theme, has this nice post about why the Fundamental Theorem of Algebra is proved using Analysis.
  5. Featured in Math Scholar's thread Simple proofs of great theorems.

Theorem no. 7: The Fundamental Theorem of Arithmetic
  1. The origins of this theorem are given expert treatment by Mary Joan Collison: "The unique factorization theorem: from Euclid to Gauss", Mathematics Magazine, Vol. 53, Issue 2, 1980, pp. 96–100; online (paywall).
  2. This theorem description replaces an older version which had a more explicit illustration of walks defined by prime factorisations. The new version has a more sophisticated plot and the accompanying text sketches the proof of the theorem and has more on Goldbach. For those who prefer something more simple-minded (less crowded!) I have left the old version here.
  3. Symbolically, this theorem asserts a unique (up to order) representation of a positive integer \(n\) as a product of powers of primes: $$n=p_1^{a_1}p_2^{a_2}\cdots p_r^{a_r},$$ (with 1 being by convention the value of the empty product). This representation is implicit or explicit in many proofs in elementary number theory. It is also explicit in various calculations, e.g. in counting certain magic squares (Theorem 129); in calculating periods of modular Fibonacci sequences (Theorem 235, notes(4)) ; and in calculating Liouville's function (see Theorem 197, notes(2))
  4. By the way, non-symbolically, the phrase 'up to order' can be removed from the statement of this theorem by writing "every positive integer is uniquely expressed as a product of non-decreasing primes".
  5. Although Goldbach does not imply that every point \((2k,2)\) will eventually appear on the walks plot illustrating this theorem, Andrzej Schinzel has shown ("Sur une conséquence de l'hypothèse de Goldbach", Bulgar. Akad. Nauk. Izv. Mat. lnst., 4 (1959) 35–38) that Goldbach does imply that every odd integer greater than 17 is a sum of three different primes, which would mean every point \((2k+1,3),\,k>8\), is plotted. Sierpinski on p. 124 of Elementary Theory of Numbers, Elsevier, 1988 adds "It follows from the results of Vinogradov that each sufficiently large odd number is such a sum". Of course H.A. Helfgott's proof of the Ternary Goldbach Conjecture confirms that every odd integer greater than 5 is a sum of three primes. It is not mentioned explicitly in Helfgott's preprint but I asked him and a result for three distinct primes is indeed implied.
  6. A lovely implementation of FTA for positive integers up to 99 has been knitted by Sondra Eklund!
  7. Evelyn Lamb created a math-poetry project called Prime Factorization as Verse which was extended on her twitter account.
  8. An intriguing exploration of iterated factorisation by Jon Awbrey can be found here, under the title Riffs and Rotes.

Theorem no. 8: The Central Limit Theorem
  1. Original source for the Turing CLT story is S. L. Zabell, "Alan Turing and the Central Limit Theorem", The American Mathematical Monthly, Vol. 102, No. 6, 1995, pp. 483–494; online (paywall), which is authoritative on the theorem's origins. The Lindeberg's CLT is Lindeberg, J . W., "Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung", Mathematische Zeitschrift, 15, pp. 211–225; online (paywall; euDML).
  2. The recommended weblink from this theorem was previously this by Thayer Watkins, which is still good, but the applets are a bit problematical now.
  3. Good modern animations of CLT can be found however: this by Michael Freeman.
  4. John D. Cook gives an elegant summary of the historical origins of the normal distribution here.
  5. There is a formal proof of the theorem, announced here by Jeremy Avigad, Johannes Hölzl, Luke Serafin.

Theorem no. 9: Fermat's Last Theorem
  1. Wiles' proof of FLT occupied a complete issue of Annals of Mathematics: Wiles' "Modular elliptic curves and Fermat's Last Theorem", vol. 141, no. 3 (1995), pp. 443–551; online (paywall), accompanied by Richard Taylor and Andrew Wiles, "Ring-theoretic properties of certain Hecke algebras", pp. 553– 572; online (paywall). Wiles' paper is prefaced by the famous Fermat quote (in Latin) ending "Hanc marginis exiguitas non caperet".
  2. The current page on FLT replaces an earlier one which had a more primitive graphic (without the benefit of Benoît Leturcq's fun Fermat sketch) and omitted the Fermat near-miss example). I have kept the old version here in case anyone prefers a less 'busy' page.
  3. Fermat near-misses are based on lots of clever algebraic number theory: see this by Noam Elkies.
  4. The reference in the illustration to probability theory is a nod to the fact that Fermat co-invented it. See this, for example, by Peter Lee.
  5. A reference to "Molina's Urns" can be found, for example, in Frederick Mosteller, Fifty Challenging Problems in Probability with Solutions, Dover reprint, 2000 (problem 56 on p. 88).
  6. Another elegant account (7.1MB pdf) of the resolution of FLT, which gives a little more on subsequent developments, can be found in this list of lectures by Karl Rubin.
  7. The role of modular forms in the proof of FLT is made explicit in an iconic presentation by Ken Ribet "The five fundamental operations of mathematics: addition, subtration, multiplication, division, and modular forms", which can be found here (dated March 2008, a 7MB pdf).
  8. The role of class numbers in the study of FLT is described by Kevin Hartnett in this Quanta article.
  9. FLT has a formal proof in Lean for Case I and for regular prime exponents. A formalisation in Lean of Wiles's proof has been proposed by Kevin Buzzard, see this and this, for example (the former is an accessible way to appreciate just what a monumental achievement is the proof of FLT!)

Theorem no. 10: Bayes' Theorem
  1. Original source for this theorem: "An Essay towards Solving a Problem in the Doctrine of Chances. By the Late Rev. Mr. Bayes, F. R. S. Communicated by Mr. Price, in a Letter to John Canton, A. M. F. R. S". Phil Trans Royal Society of London, Vol. 53, 1763, pp. 370–418; online.
  2. More on the origins of Bayes' theorem can be found in its Wiki entry. Notable is an investigation by Stephen M. Stigler, "Who Discovered Bayes's Theorem?", The American Statistician, 37 (4), 1983, 290–296; online (paywall; pdf here, February 20024). Stigler finds credible evidence that Bayes' Theorem was first discovered by Nicholas Saunderson. The article is reproduced in Stigler's book Statistics on the Table: The History of Statistical Concepts and Methods, Harvard University Press, paperback edition 2002.
  3. You can view a Bayes' theorem prior as allowing the inclusion of numerical odds for subjective assumptions. I think a Bayesian would argue that not including these odds is to make an equally subjective assumption that prior knowledge is irrelevant. This is very well argued by Mike Lee and Benedict King in this Conversation article.
  4. A nice retrospective on the history of use and abuse of Bayes' Theorem is provided by Bradley Efron in "Bayes’ Theorem in the Twenty-First Century", Science, Vol. 340, Number 6137, 2013, pp. 1177–1178; online (paywall; copy here February 2024). Also recommended, this Scientific American blog entry by John Horgan (which was paywalled at one point but seems desktop-accessbile as of February 2024).
  5. A beautiful animated illustration of conditional probability by Victor Powell is here, while Will Kurt's Bayesian blog Count Bayesie has a nice alternative to our cow counting illustration here.
  6. Bayes applied by Chris Budd to the Monty Hall problem. Norman Fenton and David Lagnado on Bayes applied in the legal profession.
  7. Robin Evans contributes this entry (with a fine A3 poster version) on Bayes to the Oxford Mathematics Alphabet.

Theorem no. 11: Lagrange's Four-Squares Theorem
  1. Original source for this theorem: Lagrange, J.-L., "Démonstration d’un théorème d’arithmétique", Nouveaux mémoires de l’Académie Royale des Sciences et Belles-lettres de Berlin, Anneé 1770, 1772, pp. 123–133; online (as reproduced in Lagrange's collected works). Definitive on the history of Lagrange's contributions is Jenny Boucard, "Lagrange and the four-square theorem", Lettera Matematica, Vol. 2, 2014, pp. 59–66; online.
  2. Very good on the evolution of the four square's problem is Mark B. Beintema and Azar N. Khosravani,"Universal forms: the four-square theorem and its generalizations", Missouri J. Math. Sci., 15(3), 2003, pp. 153–161; online. There is an attractive popular article on the history of the theorem by Anuradha S. Garge, "Lagrange's Four Squares theorem: from conjecture to proof", At Right Angles, Vol. 1, No. 2, 2012, pp. 5–9; online (complete issue, 13MB pdf; the article is extracted here, 160KB pdf, February 2024).
  3. The problem of finding a four-squares representation of a given integer is discussed in Paul Pollack and Enrique Treviño, "Finding the four squares in Lagrange's theorem, Integers, Vol. 18A (2018), paper A15; online. There is an online app by Dario Alpern here.

Theorem no. 12: The Matrix Tree Theorem
  1. Original source for this theorem: Kirchhoff, G.m "Über die Auflösung der Gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer Ströme geführt wird", Ann. Phys. Chem., 72, 1847, pp. 497–508; online (paywall; a pdf download is available via semanticscholar).
  2. The reliability calculation here, in general, is asking what is the probability that deleting \(e-n+1\) edges uniformly at random will result in a spanning tree. For a plane graph with \(e\) edges, \(n\) vertices and \(f\) faces, and having \(t\) spanning trees, the calculation becomes \(t(f-1)!(n-1)!/e!\), which neatly shows that the probability is identical for the dual graph.
  3. The weblink for this page proves MTT using the Binet–Cauchy theorem from matrix theory. A standard combinatorial approach uses induction based on deletion-contraction as in these notes by David P. Williamson. A direct combinatorial proof by Doron Zeilberger is given in section 4 of "A combinatorial approach to matrix algebra", Discrete Mathematics, Vol. 56, Issue 1, 1985, pages 61–72; online. The proof of Seth Chaiken and Daniel J. Kleitman given in "Matrix Tree Theorems", Journal of Combinatorial Theory, Series A, Vol. 24, Issue 3, May 1978, pp 377–381 is also of interest; online. A nice random walk proof is given by Michael J. Kozdron here, invoking the algorithm of David Wilson for selecting a spanning tree uniformly at random. Gil Kalai has a nice overview here.
  4. See Garrys Tee's article in vol. 30 (3.9MB pdf) of Image for more on the history and applications of determinants.

Theorem no. 13: Fermat's Little Theorem
  1. Regarding the origins of this theorem an authoritative source is Colin R. Fletcher, "A reconstruction of the Frenicle-Fermat correspondence of 1640", Historia Mathematica, Vol. 18, Issue 4, 1991, pp. 344–351; online. On the possible origins of the theorem in ancient Chinese mathematics, see this fascinating investigation by Qi Han and Man-Keung Siu, "On the myth of an ancient Chinese theorem about primality", Taiwanese J. Math., Vol. 12, Number 4, 2008, pp. 941–949; online (paywall; preprint, November, 2023).
  2. Counterexamples to the converse of Fermat's Little Theorem are called Carmichael numbers, 561 being the smallest. A good popular introduction is this Quanta article by Jordana Cepelewicz.
  3. For more on Guiga's conjecture see Takashi Agoh, "On Giuga's Conjecture", Manuscripta mathematica, Vol. 87, Issue 4, 1995, pp. 501–510; online. Also good but apparently no longer with open access options is D. Borwein, J.M. Borwein, P.B. Borwein, R. Girgensohn, "Guiga's conjecture on primality", American Mathematical Monthly, vol. 103, 1996, pp 40–50; online (paywall). The lower bounds for counterexamples (>4771 prime factors, > 19908 digits) are from this presentation (4.4MB pdf file, November 2023).
  4. An alternative strengthening of its hypothesis that makes Fermat's test necessary and sufficient is Lucas's test, see Vaughan Pratt's Theorem.
  5. The proof of Fermat's Little Theorem given in the description here is due to James Ivory, "Demonstration of a theorem respecting prime numbers", New series of The Mathematical Depository, 1 (II),1806, pp 6–8. You appear to be able to access all of this volume free online from google books. The proof is given in slightly more detail by cut-the-knot, which is where I took my version from.
  6. Euler's important generalisation of Fermat's theorem should be recorded here. Euler's totient function \(\phi(n)\), for \(n\) a positive integer, is the number of positive integers less than \(n\) and coprime to \(n\). Now for \(m\) a positive integer and \(a\) any integer coprime to \(m\) we have \(a^{\phi(m)}=1 \mbox{ mod } m\). Art of Problem Solving give a proof. For example, \(10^{\phi(9)}=10^6\) which has remainder \(1\) on division by \(9\). For prime \(p\) we have \(\phi(p)=p-1\) so that Fermat's theorem is an immediate corollary. Euler's first proof of Fermat's theorem was published in 1736. He published several other proofs, culminating in 1763 with this generalisation, published in his "Theoremata arithmetica nova methodo demonstrata"; online.
  7. This theorem is the choice of Jordan Ellenberg in Episode 4 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.
  8. Tangential but the cube images on this theorem page come from the webpage of Jessica Fridrich who is the subject of a valuable blog post at Gödel's Lost Letter.

Theorem no. 14: Cook's Theorem
  1. Original source for this theorem is: Cook, Stephen, "The complexity of theorem proving procedures", Proceedings of the Third Annual ACM Symposium on Theory of Computing, 1981, pp. 151–158; online (paywall; the paper has been TEXed into pdf by Tim Rohlfs, available here.) There is a nice 40th anniversary blog post on Lance Fortnow's blog. Another entry there explains how the name 'NP-complete' was invented in response to a poll run by Knuth in 1973.
  2. An English translation of Leonid Levin's paper, together with a thorough analysis, may be found here. The background to Levin's work in the USSR is described by B.A. Trakhtenbrot, "A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms", IEEE Annals of the History of Computing, Vol. 6, Issue 4, 1984, pp. 384–400; online (paywall; a facsimile was here, March 2024, 24MB pdf).
  3. CACM blog entry, Robin K. Hill, "Cook-Levin: The Ugly Underbelly is Good for Us" gives a very good sketch proof of this theorem.
  4. It may be asserted that Kurt Gödel was the first to ask the P vs NP question, in a letter to von Neumann in 1956. See page 250 of John W. Dawson Jr, Logical Dilemmas: The Life and Work of Kurt Gödel, A K Peters, 1997. Ash Jogalekar @curiouswavefn has tweeted a picture of the revelent passge.
  5. I provide a little more (amateur) analysis of this theorem as an example of a 'simultaneity' in mathematics here.
  6. Regarding P=NP? Gerhard Woeginger provides a valuable 'clearing house' for proof/disproof attempts.
  7. Although it is widely believed that P≠NP there are a few prominent sceptics, for example Don Knuth (see Q. 17 here) and R.J. Lipton (see this). Bill Gasarch has conducted three surveys over nearly 20 years regarding people's beliefs about P=NP. The latest (and links to previous) is described here.

Theorem no. 15: The Cauchy–Frobenius Lemma
  1. The current version of this page replaces an earlier version which concentrated on depicting permutations and the idea of fixing a point. This new version offers a basic illustration of how the lemma applies in counting, an illustration which is continued in the page for the Pólya-Redfield Enumeration Theorem. The old version of this page is retained here.
  2. Furthermore, I have abandoned the original name "The Orbit Counting Lemma" of this page since the attribution to Cauchy and Frobenius seems appropriate, and because it made for an easier correspondence with the French translation of the page. A probably definitive account of the lemma is given by Peter M. Neumann in "A lemma that is not Burnside's", The Mathematical Scientist, Vol. 4, Issue 2, July 1979, pp. 133–141; this link used to give online access directly but (Jan. 2022) access now appears to be via google docs from here (and is far from direct—click on the fifth file icon and navigate to page 133 as best you can).
  3. The lemma applies to arbitrary group actions; I preferred to limit myself to permutation groups to avoid having to define what is meant by a group action.
  4. This theorem is the choice of Mohamed Omar in Episode 10 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 16: Sperner's Lemma
  1. Original source for this theorem is: E. Sperner, "Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes", Abh. Math. Sem. Univ. Hamburg, 6, 1928, pp. 265–272; online (paywall). The relationship between Sperner's result and that of Knaster–Kuratowski–Mazurkiewicz which is more explicitly a lemma for Brouer's Fixed Point Theorem is discussed in the Wiki and Springer Encyclopedia entries for the former.
  2. My original weblink from this theorem page was to the Sperner entry at cut-the-knot. I prefer, because of java applet browser issues, to relocate this link to my notes page. I hope that at some future date I can re-instate it because some clever and altruistic people have honoured Alexander Bogomolny by giving cut-the-knot a new lease of life!
  3. Sperner's Lemma provides an elementary addition to Maryam Mirzakhani's legacy.
  4. There is a fun game based on Sperner by Kyle Burke called Atropos. Background in this Quanta article by Ben Brubaker on Burke's professor at the University of Southern California, Shang-Hua Teng.
  5. This theorem is the choice of James Tanton in Episode 27 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 17: The Well-Ordering Theorem
  1. The origins of this theorem are well summarised by Michael Hallett in the abstract of an entry in the Springer collection of Zermelo's works. The first official publication was E. Zermelo, "Beweis, daß jede Menge wohlgeordnet werden kann. (Aus einem an Herrn Hilbert gerichteten Briefe)", Mathematische Annalen, Vol. 59, 1904, pp. 514–516; online (paywall; facsimile).
  2. The Banach–Tarski paradox was published in Stefan Banach and Alfred Tarski, "Sur la décomposition des ensembles de points en parties respectivement congruentes", Fundamenta Mathematicae, Vol. 6, 1924, pp. 244–277; online. There were antecedents, see its Wiki page.
  3. As the remark on the Banach–Tarski Paradox suggests, it is infinity, rather than well-ordering or the Axiom of Choice, that can defy intuition. This 'anti-anti-Banarch–Tarski' argument is very well made by Asaf Karagila (follow the link back from the wonderful cartoon; I found this originally at Boole's Rings).
  4. There is a charming Youtube 'demonstration' of Banach–Tarski by Joel David Hamkins.

Theorem no. 18: Brouwer's Fixed Point Theorem
  1. Original sources for this theorem:
    1. Bohl, P., "Über die Bewegung eines mechanischen Systems in der Nähe einer Gleichgewichtslage", J. Reine Angew. Math., 127 (3/4), 1904, pp. 179–276; online (paywall).
    2. Jacques Hadamard, "Note sur quelques applications de l’indice de Kronecker", in Jules Tannery, Introduction à la théorie des fonctions d’une variable (Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477; online (facsimile)
    3. Brouwer, L. E. J., "Über Abbildung von Mannigfaltigkeiten", Mathematische Annalen, Vol. 71, 1912, pp. 97–115; online (paywall; facsimile).
  2. There is a comparison of Hadamard's and Brouwer's work on fixed point topology in chapter 13 of Vladimir Maz'ya and Tatyana Shaposhnikova, Jacques Hadamard, A Universal Mathematician, American Mathematical Society, 2000. They quote Donald M. Johnson: "Hadamard's Note is markedly similar to Brouwer's classic paper defining the degree of a mapping ... Yet there is hardly any doubt that Brouwer's is the superior work. Whereas Hadamard's Note stands at the end of a great line of mathematical development, Brouwer's great paper looks forward to new avenues of topological thinking".
  3. The contention that there is no constructive proof of 'BFPT' goes very deep. See this at mathoverflow, for example.
  4. A very nice discussion by Phil Wilson of Brouwer and constructivist mathematics is given here by plus magazine.
  5. The wikipedia page for this theorem has good coverage of the necessity of its hypotheses.
  6. For French readers, an attractive article from the CNRS's Images.
  7. This theorem is the choice of Francis Su in Episode 20 and of Holly Krieger in Episode 25 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 19: Dilworth's Theorem
  1. Original sources for this theorem:
    1. Dilworth, Robert P., "A decomposition theorem for partially ordered sets", Annals of Mathematics, 51 (1), 1950, pp. 161–166; online (paywall).
    2. T. Gallai and A.N. Milgram, "Verallgemeinerung eines graphen-theoretischen satzes von Redei", Acta SC. Math., Vol. 21, 1960, pp. 181–186. In don't find this paper online but it is quoted in, for example, I.Ben-Arroyo Hartman, "Variations on the Gallai-Milgram theorem", Discrete Mathematics, Vol. 71, Issue 2, 1988, pp. 95–105; online. Path cover, the subject of Gallai and Milgram has its own Wiki page.
    3. Cameron's attribution to Gallai and Milgram also appears on the former's wiki page with the citation P. Erdős: "In memory of Tibor Gallai", Combinatorica, 12 (1992), pp. 373–374; online (paywall). The earlier result of Erdős–Szekeres is given in Erdős, P. and Szekeres, G., "A combinatorial problem in geometry", Compositio Mathematica, Tome 2 (1935), pp. 463–470; online; and boasts its own wikipedia page.
  2. The number of distinct antichains in the lattice of subsets of \(\{1,\ldots,n\}\) is called the \(n\)-th Dedekind number. It is sequence A000372 at OEIS and the subject of an absorbing August 2023 Quanta article by Rachel Crowell.
  3. The current illustration of this theorem replaces one based on snooker balls which was less informative but to which I may as well retain a link.

Theorem no. 20: The Merton College Theorem
  1. Francesca Lovell-Read offers a good historical account of this theorem.

Theorem no. 21: Brun's Theorem
  1. The original source of this theorem is Brun, Viggo (1919). "La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., où les dénominateurs sont nombres premiers jumeaux est convergente ou finie", Bulletin des Sciences Mathématiques, 43: avril, pp. 100–104, mai, pp. 124–128; online.
  2. Brun's \(cx/(\ln x)^2\) bound on the twin prime count is very well motivated by James Maynard in his PROMYS Europe 2015 lecture "Patterns in the Primes" which can be accessed here.
  3. Brun's sieve received less attention in the early 1900s than it perhaps deserved. In the introduction to George Greaves, Sieves in Number Theory, we read that "The mathematical community did not immediately give Brun's results the recognition they later received. E. Landau left Brun's 1920 paper unread for a decade, apparently because he was not predisposed to believe that elementary methods as used by Brun could penetrate problems such as Goldbach's to the asserted extent". And in the introduction to Heini Halberstam and Hans-Egon Richert's classic, Sieve Methods, we read "its complicated structure and Brun's own early accounts tended to discourage closer study", and that Landau, writing an account of the method, commented "Myself I have never bothered to thoroughly work through Mr. Brun's original work" (google's translation).
  4. Euler proved the divergence of the prime reciprocals in 1737, the climax of the paper which introduced the product formula for the zeta function (see theorem 246).
  5. The first 100000 twin primes are listed here.

Theorem no. 22: Cantor's Uncountability Theorem
  1. Original source for this theorem: Cantor, G., "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", Journal für die reine und angewandte Mathematik, Vol., 1874, Issue 77, pp. 258-262; online (paywall; facsimile, there is also a link to the 1883 French translation). The article has its own Wiki page.
  2. There is an English translation of Cantor's article here at the website of James R Meyer, a resource which is controversial but has much of interest and value.
  3. There is an interesting discussion about the earliest apparence of the diagonal argument in Cantor's work here at However, it seems to be agreed that diagonalisation was already discovered by Paul du Bois-Reymond in 1875 (see, for example, the MacTutor link from his entry in the list of mathematicians; thanks to Thony Christie for alerting me).
  4. Amateur refutations of Cantor's diagonalisation proof of this theorem inspire an interesting discussion of how non-professionals learn and think about logic: Wilfrid Hodges, "An Editor Recalls Some Hopeless Papers", Bulletin of Symbolic Logic, Vol. 4, No. 1, 1998, pp. 1–16; online (paywall; a pdf copy is here, May 2024).
  5. Cantor was working in the context of a prevailing theological stance on the infinite and he engaged productively with the Catholic church. A good account is given by Chris Lambie-Hanson.
  6. This theorem is the choice of Skip Garibaldi in Episode 34 and of Yoon Ha Lee in Episode 61 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 23: Cantor's Theorem
  1. Original source for this theorem: Cantor, G.,"Über eine elementare Frage der Mannigfaltigkeitslehre", Jahresericht der Deutsch. Math. Vereing., Vol. 1, 1891, pp. 75–78; online (facsimile).
  2. The popular story has it that thinking about infinity, and attacks by others (notably Kronecker) on his thinking about infinity, drove Cantor insane. A much less sensational view is given in this blog entry by Richard Zach and this by Thony Christie.

Theorem no. 24: Kuratowski's Theorem
  1. Original source for this theorem: it was announced in K Kuratowski, "Sur les courbes gauches", Annales Polonici Mathematici, 8, 1929, p. 324. The publication, with proof, followed in "Sur le problème des courbes gauches en topologie", Fundamenta Mathematica, 15, 1930, pp. 271–283; online. The abstract of Frink and Smith's unpublished proof appears here (Bull. AMS, 36, 3, p. 214) under 'Abstracts of papers', where it is no. 179 (but merely says "One of the results of this paper is a simple necessary and sufficient condition that an arbitrary linear graph be mappable on a plane.")
  2. The multiple discoveries and proofs of this theorem are wonderfully charted in John W Kennedy, Louis V Quintas, Maciej M Sysłois, "The theorem on planar graphs", Historia Mathematica, Vol. 12, No. 4, 1985, 356–368; online.
  3. Bill Tutte, under the Blache Descartes pseudonymn wrote a little poem about the non-planarity of \(K_{33}\) which may be read on p. 17 of this fine tribute (pdf 180KB) by Graham Farr and James Oxley (there was a copy on the Bill Tutte Memorial facebook page but when I checked in June 2024 it wouldn't display. You can try here and meanwhile the facebook page itself has many interesting things).

Theorem no. 25: Wagner's Theorem
  1. Original source for this theorem is Wagner, K., "Über eine Eigenschaft der ebenen Komplexe", Math. Ann., 1937, 114: pp. 570–590; online (paywall), at Göttinger Digitalisierungszentrum.

Theorem no. 26: Euler's Polyhedral Formula
  1. Original sources for this theorem:
    1. Euler, L., "Elementa doctrine solidorum", Novi Commentarii academiae scientiarum Petropolitanae 4, 1758, pp. 109–140; online (where it is suggested that the paper was read to the Berlin Academy on November 26, 1750).
    2. Legendre, A.-M., Éléments de géométrie, Paris, 1794; online. Very authoritative on Legendre's contributions to geometry, including his proof of Euler's formula, is Giora Hon and Bernard R. Goldstein, "Legendre’s revolution (1794): the definition of symmetry in solid geometry", Archive for History of Exact Sciences, Vol. 59, 2005, pp. 107–155; online (paywall; pdf download, June 2024).
    3. Von Staudt, G., Geometrie der Lage, Nürenberg, 1847. There is more detail in this excellent AMS Feature column by Joe Malkevitch where credit for the proof as presented on my page is given to a popular book by Hans Rademacher and Otto Toeplitz.
  2. Joe Malkevitch's column offers a good overview generally of this theorem and has extensive references. And he has a Part II which describes many applications and ramifications of the formula. Plus magazine has this lovely account of Euler's formula by Abigail Kirk.
  3. It has been argued that Descartes discovered the polyhedral formula before Euler although his version, which does not recognise the significance of the polyhedral 'edge', can perhaps not be considered equivalent. Tony Phillips introduces the topic; it is also covered in Dave Richeson's book (our theorem page's recommended reading) which in turn refers us to P. J. Federico's whole book dedicated to this one question.
  4. A classic of the philosophy of mathematics is dervied from Euler's formula: Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, edited by Worrall and Zahar, Cambridge University Press, 1976. A good account may be found here at the Stanford Encyclopedia of Philosophy.

Theorem no. 27: The Pythagorean Theorem
  1. A nice summary of the theorem's history is provided by Manjul Bharagava here. It would seem that even 'Pythagorean' is a misnomer. Piers Bursill-Hall gave me the following valuable pointers:
           "work by David Fowler (U.Warwick) and Wilbur Knorr (U.Stanford) more than a couple of decades ago demonstrated very convincingly what had been at least implied since German philologists and classicists in the late 19th century debunked most of the mythology around Pythagoras. A good place to start would be the references and footnotes in David Fowler's The Mathematics of Plato's Academy: A New Reconstruction or Knorr's The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry. It ought not to be a controversial, surprising, or new item to mathematicians ... although sadly that is not the case."       
  2. The theorem is equivalent to Euclid's notorious 5th axiom, the Parallel Postulate, in the sense that each may be derived directly from the other, a fact which seems to date back to Legendre. More details here. It is also equivalent to Heron's formula (see Theorem no. 76) as revealed in a beautiful article by Vaughan Pratt, "Factoring Heron," The College Mathematics Journal, Vol. 40, no. 1, January 2009, pp. 15–16; online.
  3. The fact that the theorem is Prop. 47 of Book 1 of Euclid makes this diagram by Thomson Nguyen of dependencies in Book 1 of interest!
  4. There is a beautiful account by Steven Strogatz of a particularly elegant proof of the Pythagorean Theorem attributable to and, Strogatz argues, bearing all the hallmarks of, Albert Einstein.
  5. Krzysztof Apt's elegant essay on Dijkstra's work "Edsger Dijkstra, the man who carried computer science on his shoulders", Inference, vol. 5, issue 3, 2020; online, gives some background to Dijkstra's discovery of his generalisation of the Phythagorean theorem. Further interesting commentary by Jan Stevens can be found here.
  6. An intriguing contribution from two New Orleans high school students Ne’Kiya D Jackson, Calcea Rujean Johnson is a proof of Pythagoras using trigonometry (the rule of sines) in a way which avoids (indirect) appeal to Pythagoras. See this from the UK Guardian. A follow-up. Some incisive analysis by Tony Forbes appears in issue 313 of M500.
  7. A nice blog entry from John D. Cook on a 'unified Pythagorean theorem" which applies in non-Euclidean cases as well as the Euclidean.
  8. This theorem is the choice of Henry Fowler in Episode 7 and of Fawn Nguyen in Episode 39 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 28: Ramsey's Theorem
  1. Original sources for this theorem:
    1. Ramsey, F., "On a problem of formal logic", Proc. London Math. Soc., Vol. s2-30, Issue 1, 1930, pp. 264–286; online (paywall; facsimile 1.4MB pdf, June 2024).
    2. Paul Erdős and George Szekeres, "A combinatorial problem in geometry", Compositio Mathematica, 2, 1935, pp. 463–470; online.
    3. Thoralf Skolem also gives an early proof of Ramsey's theorem in "Ein kombinatorischer Satz mit Anwendung auf ein logisches Entscheidungsproblem", Fundamenta Mathematicae, 20, 1933, pp. 254–261; online. This is not an independent discovery though—Skolem opens by citing Ramsey's paper. However, the values of the smallest Ramsey numbers were computed in R. E. Greenwood and A. M. Gleason, "Combinatorial relations and chromatic graphs", Canadian Journal of Mathematics, Vol. 7, 1955, pp. 1– 7; online. This paper does not cite Ramsey but rather "... a question in the William Lowell Putnam Mathematical Competition held in March 1953."
    4. The definitive source for (updated) values of Ramsey numbers is the dynamic survey Stanisław Radziszowski, "Small Ramsey numbers", Electronic J. Combinatorics; online.
  2. Improvements on the original Erdős–Szekeres upper bound (stated, as in our page, for \(R(s+1,t+1)\), to simplify the binomial coefficient) are reviewed in this latest (May 2020) advance by Ashwin Sah. The latest bounds (March 2017) for \(R(5)\) are due to Vigleik Angeltveit and  Brendan D. McKay. A big breakthrough came in March 2023 when Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe announced a reduction in the asymptotic upper bound for \(R(k)\) from \(4^k\) to \((4-\varepsilon)^k\). A post by Gil Kalai gives more details and Leila Sloman has this for Quanta magazine. The result has been verified formally, see this guest post by Bhavik Mehta for the Xena blog.
  3. For lower bounds see Gil Kalai's blog entry on a breakthrough (September 2020) by Asaf Ferber and David Conlon. And in June 2023, hard on the heels of the March upper bound announcement (see note (2)) came another Gil Kalai post.
  4. An account of Erdős and Ramsey theory is given by Ronald L. Graham and Joel Spencer in the centennial reflections here (from p. 132).
  5. A very interesting account of early precursors to Ramsey's theorem is offered by the Computational Complexity blog. This is also the focus of Alexander Soifer, The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators, Springer, 2008.
  6. More on the famous Erdős quote by Evelyn Lamb.
  7. Veselin Jungic at Simon Fraser has a podcast on Ramsey theory: No Strangers At This Party.
  8. This theorem is the choice of Yen Duong in Episode 31 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 29: Gauss's Law of Quadratic Reciprocity
  1. Özlem Imamoğlu's review, Bull. Amer. Math. Soc., Vol. 44, No. 4, 2007, 647–652, online, of Franz Lemmermeyer's Reciprocity Laws: From Euler to Eisenstein, Springer, 2000, is a superb mini-essay on reciprocity beyond quadratic.
  2. Eisenstein's famous geometric recasting of Gauss's 3rd proof of his law is carefully explained in Reinhard C. Laubenbacher and David J. Pengelley, "Eisenstein's misunderstood geometric proof of the Quadratic Reciprocity Theorem", The College Mathematics Journal, Vol. 25, No. 1, 1994, pp. 29–34; online (paywall; a copy can be found here, June 2024). By the same authors is a dramatisation: "Gauss, Eisenstein, and the "third" proof of the quadratic reciprocity theorem: ein kleines schauspiel", The Mathematical Intelligencer, Vol. 16, 1994, pp. 67–72; online (paywall; copy here, June 2024). John Baez (whose grasp on all this is certainly better than mine!) wrote "I don’t understand what makes Eisenstein’s proof tick, even after reading this play about it" in this blog entry which is followed by some valuable comments.
  3. Max G. Levy has this excellent Quanta Magazine article.
  4. Gauss found a lovely connection between quadratic reciprocity and the discrete Fourier transform - see this by John D. Cook.
  5. This reply by Alon Amit on Quora is a helpful reflection on the 'depth' of this result.

Theorem no. 30: The Law of Large Numbers
  1. The theorem is described here in elementary terms, as would have been understood by Laplace himself. An excellent modern account in terms of measure theory is given by Terence Tao here.
  2. I cannot resist linking to "The Law of Small Numbers", Jonathan Kujawa's elegant centenary homage to Richard Guy in 3quarksdaily.

Theorem no. 31: Benford's Law
  1. The Wikipedia entry has a good entry on forensic aspects of Benford. Another compelling forensic application is Daniel Gamermann and Felipe Leite Antunes, "Evidence of Fraud in Brazil's Electoral Campaigns Via the Benford's Law", online. Another application is Vadim S. Balashov, Yuxing Yan and Xiaodi Zhu, "Using the Newcomb–Benford law to study the association between a country’s COVID-19 reporting accuracy and its development", Sci Rep 11, 22914 (2021); online (thanks to Mario Cortina Borja for this).
  2. An interesting occurrence of Benford is in the frequencies of leading digits in base 10 representations of powers, e.g. \(2^k,k=0,1,\ldots\). See Theorem 299 (notes(5)) and also "A simple answer to Gelfand’s Question" by Jaap Eising, David Radcliffe and Jaap Top, The American Mathematical Monthly, March 2015, pp. 234–245; online (paywall; a copy is here, July 2024). Tangentially, we can ask whether prime numbers obey Benford.

Theorem no. 32: The Green–Tao Theorem on Primes in Arithmetic Progression
  1. Original source for this theorem is Green, Ben and Tao, Terence, "The primes contain arbitrarily long arithmetic progressions", Annals of Mathematics, Vol. 167 , no. 2, 2008, pp. 481–547; online.
  2. Green and Tao's achievement is described by Bryna Kra as "an amazing fusion of methods from analytic number theory and ergodic theory" in his technical overview of their proof "The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view", Bull. Amer. Math. Soc. 43 (2006), 3–23; online. There is a nice overview by Ben Green here (p. 10, 11MB pdf file). Tao has collected some survey-type presentations at various levels here.

Theorem no. 33: The Prime Number Theorem
  1. See also Benjamin Fine and Gerhard Rosenberger, "An Epic Drama: The Development of the Prime Number Theorem", Scientia Series A: Mathematical Sciences, Vol. 20 (2010), 1–26; online.
  2. A fine general historical account of the prime number theorem by Tom M. Apostol, "A centennial history of the prime number theorem", Engineering and Science, No. 4, 1996, pp. 19–28, is online here (3.4MB pdf). The classic account by Don Zagier of "Newman's short proof of the prime number theorem", American Mathematical Monthly, vol. 104 (1997), pp. 705–708, is online here.
  3. The TME-EMT project has a list of explicit bounds on primes,and much else besides! (Thanks @JoshuaZed1 for this.)
  4. There is a brief discussion of heuristic explanations for the Prime Number Theorem (notably the one by Greg Martin) here.
  5. If \((\bar{x},\bar{y})\) is the centre of mass of the arc of \(y=\log(x)\) in the interval \([1,x]\) then \(\frac12\pi(x)\) is asymptotic to \(\bar{x}/\bar{y}\) (see M500 magazine, issue 260, pp. 10–12).
  6. Regarding the famous 'elementary proof' of PNT see Norman Levinson's "A motivated account of an elementary proof of the prime number theorem", American Mathematical Monthly, vol. 76, 1969, pp. 225–245; online. The unfortunate associated priority dispute is meticulously documented by Dorien Goldfeld in "The elementary proof of the prime number theorem: an historical perspective", in David Chudnovsky, Gregory Chudnovsky and Melvyn Nathanson (eds.), Number Theory New York Seminar 2003, Springer, 2004; online via Goldfeld's webpage (under preprints). It includes the observation that Tchebychef had given an elementary proof in 1852 that \(x/\log x\) is the correct order of magnitude for \(\pi(x)\). Both the proof and dispute are given a non-technical overview by Joel Spencer and Ronald Graham, "The Elementary proof of the prime number theorem", Mathematical Intelligencer, vol. 31 (3), June 2009, 18–23.
  7. A number of other elementary proofs of PNT have been found. The latest (June 2020) is by Florian Richter and his paper gives a short history of PNT and elementary proofs of it.
  8. There are formal proofs of PNT:
    1. the Erdős–Selberg elementary proof: Jeremy Avigad, Kevin Donnelly, David Gray, and Paul Raffand, "A formally verified proof of the prime number theorem", ACM Transactions on Computational Logic, Vol. 9 Issue 1, December 2007; online preprint (and see here for a nice overview presentation by Avigad) and
    2. the classical complex analysis proof: John Harrison, "Formalizing an analytic proof of the prime number theorem", Journal of Automated Reasoning, vol. 43, pp. 243–261, 2009; online.
  9. Posted on twitter by Tamàs Görbe this attractive the corollary of the PNT: \(\lim_{n\rightarrow\infty} (p_1\times \ldots \times p_n)^{1/p_n}=e.\) \begin{align*}\textrm{Exponentiate both sides of: } \frac{1}{p_n}\sum_{k=1}^{n}\log p_k &\sim \frac{1}{n\log n}\sum (\log k+\log\log k) \hspace{.3in}\textrm{ (PNT)}\\ &\sim \frac{1}{n\log n}(n\log n - n) \hspace{.3in}\textrm{ (Stirling)}\\ &\sim 1. \end{align*}
  10. Find \(\pi(x)\) for \(x\leq 10^{13}\) at

Theorem no. 34: The First Isomorphism Theorem
  1. The wiki entry for the isomorphism theorems gives as source Emmy Noether's paper "Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern", Mathematische Annalen, vol. 96 (1927) pp. 26–61; online (paywall); at Göttinger Digitaisierungszentrum.

Theorem no. 35: The Second Isomorphism Theorem
  1. This page replaces an earlier version combining the 2nd and 3rd isomorphism theorems with an illustration based on their superficial similarity to rules for arithmetic with fractions. I've left a copy here (opens in new tab). The 3rd isomorphism theorem is now presented seperately as Theorem no. 253.
  2. The example given is a special case of the application given by P.M. Cohn in Algebra, Volume 1 (which I assume is carried over to Classic Algebra although I haven't checked): if a subgroup \(H\) of \(\mbox{Sym}_n\) has any odd permutations then its even permutations form a normal subgroup of index 2 in \(H\).
  3. Another depiction of the Cayley table of Frobenius 20, together with much other valuable information is given here by the beguiling website

Theorem no. 36: Euler's Identity
  1. The relevant Wiki entry offers a good quote from Robin Wilson on the origins of this identity. See also notes to Theorem 206.
  2. The MacTutor Archive entry for Benjamin Peirce records his charming comment on Euler's identity: "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it is the truth."
  3. There is a nice Devlin's Angle post addressing the subject of beauty in mathematics and in Euler's identity in particular.
  4. The Tau icon symbol used in the illustration of this theorem is on loan from Michael Hartl, with thanks.

Theorem no. 37: Girard's Theorem
  1. Fix the area \(T\) of a spherical triangle and invert Girard's formula to give \(A+B+C=T/r^2+\tau/2\). Now let radius \(r\) tend to infinity: we recover a triangle in the Euclidean plane whose angles sum to \(\tau/2\), as expected.
  2. Attribution of this theorem to Harriot can be found in chapter 2 of Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Vintage, 2005; and in chapter 10 of David S. Richeson, Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press, 2008. I have seen it given to Legendre (e.g.) but Legendre's result, published in 1798, much later than Girard, approximates the difference between angles in a spherical triangle and angles in a plane triangle having the same side lengths. A good account is here. (Legendre did not, in any case, claim the result as his.)

Theorem no. 38: Lucas' Theorem
  1. Romeo Meštrović has compiled a fine survey of applications and extensions of Lucas's theorem.
  2. A generalisation of Lucas's theorem is given by Andrew Granville in his dynamic e-survey Arithmetic properties of Binomial Coefficients.

Theorem no. 39: Pascal's Rule
  1. Pascal presented his triangle in "Traité du triangle arithmetique" published posthumously in 1665; online.
  2. Pascal's triangle as it is usually displayed has sides which are parabolic, that is, quadratic in \(n\). The easiest way to confirm this is perhaps to estimate the sum of the digits in the \(n\)-th row using the normal curve approximation. This gives \({n\choose k}\approx\dfrac{2^{n+1}}{\sqrt{\tau n}}e^{-(2k-n)^2/2n}\). Taking logs and summing over \(k\) gives highest terms of order \(n^2\).
  3. There is one version of Pascal's triangle which is indeed triangular: where the entries are reduced modulo 2. In this case the pattern which emerges is a version of Sierpinski's gasket, see Wolfram, S., "Geometry of binomial coefficients, American Mathematical Monthly, vol. 91, no. 9, 1984, pp. 566–571; online. Indeed, similar patterns emerge for divisibility of entries by any integers: see this entry.
  4. Amazingly a simple relationship between Pascal's triangle and \(e=2.71828\ldots\) appears to have been noticed for the first time, by Harlan J. Brothers, only in the twenty-first century.
  5. Some slides on Pascal's triangle that I prepared (in French but mostly pictures and equations, 1MB pdf download). Includes some interesting elementary blunders made by ChatGPT. This is free-to-use May 2023 ChatGPT: more than likely it will act smarter by the time you read this. But not because it is smarter! I am reminded of the anecdote in Ulam's memoirs
           Hirniak would tell Banach, for instance, that there were still some gaps in his proof of Fermat's problem. Then he would add, "The bigger my proof, the smaller the hole. The longer and larger the proof, the smaller the hole. To a mathematician this constitutes an amusing formulation.       
    Stanislaw Ulam, The Adventures of a Mathematician, chapter 2.
  6. A cute take on Pascal from XKCD.

Theorem no. 40: Stirling's Approximation
  1. A very nice introduction to Stirling's approximation is Finbarr Holland, "A leisurely elementary treatment of Stirling’s formula", Irish Mathematical Society Bulletin, 77, Summer 2016, pp. 35–44; online. John Baez has this excellent blog post.
  2. A good source of information on the central binomial coefficients is the corresponding entry at, where the sequence is no. 984.
  3. A nice application of Stirling in number theory may be found at (Theorem 33, notes(8))

Theorem no. 41: Lagrange's Theorem
  1. There is a very fine presentation "Some prehistory of Lagrange’s Theorem in group theory: 'The number of values of a function'" by Peter M. Neumann for the Mathematical Association (whose sometime president he was). I didn't find a link on the MA's website but a pdf download is here (.8MB). Another excellent historical source is Richard L. Roth, "A History of Lagrange's Theorem on Groups", Mathematics Magazine, Vol. 74, No. 2, 2001, pp. 99–108; online (paywall; reprint here November 2023).

Theorem no. 42: Zeckendorf's Theorem
  1. It would appear that Zeckendorf's theorem was first published in C. G. Lekkerkerker, "Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci", Simon Stevin, 29 (1951-1952), 190–195. I have not seen this paper but Daykin in a 1960 paper (see note 3 below) says "Zeckendorf's proof is given by C. G. Lekkerkerka (sic) in [1]," ([1] being the Simon Stevin paper). Zeckendorf himself published his theorem in 1972 in "Representation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas", Bull. Soc. Royale Sci. Liege 41 (1972) 179–182. I haven't seen this paper either (the Bulletin de la Société Royale des Sciences de Liège is online but not all issues appear to be digitised).
  2. A good presentation of Zeckendorf's and Lekkerkerker's theorem is given here by Steven J. Miller. Another good source on Lekkerkerker's theorem is Jukka Pihko, "On Fibonacci and Lucas representations and a theorem of Lekkerkerker", Fibonacci Quarterly, vol. 23, no. 3 (1988), 256–261, online here.
  3. The exact value of the average number of Zeckendorf summands, over the interval \([F_{n+1},F_{n+2})\), as approached asymptotically by Lekkerkerker's theorem, is \(L_n=1+\varepsilon(n)/F_n\), where \(\varepsilon(n)=\sum_{k=0}^{\lfloor(n-1)/2\rfloor}k{n-1-k\choose k}\) (see Steven J. Miller's presentation). This apparently allows us to write, via the identity \(\varphi^2=1+\varphi\), the error in Lekkerkerker's ratio as the constant \(3/5\), thus: \(\lim_{n\rightarrow\infty}\left(L_n-2n/(5+\sqrt{5}\,)\right)=3/5\). The function \(\varepsilon(n)\) can itself be written entiredly in terms of Fibonacci numbers: $$\varepsilon(n)=\left\{\begin{array}{ccl} \left(\frac{n}{2}-\frac25\right)F_n-\frac{n}{10}\left(F_{n-1}+F_{n+1}\right) & &n \mbox{ even}\\ -\frac15F_{n-1}+\frac{n-1}{5}(F_{n-2}+F_n)&&n \mbox{ odd}, \end{array}\right.$$ alternatively, $$\varepsilon(n)=\left\{\begin{array}{ccl} \left(\frac{n}{2}-\frac25\right)F_n-\frac{n}{10}F_{2n}/F_{n} & &n \mbox{ even}\\ -\frac15F_{n-1}+\frac{n-1}{5}F_{2n-2}/F_{n-1}&&n \mbox{ odd},\end{array}\right.$$ the ratio \(F_{2T}/F_T\) also being the \(T\)-th Lucas number (A000032).
  4. David Daykin's paper is "Representation of Natural Numbers as Sums of Generalised Fibonacci Numbers", J. London Math. Soc., (1960) s1-35 (2): 143-160. The first page is free-access here. A follow-up paper by Daykin appeared in Fibonacci Quarterly in 1969 and can be viewed here. A brief account of Daykin's uniqueness result is given in J. L. Brown, Jr., "Zeckendorf's theorem and some applications", Fibonacci Quarterly, vol. 2, no. 3, 1964, 163–168; online here.
  5. Garry J. Tee has contributed some interesting remarks on Zeckendorf-based arithmetic, which he investigated in "Russian Peasant Multiplication and Egyptian Division in Zeckendorf Arithmetic", Australian Mathematical Society Gazette, vol. 30, no. 5, 2003, 267–276:
           "My algorithms for arithmetic in Zeckendorf arithmetic are much more efficient than any published previously, but they still cost very  much more than binary arithmetic. I commented that, if more efficient algorithms for Zeckendorf addition and subtraction could be devised, then they could be used to give much more efficient algorithms for Zeckendorf multiplication and division.
    The 2013 paper  by Conner Ahlbach, Jeremy Usatine, Christiane Frougny and Nicholas Pippenger on Efficient Algorithms for Zeckendorf Arithmetic, Fibonacci Quarterly, 51(3):249–255  does give much more efficient algorithms for Zeckendorf addition and subtraction - but their main emphasis is on the depth of circuitry required."
    The paper can be viewed in pdf form (≈1.6MB) here (with kind permission of the Australian Mathematical Society).
  6. The universality of the Fibonacci sequence, restricted to just 1,1,2,3,5, is exploited in a clever clock by Philippe Chrétien, described here by Alex Bellos.
  7. There is a less sophisticated version of Colm Mulcahy's card trick here by Kiran Ananthpur Bacche.
  8. This theorem is the choice of Pamela Harris in Episode 64 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem blog. She talks about generalising Zeckendorf, work which can be read about here, for example.

Theorem no. 43: The DPRM Theorem
  1. Scholarpedia's article on Matiyasevich theorem lists four articles which together comprise the original proof of this theorem, together amounting to less than 40 pages.
  2. Yuri Matiyasevich, Hilbert's 10th Problem, MIT Press, 1993, gives a complete self-contained exposition of the proof of DRPM.
  3. Martin D. Davis, "Hilbert's tenth problem is unsolvable", The American Mathematical Monthly, vol. 80, (1973), pp. 233–269; online (paywall; pdf here, July 2024), is an excellent account of the proof of DPRM.
  4. A particularly charming and accessible example of a Diophantine set is the set of Fibonacci numbers: James P. Jones, "Diophantine Representation of the Fibonacci Numbers", Fibonacci Quarterly, Feb. 1975, 84–88; online here (3rd from bottom). Jones is also one of the team who produced a particularly compelling prime polynomial, via the fact that the set of primes is Diophantine: Douglas Wiens, James P. Jones, Daihachiro Sato, Hideo Wada, "Diophantine representation of the set of prime numbers", The American Mathematical Monthly, vol. 83, 1976, pp. 449-464; online (paywall; pdf here, July 2024).
  5. There is a machine proof of this theorem: Dominique Larchey-Wendling, Yannick Forster, "Hilbert's Tenth Problem in Coq"; arxiv.
  6. Jonathan Pila contributes this entry (with a fine A3 poster version) on Diophantine Equations to the Oxford Mathematics Alphabet.
  7. Regarding the Pell equation Jordana Cepelewicz offers this valuable Quanta article.

Theorem no. 44: Pappus's Theorem
  1. Same comment as for Theorem 55 regarding Java; the app for this theorem, if you want to try, is here.
  2. See note (5) for Theorem 215 regarding Pappus's theorem becoming involved in twentieth century mathematics

Theorem no. 45: Binet's Formula
  1. The role of the Fibonacci sequence in the resolution of Hilbert's tenth problem is given a characteristically beguiling treatment by Evelyn Lamb here (paywall).
  2. It may be observed that computing \(F_n\) for negative values of \(n\) continues to 'work', extending the Fibonacci sequence to the left: \(\ldots, 4,-3,2,-1,1,0,1,1,2,\ldots\).
  3. The 'square spiral' illustrating this formula is the basis for a lovely curve construction discovered by Edmund Harriss and described here. The spiral is interestingly animated in this exploration by Matt Enlow as part of Aperiodical's The Big Internet Math-Off 2024.
  4. Audrey G. Bennett traces the Fibonacci sequence and spiral back to African architects and weavers in this Conversation article.
  5. A nice Quanta article by Alex Stone explores some recent discoveries in the world of recursively defined sequences (e.g. Somos sequences) .

Theorem no. 46: Cameron's Theorem on Distance-Transitive Graphs
  1. Original sources for this theorem:
    1. N. L. Biggs and D. H. Smith, "On trivalent graphs", Bull. London Math. Soc., Vol. 3, Issue 2, 1971, pp. 155–158; online (paywall).
    2. P.J. Cameron, "There are only finitely many finite distance-transitive graphs of given valency greater than two", Combinatorica, Vol. 2, No. 1, 1982, pp. 9–13; online (paywall).
  2. Cameron's proof of this theorem depends on his proof, with Preager, Saxl and Seitz, of the Sims Conjecture which in turn relies on the Classification of the Finite Simple Groups. A CFSG-free proof was suggested by Cameron and completed by Richard Weiss, "On distance-transitive graphs", Bull. London Math. Soc., Vol. 17, Issue 3, 1985, pp. 253–256; online (paywall). The paper proving the Sims Conjecture is cited in (Theorem 65, notes(1))
  3. Questions regarding infinite distance-transitive graphs are generally open, except in the locally finite case which was solved by H. Dugald Mcpherson in 1982. This work and progress on the non-locally finite case, as of 1998, is described by Cameron in "A census of infinite distance-transitive graphs", Discrete Mathematics, Volume 192, Issues 1–3, 1998, pp 11–26; online.
  4. Our illustration of this theorem shows eight of the twelve 3-regular distance transitive graphs. The full list is given in the relevant wikipedia entry, with pictures of each graph.

Theorem no. 47: The Binomial Theorem
  1. The definition of binomial coefficients to allow for arbitrary complex powers of the binomial can be generalised still further to allow both parameters to be complex, as explained here by John D. Cook. (This does not impact on the Binomial Theorem whose statement only features the 'top' parameter.)
  2. The original weblink for this theorem was an absorbing paper by Lawrence Neff Stout (1948–2012): "Aesthetic Analysis of Proofs of the Binomial Theorem" which was slated for but does not appear in, The Humanistic Mathematics Network Journal. It is available at and I have uploaded a temporary copy here for quick reference.
  3. Steven Strogatz has a lovely article on Newton's discovery of binomial power series for Quanta magazine.

Theorem no. 48: Beineke's Theorem on Line Graphs
  1. Original source for this theorem: Lowell W.Beineke, "Characterizations of derived graphs", J. Combinatorial Theory, Vol. 9, Issue 2, 1970, pp. 129–135; online. Beineke announced the result in 1968, according to this Wiki entry.
  2. The attribution to N. (presumably Neil) Robertson is found on p. 74 of Frank Harary, Graph Theory, Westview Press, new edition, 1994 and is confirmed by Hong-Jian Lai and Ľubomír Šoltés in their 2001 paper (see note (3)).
  3. The number of forbidden subgraphs for characterising line graphs can be lowered under slightly stronger conditions. Thus Ľubomír Šoltés proved in 1994 that, for a graph on at least 9 vertices, forbidden subgraphs V and IX can be ignored, bringing the number down to 7. Then in 2001, with Hong-Jian Lai, he proved that just forbidden subgraphs I, VI and VIII are enough, provided that the graph being tested has minimum degree 7 and is not isomophic to two complete graphs sharing an edge (e.g. subgraph VII): Hong-Jian Lai and Ľubomír Šoltés, "Line Graphs and Forbidden Induced Subgraphs", Journal of Combinatorial Theory, Series B, Vol. 82, Issue 1, 2001, pp. 38–55; online.

Theorem no. 49: Netto's Conjecture (Dixon's Theorem)
  1. Original sources for this theorem:
    1. Eugen Netto's conjecture appears in his 1882 book Substitutionentheorie und ihre Anwendung auf die Algebra, for which there is a page in the Mactutor Archive. More details can be found in section 2.1 of the excellent survey Timothy C. Burness, "Simple groups, generation and probabilistic methods, in C. M. Campbell, C. W. Parker, M. R. Quick, E. F. Robertson and C. M. Roney-Dougal (eds.), Groups St Andrews 2017 in Birmingham, LMS Lecture Notes Series, Volume 455, CUP, 2019.
    2. John D. Dixon, "The probability of generating the symmetric group", Mathematische Zeitschrift, Vol. 110, 1969, pp. 199–205; online (paywall).
    3. Lázsló Babai, "The probability of generating the symmetric group", J. Combin. Theory (Ser. A), Vol. 52, Issue 1, 1989, pp. 148–153; online.
  2. The sharpest asymptotics for the probability of generating \(S_n\) and \(A_n\) are given by Dixon J.D., "Asymptotics of generating the symmetric and alternating groups", Electronic J. Comb., Vol. 12, 2005, article R56; online. Dixon relies on Babai's proof of his conjectured asymptotic, so there is still a dependence on the classification of the finite simple groups; a CSFG-free proof is provided in Sean Eberhard and Stefan-Christoph Virchow, "The probability of generating the symmetric group", Combinatorica, Vol. 39, 2019, pp. 273–288; online (paywall; arxiv).
  3. A good overview of probabilistic group theory by John Dixon is given in this 2004 preprint.
  4. Bounds, as opposed to asymptotics, for the probability of generating \(S_n\) and \(A_n\) are given in Attila Maróti and M. Chiara Tamburini, "Bounds for the probability of generating the symmetric and alternating groups", Archiv der Mathematik, Vol. 96, Issue 2, 2011, pp 115–121; online (paywalled, preprint May 2024). These have been improved in Luke Morgan and Colva M. Roney-Dougal, "A note on the probability of generating alternating or symmetric groups", Archiv der Mathematik, Vol. 105, Issue 3, 2015, pp 201–204; online (paywall; arxiv).
  5. The plots on this page use the elements of \(S_n\) ordered by parity (even permutations before odd) then by number of moved points, then by value of first moved point. For \(S_4\), for example, this gives the listing \((),(1 2 3),(1 2 4),(1 3 2),(1 3 4),(1 4 2),(1 4 3),(2 3 4),(2 4 3),(1 2)(3 4),(1 3)(2 4),(1 4)(2 3),(1 2),(1 3),(1 4),(2 3),(2 4),(3 4),(1 2 3 4),(1 2 4 3),(1 3 4 2), (1 3 2 4),(1 4 3 2),(1 4 2 3) \).
  6. The numbers of permutation pairs generating \(S_n\) is sequence A071605 at
  7. An earlier version of this page which explained permutation multiplication, rather than depicting the distribution of pairs generating \(S_n\) and \(A_n\), has been preserved here.

Theorem no. 50: The Euler–Hierholzer "Bridges of Königsberg" Theorem
  1. An account of this theorem is to be found on page 33ff of the 2nd edition of Edouard Lucas's Récréations Mathématiques, vol. 1, published in 1891, the year of Lucas's tragically early death. The sufficiency of all degrees even is dealt with in a note on page 223 and is, according to this Wikipedia entry, essentially Hierholzer's 1873 argument. Lucas lists in his bibliography the article of Fleury giving an alternative method of construction. This article is a response to Lucas's Récréations Mathématiques entry and proposes a first solution to the construction problem in the version of drawing a figure in one continuous line. I have not seen Lucas's 1st edition to see if his entry differs in the light of Fleury's article. You can see Lucas's 2nd edition online here and Fleury's article is online here (p.257ff).
  2. There is a nice real-life application to snow ploughing described here (which is actually the Chinese Postman problem but Euler tours are the background).
  3. And the traditional puzzle is solved, with very nice graphics, for New York here.
  4. It should be noted that this theorem may at some point spontaneously disappear from this website: xkcd explains!

Theorem no. 51: A Theorem of Melody Chan on Group Actions
  1. Original source for this theorem (which is the weblink from the theorem page) is Melody Chan, "The maximum distinguishing number of a group", Electronic J. Comb., vol. 31 (2006), paper R70; online. The theorem in question is Theorem 3.1.
  2. Chan's theorem concerns group actions with distinguishing number 2, a subject taken further in, for example, Marston Conder and Thomas Tucker, "Motion and distinguishing number two", Ars Mathematica Contemporanea, vol. 4, no. 1 (2011), pp. 63–72; online.

Theorem no. 52: The Robertson--Seymour Graph Minors Theorem
  1. The original sources for this theorem are detailed in its Wiki page. The article directly addressing the theorem presented here is Neil Robertson and P.D. Seymour, "Graph Minors. XX. Wagner's conjecture", J. Comb. Theory, Series B, Vol. 92, Issue 2, 2004, pp. 325–357; online.
  2. There is a very nice semi-technical overview of 'Robertson-Seymour' theory by Lovász here. Graph minors theory is also very well explained in a series of blog posts by Jim Belk.

Theorem no. 53: Bailey's Theorem on Latin Squares
  1. The original source for this theorem is R.A. Bailey, "Quasi-Complete Latin Squares: Construction and Randomization", Journal of the Royal Statistical Society. Series B (Methodological) Vol. 46, No. 2 (1984), pp. 323–334; online (paywalled).
  2. Bailey's conjecture on terraces has been shown by Matt Ollis to hold for groups of all orders up to 511 with the possible exception of 256 and 384. Ollis also has this on the conjecture for abelian groups: "On terraces for abelian groups", Discrete Mathematics, Vol. 305, Issues 1–3, 6 December 2005, pp 250–263; online.
  3. The original web link for this theorem description was this entry at the Encyclopedia of Design Theory which remains a valuable resource but now hosted, and perhaps ephemerally, by Leonard Soicher at Queen Mary University of London, the domain having ceased to exist.

Theorem no. 54: The Bose Equivalence Theorem in Design Theory
  1. The independent publications of this theorem are
    1. E.H. Moore, Tactical memoranda I–III. Amer. J. Math. vol. 18, no. 3, 1896, pp. 264–303; online;
    2. Raj Chandra Bose, "On the application of the properties of Galois fields to the problem of construction of hyper-Graeco Latin squares”, Sankhyā, vol. 3, pt. 4, 1938, 323–38; online (paywalled);
    3. W.L. Stevens, The completely orthogonalized Latin Square", Annals of Eugenics, vol. 9, issue 1, 1939, pp. 82–93; online.

Theorem no. 55: Miquel's Triangle Theorem
  1. Original source for this theorem: Miquel, A., "Théorèmes de géométrie", Journal de Mathématiques Pures et Appliquées, Tome 3, 1838, pp. 485–487; online (facsimile by Gallica). The theorem is the 2nd of two corollaries (réciproques) to Theorem 1. The figures appear at the end of the volume with figure 1 (pertaining to the Theorem 1, appearing here).
  2. I used to link to the geometry app that created this theorem's illustration. But guaranteeing that a Java app will work for everyone's favourite browser has become too painful. And the app wasn't all that exciting anyway, so it is no longer maintained. But you can look at it here — last time I checked it worked in Internet Explorer.

Theorem no. 56: Morley's Miracle
  1. Original source for this theorem is F. Morley, "On the metric geometry of the plane n-line", Trans. Amer. Math. Soc., Vol. 1, No. 2, 1900, pp. 97–115; online. However, there is famously no mention in this paper of equilateral triangles, or triangles of any sort. A fine discussion of how and why, and when, the triangle theorem emerged is given beginning here, at cut-the-knot (the main article is the weblink from the theorem page). Another important source is Cletus O. Oakley and Justine C. Baker, "The Morley trisector theorem", The American Mathematical Monthly, Vol. 85, No. 9, 1978, pp. 737–745; online (paywall). Clark Kimberling has a valuable page on Morley which is also a good source for this theorem.
  2. This theorem page replaces a previous one which linked to an animation created using David Joyce’s Geometry Applet package. Such (Java) animations have become problematical for many browers and anyway much better animations than mine are easy to find on the web.The app for this theorem, if you want to try to see it it works, is here, and the original theorem page has been retained here. In passing, the addition of angle bisectors in the new illustration, the depicted triangle having vertices \((4,2), (1,9), (10,1)\), suggests some common bisector-trisector intersections but I suppose these are coincidental. The vertices \((4, 2), (7, 8),(12, 1)\), for example, show no such intersections.
  3. One of Morley's sons, Frank V. Morley, was also a mathematician but returned to England and became a director, with Geoffrey Faber and T.S. Eliot, of Faber & Gwyer, later Faber & Faber ("Morley, Faber and Eliot would sometimes communicate in exchanges of light verse." John Mullan writing in The Guardian, September 25, 2004, here (paragraph 6).

Theorem no. 57: The Birkhoff–von Neumann Theorem
  1. Original sources for this theorem:
    1. Birkhoff, Garrett, "Tres observaciones sobre el algebra lineal", Univ. Nac. Tucumàn. Revista A., 5, 1946, pp. 147–151; not online, I think.
    2. J.. von Neumann, "A certain zero-sum two-person game equivalent to an optimal assignment problem", Ann. Math. Studies, Vol. 28, Contributions to the Theory of Games (AM-28), Volume II, 1953, pp. 5–12; online (paywall).
  2. The convex polytope exhibited in this theorem is usually referred to as the Birkhoff polytope, despite the fact that it appears to have been known about at least fifty years earlier. It has a wikipedia page where more on the historical background can be found.

Theorem no. 58: Galois' Theorem on Finite Fields
  1. The recommended weblink for this page was previously some notes by Peter Cameron on finite fields, posted at This website is being hosted by Leonard Soicher at Queen Mary University of London (which is the link given in the previous sentence). The notes on finite fields are thus still available but I have preferred to use a more permanent link on the theorem page itself.
  2. It would be ahistorical to say something like "Galois showed constructively that finite fields have prime power order; and Eliakim Moore showed that this construction was the only one possible". Moore, a pioneer of abstract algebra proved the structure theorem which says "this is what finite fields look like". But this answers a question which would not have occurred to Galois! A detailed study of the work which led up to and beyond Moore is given by Frederic Brechenmacher in "A history of Galois fields", 2012, <hal-00786662>. The definitive analysis of the Galois archive is of course Peter M. Neumann's The Mathematical Writings of Evariste Galois, European Mathematical Society, 2011.
  3. There is a wonderful knitted GF(16) by the woolythoughts blog.

Theorem no. 59: Germain's Theorem
  1. A detailed account of Germain's work on Fermat's Last Theorem has been given by Roger Thompson in the October and December 2016 issues of M500 magazine (pdf 600KB downloads).
  2. Although Germain's Theorem does not play a direct part in the eventual proof of FLT (Theorem 9) it has deep connections with other parts of number theory which are still of interest. For instance, it is connected with period lengths of modular Fibonacci sequences via 'Wall's Question' (see Theorem 235, notes(3)). There is also a connection to cryptography via the idea of 'strong' and 'safe' primes. See this, for example.
  3. The number and distribution of Germain primes are of interest in their own right, see here, for example.
  4. There is an interesting side story on the Germain–Gauss correspondence in this prize-winning essay by William C. Waterhouse. More on Gauss and Germain and on her work in number theory can be found in Raymond Flood's Gresham College lecture on the subject. Evelyn J. Lamb has this evocative piece which also has some useful onward links.
  5. There is a nice characterisation of Germain primes by Paolo Leonetti: "A characterization of Sophie Germain primes", Int. J. Number Theory, Vol. 14, No. 3, 2018, pp. 653–660; online (paywall; arxiv).

Theorem no. 60: The Strong Perfect Graph Theorem
  1. The qualifier 'strong' distinguishes this theorem from the Perfect Graph Theorem, also conjectured by Claude Berge and proved by László Lovász in 1972.
  2. Original source for this theorem: Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas, "The strong perfect graph theorem", Annals of Mathematics, Vol. 164, Issue 1, 2006, 51–229. Online version.
  3. The 2001 Strong Perfect Graph Theorem for square-free graphs (no induced 4-cycles) of Michele Conforti, Gérard Cornuéjols and Kristina Vušković is proved in "Square-Free Perfect Graphs", Journal of Combinatorial Theory B, 90 (2004) 257–307; online.
  4. Colouring a perfect graph with a number of colours equal to its clique number has remained a challenge. Progress as of 2015 is well-described in this Quanta magazine article by Natalie Wolchover. Thanks to @livecitizen1 for drawing my attention to this. The work in question was published in 2019: Maria Chudnovsky, Irene Lo, Frédéric Maffray, Nicolas Trotignon and Kristina Vuškovic, "Coloring square-free Berge graphs", Journal of Combinatorial Theory, Series B, Vol. 135, 2019, pp. 96–128; online.

Theorem no. 61: Moufang's Theorem
  1. Original source for this theorem: Moufang, R., "Zur Struktur von Alternativkörpern", Math. Ann., 110, 1935, pp. 416–430; online (paywall; facsimile). For historical context, see the excellent article Hala Orlik Pflugfelder, "Historical notes on loop theory", Commentationes Mathematicae Universitatis Carolinae, Vol. 41, Issue 2, 2000, pp. 359–370; online.
  2. For a proof of Moufang's theorem see Aleš Drápal, "A simplified proof of Moufang's theorem", Proc. AMS, Vol. 139, No. 1, 2011, 93–98; online.
  3. The smallest non-associative Moufang loop has order 9 — see Theorem no. 114; the octonions form a loop of order 16 (the negated elements are omitted from our multiplication table for conciseness) and there are four other non-associative Moufang loops of this order, see the paper by Orin Chein here. The sequence of numbers of non-associative Moufang loops is A090750.
  4. The question of whether non-Moufang loops may still obey Moufang's Theorem is addressedby Izabella Stuhl, "Moufang’s theorem for non-Moufang loops", Aequationes mathematicae, Vol. 90, 2016, pp. 329–333; online (paywall; arxiv).

Theorem no. 62: The Marriage Theorem and The Frobenius–Kőnig Theorem
  1. Original source for Hall's marriage theorem is P. Hall, "On representatives of subsets", J. London Math. Soc., Vol. s1-10, Issue 1, January 1935, pp. 26–30; online (paywall). Hall distinguishes his result from Kőnig's (graph-theoretic) version because he is choosing representatives from sets which are not necessarily of the same size. So our presentation is, superficially, a simplification.
  2. Original source for Kőnig is D. Kőnig, "Über Graphen und ihre Anwendungen auf Determinantentheorie und Mengenlehre", Math. Annalen, Vol. 77, Issue 4, December 1916, pp. 453–465; online (paywall); at Göttinger Digitaisierungszentrum.
  3. A valuable source on Kőnig's work in graph theory is the doctoral dissertation (Université Paris-Diderot - Paris VII, 2010) of Mitsuko Wate Mizuno, "The works of KONIG Dénes (1884–1944) in the domain of mathematical recreations and his treatment of recreational problems in his works of graph theory"; online. I read there that Kőnig's paper (note 2) is given an English translation in Norman L. Biggs, E. Keith Lloyd and Robin J. Wilson, Graph Theory, 1736-1936, Clarendon Press, 1986. I have not seen this.
  4. Peter Cameron explains Philip Hall's interest in this theorem as a group theorist.
  5. The gender implications of the name 'marriage theorem' made the news in Australia in 2017: Morgan, R. (2017) UNSW lecturer discouraged use of the term ‘marriage’ in mathematics theorem, SBS News. This and issues of mathematical nomenclature more generally are discussed by Tony Mann in "Mathematical Results – do names matter?", Mathematics Today, Institute of Mathematics and its Applications, April 2022; online.

Theorem no. 63: Thales' Theorem
  1. The story of Thales sacrificing an ox is apocryphal. It has also been attached to Pythagoras. See here for more details.
  2. I like this colourful description by Paula Beardell Krieg of applying Thales to find a circle centre.

Theorem no. 64: Neumann's Separation Lemma
  1. Peter Neumann published his lemma in "The structure of finitary permutation groups", Archiv der Mathematik, Vol. 27, Issue 1, 1976, 3–17; online; appearing in December. The extension to finite permutation groups beat it into print, appearing in October: B.J. Birch, R.G. Burns, Sheila Oates Macdonald and Peter M. Neumann, "On the orbit-sizes of permutation groups containing elements separating finite subsets", Bull. Austral. Math. Soc., Vol. 14, Issue 1, 1976, 7–10; online. I think Peter Neumann must have told me that both results were discovered in 1974.
  2. A charming little refresher on group actions by Justin Chen for Plus magazine (with links to other parts of group theory).

Theorem no. 65: Sims' Conjecture
  1. Original source for this theorem: P. J. Cameron, C. E. Praeger, G, J. Saxl and G. M. Seitz, "On the Sims Conjecture and distance transitive graphs", Bull. London Math. Soc, Vol. 15, Issue 5, 1983, pp. 499–506; online (paywall).
  2. The theorem is very well placed in context by Cheryl Praeger: "Using the finite simple groups", Asia Pacific Mathematics Magazine, Vol. 1, No. 3, 2011, pp. 7–10; online.
  3. See notes to Theorem 46 for an important application of Sims' conjecture.
  4. A little obituary footnote from Peter Cameron's blog.

Theorem no. 66: An Erdős–Ko–Rado Theorem on Intersecting Permutations
  1. Original sources for this theorem:
    1. Péter Frankl and Mikhail Deza, "On the maximum number of permutations with given maximal or minimal distance", J. Combin. Theory Ser. A, Vol. 22, Issue 3, 1977, pp. 352–360; online.
    2. P. J. Cameron and C. Y. Ku, "Intersecting families of permutations", European J. Combin., Vol. 24, Issue 7, 2003, pp. 881–890; online.
    3. B. Larose and C. Malvenuto, "Stable sets of maximal size in Kneser-type graphs", European J. Combin., Vol. 25, Issue 5, 2004, pp. 657–673; online.
  2. The woolythoughts blog offers an alternative view of \(S_4\). They have also done \(S_5\).

Theorem no. 67: Reidemeister's Theorem
  1. Original sources for this theorem:
    1. Reidemeister, Kurt, "Elementare Begründung der Knotentheorie", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5 (1), 1927, pp. 24–32; online (paywall).
    2. J.W. Alexander and G.B. Briggs, "On types of knotted curves", Annals of Mathematics, Vol. 28, No. 1/4,1926–1927, pp. 562–586; online (paywall; 1.8MB pdf, April 2024).
  2. The independent discovery of the Reidemister moves is analysed in M. Epple, “Knot invariants in Vienna and Princeton during the 1920s: epistemic configurations of mathematical research”, Science in Context, Vol. 17, No. 1/2, 2004, pp. 131–164; online (paywall; reprint on the webpages of the late, great Andrew Ranicki whose fine page of knot theory links remains live, April 2024).
  3. Valuable resources for visualising knots and for knot theory generally are to be found at the knotplot website.
  4. On knot theory in general, this Quanta Magazine article by David S. Richeson can be confidently recommended.
  5. There are deep connections between knot theory and theoretical physics (a little on this is offered in note (1) to theorem 240). How the Reidemeister moves are connected to the gauge groups of quantum field theory is very clearly presented in this Motion Mountain post.

Theorem no. 68: The Stable Marriage Theorem
  1. Original source for this theorem: Gale, D. and Shapley, L. S., "College admissions and the stability of marriage", Rand Report no. P2240, 1961; online. Published in American Mathematical Monthly, 69 (1), 1962, pp. 9–14; online (paywall; but pdf downloads are easy to find by searching for the paper title).
  2. Closely related is the economist Thomas Schelling's work on segregation and diversity. A compelling illustration of these ideas is provided in animated form by The Parable of the Polygons by Vi Hart and Nicky Case.
  3. An excellent account of Lloyd Shapley and his work is given here by Joseph Malkevitch.
  4. An account of real-life mate-finding algorithm is given here by Plus magazine.
  5. There is an intriguing connection between the stable marriage problem and Soduku discovered by Tanya Khovanova and a group of students and described by her on her blog.
  6. See note (5) for The Marriage Theorem regarding the name of this theorem.

Theorem no. 69: Arrow's Impossibility Theorem
  1. Original sources for this theorem: Arrow, Kenneth J., "The possibility of a universal social welfare function", RAND Report P-41, 1948; online; Arrow, Kenneth J.,"A Difficulty in the Concept of Social Welfare", J. Political Economy, Vol. 58, No. 4, 1950, pp. 328–346; online (paywall; but copies are not hard to find online, e.g. on the Wiki page for the theorem, July 2023).
  2. Arrow's theorem as a result in social science belongs to the general area of social choice theory. From this perspective, a good account (English and German versions) is given here (Snapshot no. 9/2015) by Victoria Power (March 2024, links to Snapshots seem not to work, but try the main page). To see where cutting-edge theoretical reserach has taken this see Benedict Eastaugh, "Arrow's theorem, ultrafilters, and reverse mathematics", The Review of Symbolic Logic, in press, 2024; online (paywall; arxiv).
  3. And something less serious!
  4. This theorem is the choice of Belin Tsinnajinnie in Episode 56 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 70: The 1-2-3 Conjecture
  1. Original source for this theorem: Michal Karoński, Thomasz Łuczak and Andrew Thomason, "Edge weights and vertex colours", J. Combin. Theory Ser. B, 91 (1), 2004, pp. 151–157; online. Subsequent progress up to 2012 is charted in Ben Seamone, "The 1-2-3 Conjecture and related problems: a survey", arxiv (which is the weblink from the theorem page).
  2. Theorem under Construction iconThis was a 'theorem under construction' until the 2024 publication of Ralph Keusch's proof that \(K=3\) (which was posted to the arxiv in 2023). It became the 3rd 'declassified' theorem, following Kepler's Conjecture and the Erdős Discrepency Problem. Ralph Keusch, "A solution to the 1-2-3 conjecture", J. Combinatorial Theory, Series B, Vol. 166, 2024, pp. 183–202; online (paywall; but see the arxiv link above).
  3. While, following Keusch, we know that \(K=3\) for all (connected) graphs it is NP-complete to decide for which graphs \(K=2\) holds. A. Dudek and D. Wajc. "On the complexity of vertex-coloring edge-weightings", Discrete Mathematics & Theoretical Computer Science, Vol. 13:3, 2011, pp. 45–50; online..
  4. It is perhaps not quite trivial to confirm that increasing edge multiplicities is always guaranteed to produce a degree colourable graph. I think an induction argument will do: delete a vertex; increase multiplicities in the vertex-deleted graph for degree colourability; replace the deleted vertex with sufficiently high identical multiplicities on its incident edges.
  5. Jakub Przybyło has proved that the conjecture holds for regular graphs of sufficiently high degree and that \(K\leq 4\) for all regular graphs of degree 2 or more: "The 1–2–3 Conjecture almost holds for regular graphs", J. Comb. Theory, Series B, Vol. 147, 2021, pp. 183–200; online (paywall; arxiv). More recently he has this: "The 1-2-3 conjecture holds for graphs with large enough minimum degree", Combinatorica, Vol. 42, supplement issue 2, 2022, pp. 1487–1512; online (paywall; arxiv).
  6. Ralph Keusch has published an unconditional proof of \(K\leq 4\): Keusch, R.," Vertex-coloring graphs with 4-edge-weightings", Combinatorica, May, 2023; online (paywall; arxiv). For his subsequent resolution of the conjecture see Note (2).

Theorem no. 71: Sharkovsky's Theorem
  1. Original source for this theorem: Sharkovsky, A. N., "Co-existence of cycles of a continuous mapping of the line into itself", Ukrain. Mat. Zh., 16 (1), 1964, pp. 61–71. The Russian citation can be found on the Ukranian Wiki page for the theorem. There is an English translation of the paper: J. Tolosa, International Journal of Bifurcation and Chaos, Vol. 05, No. 05, 1995, pp. 1263–1273; online (paywall). Various online links to the Russian original appear to be broken.
  2. @jakebarrrett has alerted me to the fact that a number of elementary proofs of this theorem are available. See, e.g., Chapter 1 of Louis Block & William Coppel, Dynamics in One Dimension, Springer, 2009. Also some notable treatments by Bau-Sen Du which can be accessed online via his arxiv listing.
  3. The crucial (but weaker) result that period 3 implies periods of all orders is discovered independently by Tien-Yien Li and James A. York, "Period Three Implies Chaos", The American Mathematical Monthly, Vol. 82, No. 10 (Dec., 1975), pp. 985–992; online (paywalled); there is a good description of their theorem by Padraic Bartlett here. Sharkovsky published his result in Russian at the height of the war and it did not receive widespread attention until the 1970s—see his MacTutor entry.
  4. A number of valuable reprints of classics from the dynamical systems literature are available here, including Robert May's 1976 Nature article dealing with the logistic function.
  5. The Thatsmath blog has a very nice entry on this theorem.

Theorem no. 72: MacWilliams' Identity
  1. Original source for this theorem: F.J. MacWilliams, "A theorem on the distribution of weights in a systematic code", Bell System Techn. J. , Vol. 42, Issue 1, 1963, pp. 79–94; online (paywall; facsimile).
  2. An alternative to the proofs found in the theorem page weblink (notes by Jonathan I. Hall) is to view MacWilliams's theorem as a consequence of the Poisson summation formula. See, for example, the presentation "Characters and the MacWilliams identities" by Jay Wood here. Jay Wood has a very interesting talk "Failures of the MacWilliams identities" at the Banff International Research Station for Mathematical Innovation and Discovery workshop Algebraic Methods in Coding Theory and Communication 22w5180 (click on 'workshop videos').
  3. There is a one-variable version of this theorem corresponding to the nonhomoeneous polynomial obtained by setting \(y=1\) in the definition of \(W_C(x,y)\). The identity becomes \(W_{C^{\perp}}(x)=|C|^{-1}W_C(1-x,1+(q-1)x)\) which, using the single-argument version of \(W_C\) becomes \(W_{C^{\perp}}(x)=\dfrac{1}{|C|}(1+(q-1)x)^nW_C\left(\dfrac{1-x}{1+(q-1)x}\right)\).
  4. There is a very nice application of MacWilliams, due to Assmus and Maher, to prove nonexistence of projective planes of order 6 modulo 8. An excellent presentation of this work is given by Matroid Union.

Theorem no. 73: Goodstein's Theorem
  1. Original source for this theorem: Goodstein, R., "On the restricted ordinal theorem", Journal of Symbolic Logic, 9 (2), 1944, pp. 33–41; online (paywall; there is a reprint here no. 3 under 'Anderen', February 2024).
  2. For some remarks on Goodstein's theorem in the context of the search for independence results for Peano arithmetic see Michael Rathjen, "Goodstein Revisited", Annals of Pure and Applied Logic, in press; arxiv. An analysis of the growth of Goodstein's function is presented in Andrés Eduardo Caicedo, "Goodstein's function", Rev. Colombiana Mat., Vol. 41, no. 2, 2007, pp. 381–391; online.

Theorem no. 74: Gödel's First Incompleteness Theorem
  1. The Wiki entry for Gödel's theorems is very good on original sources, translations etc.
  2. The theorem can be stated as "a consistent mathematical system contains assertions for which neither the assertion nor its negation can be proved from the axioms of the system". Indeed this is how Gödel himself stated his theorem, avoiding reference to mathematical truth. The version which says "contains truths which cannot be proved" is equivalent by virtue of the fact that, given consistency, non-provability of a negated assertion is synonymous with truth of the assertion. See Peter Cameron's article in Gowers et al (eds.), The Princeton Companion to Mathematics, Princeton University Press, 2008, a preprint of which is here.
  3. Mark Chu-Carroll's blog Good Math, Bad Math has posted a very nice 4-part introduction to Gödel I: the final part, which indexes the other three, is here. A good overview with a philosophical slant is this from The Times Literary Supplement by Juliette Kennedy, which is now behind a paywall but Kennedy's webpage has much else of value. Natalie Wolchover has an excellent Quanta article on the proof of Gödel's theorem.
  4. A fascinating self-contained exposition of Gödel's two incompleteness theorems is given by Stanisław Świerczkowski here (Dissertationes Math., 422, 2003, click on the 'POBIERZ ZA DARMO' button) in terms of the theory of 'hereditarily finite sets'. These were the proofs which were chosen as suitable for machine checking in 2015: Lawrence C. Paulson (2015). "A mechanised proof of Gödel’s incompleteness theorems using Nominal Isabelle", Journal of Automated Reasoning. 55, no. 1: 1–37
  5. A short sketch proof of Gödel's theorem by Raymond Smullyan (c.f. 5000 B.C. and Other Philosophical Fantasies, St. Martin's Press, 1983) is reproduced here.
  6. Marianne Freiberger has written an excellent account for Plus magazine of Harvey Friedman's work developing non-artificial examples of Gödel's theorem in action.
  7. The link between Gödel's theorem and proofs of undecidability (e.g. the halting problem) is subtle and is very well explored by Jørgen Veisdal here.
  8. An exchange between Freeman Dyson and Saunders Mac Lane in New York Review of Books offers fascinating and first-hand insights into Hilbert and Gödel's endeavours in the 1920s and 30s: "A Matter of Temperament", Saunders Mac Lane, reply by Freeman Dyson, October 5, 1995 issue; online.

Theorem no. 75: Gödel's Second Incompleteness Theorem
  1. The version of Ramsey's theorem used by the aliens in the illustration on the theorem page is the original (Theorem B) from his 1930 paper: see note (1) to Ramsey's Theorem.
  2. George Boolos, "Gödel's second incompleteness theorem explained in words of one syllable", Mind, Vol. 3, No. 409, 1994, pp. 1–3; online (paywall but the preview shows all the 1-syllable bit; the whole thing can be found on the web, here for instance February 2024).
  3. The second incompleteness theorem was independently discovered by John Von Neumann who, it appears, swiftly saw how it followed as a corollary of the first theorem, see p. 162 of Rebecca Goldstein, Incompleteness: The Proof and Paradox of Kurt Gödel, W. W. Norton & Company, 2005.
  4. A little footnote regarding SETI (featured somewhat gratuitously in this theorem page's illustration).

Theorem no. 76: Brahmagupta's Formula
  1. Original source for this theorem, apart from the Brāhmasphuṭasiddhānta (Wiki) and the origins of Heron's formula (Wiki): Coolidge, J. L., "A Historically interesting formula for the area of a quadrilateral", The American Mathematical Monthly, 46 (6), 1939, pp. 345–347; online (paywall).
  2. Coolidge attributes an equivalent to his quadrilateral area formula to Carl Anton Bretschneider and Friedrich (I think) Strehlke, both 1842. Wikipedia has an entry on Bretschneider's Formula which also credits (also 1842) von Staudt: $$K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\theta},$$ where \(\theta\) is half the sum of either pair of opposite angles (in a cyclic quadrilateral opposite angles sum to \(180^{\circ}\) with the half angle giving zero cosine).
  3. Using the notation of the theorem description, Ptolemy's Inequality says \(ad+bc\geq ef\) with equality if and only \(abcd\) is cyclic. So the subtracted term in Coolidge's square root is always nonnegative (this is immediate from Bretschneider's Formula).
  4. A clever trapezoid version of Heron's formula due to Miguel Ochoa Sanchez can be found at Also regarding Heron, see note (2) for Pythagoras. And from Colin Beveridge an elegant Aperiodical post "From Zero to Hero: a Euclidean proof", one that only uses synthetic geometry.
  5. Fascinating material on Indian mathematicians' investigations of cyclic quadrilaterals may be found in Radha Charan Gupta, "Parameśvara's rule for the circumradius of a cyclic quadrilateral", Historia Mathematica, Vol. 4, Issue 1, 1977, pp. 67–74; online here. Parameśvara's (also known as L'Huilier's) rule states that: circumradius of cyclic quadrilateral with sides \(a,b,c,d\) is obtained on dividing \(\sqrt{(ab+cd)(ac+bd)(ad+bc)}\) by \(4\times\) area of quadrilateral.
  6. Another open archive article from Historia Mathematica deals directly with Brahmagupta's formula: Satyanad Kichenassamy, "Brahmagupta’s derivation of the area of a cyclic quadrilateral", Historia Mathematica, Vol. 37, Issue 1, 2010, pp. 28–61.
  7. As distinct from Brahmagupta's Formula, his Theorem usually refers to another result about a cyclic quadrilateral : if its diagonals are orthogonal and a line joins their intersection to a side at right angles, then the continuation of this line bisects the side opposite. Satyanad Kichenassamy is again the expert: "Brahmagupta’s propositions on the perpendiculars of cyclic quadrilaterals", Historia Mathematica, Vol. 39, Issue 4, 2012, pp. 387–404; online.

Theorem no. 77: Pick's Theorem
  1. Original source for this theorem: Pick, Georg, "Geometrisches zur Zahlenlehre", Lotos - Zeitschrift für Naturwissenschaften 47. Neue Folge XIX. Band., 1899, pp. 311–319; online (facsimile)
  2. Yiwang "Evan" Chen has a good presentation about Pick here (1.1MB pdf) and there is an interesting investigation in Christian Haase and Josef Schicho, "Lattice polygons and the number 2i+7", Amer. Math. Monthly, Vol. 116, No. 2, 2009, pp. 151–165; online (paywall; arxiv).
  3. Pick's theorem extends, of course, in numerous ways. A nice version that applies to polygons with holes and disconnections is discussed by Tony Forbes in M500, Issue 253, pp. 6–7; online. Halil Rıdvan Öz has a paper "Extension of Pick's Theorem to Spherical Geometry using Girard's Theorem", Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Vol. 13, Issue 4, 2023, pp. 2915–2925; online.

Theorem no. 78: The Three-Distance Theorem
  1. The theorem was proved independently and simultaneously by at least three people, as is well-documented in its Wikipedia page. Although somtimes referred to as 'the Steinhaus Conjecture', it doesn't seem to have survived long enough to warrant the title. Stanisław Świerczkowski sent me the following interesting background information on the history and his proof of this theorem:
           "The Three Distances Theorem was conjectured by Hugo Steinhaus. When I gave him the proof, he checked it and asked me to write a report for the Academy of Sciences. He presented this report to the Academy (only members could do that) whereupon it was published in the BULLETIN DE L’ACADEMIE POLONAISE DES SCIENCES, Cl. III – Vol. IV, No.9, 1956. The date of his presentation to the Academy is: 29 June 1956. I have here an offprint. If you wish to look at it, and it is not in your library, I could scan it and send you an image by email. About that time Vera Sós and her husband Prof. Turán visited our University, so they certainly heard about the Three Distances result directly from my colleagues or from me. Of course, Erdős was visiting us too many times. I remember Vera quite well. In the announcement in the BULLETIN DE L’ACADEMIE POLONAISE mentioned above, there are four related theorems, and I postponed publishing the proofs of these to a later paper. By the time I wrote it, Vera Sós published her proof of the Three Distances Theorem, so I found it simpler to just refer to her paper. My proof (which I do not recall) almost certainly found its way into my Ph.D. thesis on the subject of cyclically ordered groups. This dissertation was submitted to the Polish Academy of Sciences. A recent inquiry disclosed that they cannot find it."       
    These recollections can also be found in Świerczkowski's autobiography Looking Astern, which is available via his entry at the MacTutor archive (under Additional Resources). Notwithstanding, there is indeed a published proof by him: Świerczkowski, S., "On successive settings of an arc on the circumference of a circle", Fundamenta Mathematicae, 46 (2), 1958, pp. 187–189; online. In his paper Świerczkowski refers to yet another independent proof, unpublished, by Peter Szüsz, using continued fractions.
  2. There are generalisations of the theorem, see for instance J.F. Geelen and R.J. Simpson, "A two dimensional Steinhaus theorem", Australasian J. Combinatorics, vol. 8, 1993, 169–197, available online here.
  3. The theorem is often visualised as points plotted around a circle of unit circumference, see this blog entry from Σidiot's blog, for example.

Theorem no. 79: The Fifteen Theorem
  1. Original sources: the weblink from the theorem page is to preprints of two chapters from Eva Bayer-Fluckiger and David Lewis (eds.), Quadratic Forms and Their Applications, American Mathematical Society, 2001. The first, by Conway, sets the scene. The second, by Manjul Bhargava, gives the first published proof of the 15 Theorem. It references Schneeberger's 1995 PhD dissertation which contains perhaps the first statement (and sketch proof) of the theorem.
  2. Manjul Bhargava won a Fields Medal in 2014 in part for his work on quadratic forms, including his proof with Jonathan Hanke of the Conway–Schneeberger 290 Conjecture. The citation is here, and he is interviewed by plus magazine on the subject here.
  3. The nine values specified in the statement of the 15 Theorem are entry A030050 at OEIS, while the 29 values for positive definite quadratic forms (the 290 Conjecture) in general are entry A030051.

Theorem no. 80: One-Factorisation of Regular Graphs
  1. This conjecture should probably be taken as originating with Amanda Chetwynd and Anthony Hilton's 1985 paper "Regular graphs of high degree are 1-factorizable", Proc. London Math. Soc., 50 (1985), 193–206; online (paywall). In Stiebitz et al, Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture, Wiley-Blackwell, 2012, on p. 259, we find the comment that: "it [the one-factorisation conjecture] 'was going around' in the early 1950s, Hilton [136] was told by G.A. Dirac." And then, "fifteen years before the one-factorization conjecture was published by Chetwynd and Hilton, Nash-Williams [233] proposed a much stronger conjecture, which is also known as the hamiltonian factorization conjecture: Let \(G\) be a \(\Delta\)-regular graph on \(2n\) vertices, where \(\Delta\geq n\). Then \(G\) has a hamiltonian factorization, i.e., \(G\) is the edge-disjoint union of \(\Delta/2\) Hamilton cycles if \(\Delta\) is even or \((\Delta-1)/2\) Hamilton cycles and a linear factor if \(\Delta\) is odd." (The factorisation of \(K_2\times K_5\) is our theorem illustration is an example). However, Anthony Hilton has told me that he has never seen any mention of his conjecture published prior to 1985 and considers it unlikely that it would have been thought about in the 1950s. Certainly Vizing's Theorem was not known until the 1960s. Ref [233] is C. St. J. A. Nash-Williams, "Hamiltonian lines in graphs whose vertices have sufficiently large valencies", in Combinatorial theory and its applications, III, North-Holland 1970, 813–819. The recent work of Csaba et al (see main theorem description) also resolves Nash-Williams' conjecture for large \(n\).
  2. Csaba et al is here (preprint) and has been published electronically (pay-walled) as "Proof of the 1-factorization and Hamilton Decomposition Conjectures", Memoirs of the American Mathematical Society, 2016, Vol. 244, Number 1154.

Theorem no. 81: The Lutz–Nagell Theorem
  1. Original sources for this theorem:
    1. Trygve Nagell, "Solutions de quelques problèmes dans la théorie arithmétique des cubiques planes du premier genre", Wid. Akad. Skrifter Oslo, no 1, 1935, pp. 1–25; not online, it seems; zbMATH Open entry.
    2. Élisabeth Lutz, "Sur l'équation y2 = x3 - Ax - B dans les corps p-adiques", J. Reine Angew. Math., 177, 1937, pp. 237–247; online (paywall; facsimile copy).
  2. When I first posted this theorem I ignorantly assumed that my example elliptic curve \(y^2=x^3-43x+166\) crossed the horizontal axis at \((-8,0)\), giving a rational point of order 2 to illustrate the second case of the theorem. Shaun Stevens was kind enough to put me right, pointing out that this would imply a torsion group of order at least \(2\times 7\), contradicting the maximum order of 12 asserted by Mazur's Torsion Theorem.
  3. Tony Forbes' notes on "Elliptic curves, Factorization and Primality Testing" are very good for context on this theorem (200KB pdf download which will probably open in a separate application).
  4. Jennifer Balakrishnan contributes this entry (with a fine A3 poster version) on elliptic curves to the Oxford Mathematics Alphabet.

Theorem no. 82: Gruenberg's Theorem on Nilpotent Groups
  1. Original source for this theorem: K. W. Gruenberg, "Residual Properties of Infinite soluble groups", Proc. London Math. Soc., Vol. s3-7, Issue 1, 1957, pp. 29–62; online (paywall).
  2. The Heisenberg group \(H(n)\) is a group of upper triangular matrices under multiplication; thus for \(n=3\): $$\left(\begin{array}{ccc} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1\end{array}\right)\times \left(\begin{array}{ccc} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1\end{array}\right) =\left(\begin{array}{ccc} 1 & a+x & c+z+ay \\ 0 & 1 & b+y \\ 0 & 0 & 1\end{array}\right). $$ The multiplication looks a bit obscure when expressed in terms of triples, as in our theorem description. However it is a natural convenience in the analysis of this group, c.f. this interesting paper.

Theorem no. 83: The Delsarte–Goethals–Seidel Theorem
  1. The original source for this theorem is P. Delsarte, J. M. Goethals and J. J. Seidel, "Spherical codes and designs", Geom. Dedicata, Vol. 6, No. 3, 1977, pp. 363–388; online (paywall).
  2. A fascinating (but 11MB!) account of Delsarte's approach is Florian Pfender and Günter M.Ziegler, "Kissing numbers, sphere packings, and s ome unexpected proofs", Notices of the AMS, vol. 51, no. 8, 2004, pp. 873–883; online.
  3. Spherical designs owe their origin, essentially, to the 1977 paper of Delsarte, Goethals and J. J. Seidel. They are an important topic in combinatorial design theory, and have their own Wiki page.

Theorem no. 84: The Five Circle Theorem
  1. Original source for this theorem: Miquel, A., "Théorèmes de géométrie", Journal de Mathématiques Pures et Appliquées, Tome 3, 1838, pp. 485–487; online (facsimile by Gallica, the figures appear at the end of the volume with figure 3 (pertaining to the five-circle theorem, appearing here).
  2. Same comment as for Theorem 55 regarding Java; the app for this theorem, if you want to try, is here.
  3. The web link for this theorem, a presentation by Wolfgang Schief is complemented by an an article: W.K Schief and B.G Konopelchenko, "A novel generalization of Clifford's classical point–circle configuration. Geometric interpretation of the quaternionic discrete Schwarzian Kadomtsev–Petviashvili equation", 250, Vol. 465, Issue 2104, 2019, pp. 1291–1308; online (paywall), arxiv.
  4. A converse to the five-circle theorem by Frank Morley is discussed in Tobias Dantzig, "Elementary Proof of a Theorem Due to F. Morley", The American Mathematical Monthly, Vol. 23, No. 7 (Sep., 1916), pp. 246–248; online. (Tobias was the father of George of simplex method fame).

Theorem no. 85: Cayley's Theorem
  1. Original source for this theorem: A. Cayley, "On the theory of groups, as depending on the symbolic equation θn = 1", Philosophical Magazine, Series 4, Vol. 7, 1854, Issue 42, pp. 40–47; online (paywall). There is a very good companion to this paper: David J. Pengelly, "Arthur Cayley and the first paper on group theory", in From Calculus to Computers: Using the Last 200 Years of Mathematics History in the Classroom, Amy Shell-Gellasch and Dick Jardine (eds.), Mathematical Association of America, 2005. Although the above source is what concerns us here, Cayley's paper is in three parts, all with the same name, and appearing in the Philosophical Magazine: Part II is Vol. 7, 1854, Issue 47, pp. 408–409; online (paywall); Part III is Vol. 18, 1859, Issue 117; online (paywall).
  2. Our description and illustration of this theorem are limited to finite groups but it applies equally to infinite groups.
  3. Peter Neumann's assessment of Cayley's theorem is an example of what is known as Gromov's dichotomy "Any proposition concerning all countable groups is either trivially true or false". I wrote down Neumann's remarks at a talk he gave at Queen Mary University of London during a celebration to mark the 60th birthday of R.A. Bailey:
    Poster for Bailey fest 2007      Notes on a talk by Peter Neumann
  4. Jonas Karlsson has sent me the following valuable contextual remarks on Cayley's theorem:
           Regarding Peter Neumann's somewhat dismissive comment on Cayley's theorem, I think it's worth pointing out that said theorem is a special case of the Yoneda embedding. That is, regarding a group G as a one-object category, a set-valued presheaf on the group is a set with a G-action, and the content of the Yoneda lemma is that regarding the group itself as a G-set (under translation) constitutes an embedding, i.e. the map is injective; which is precisely Cayley's theorem.
    As for the Yoneda lemma itself, its proof may be a triviality but the lemma is what legitimizes the functor-of-points approach to algebraic geometry, and this viewpoint seems to be spreading to other areas of modern geometry as well. From this perspective, the theorem was remarkably prescient!
    Some other remarks on generalising Cayley are given here by Terence Tao.
  5. Again, although it may be regarded as trivial that a group of order \(n\) has a faithful permutation representation of degree \(n\), the question whether it has one of smaller degree has been the subject of much research: see this on mathoverflow and this on math stackexchange, for example. Peter Cameron's blog has an entry on the analogous question for semigroups.

Theorem no. 86: Noether's Symmetry Theorem
  1. Original source for this theorem: Noether, E., "Invariante Variationsprobleme" , Nachr. König. Gesell. Wissen. Göttingen, Math.–Phys. Kl.(1918), pp. 235–257; online at Göttinger Digitalisierungszentrum. A 1971 English translation by M.A. Tavel has been put on the arxiv here by Frank Y. Wang. I understand that the translation given in our recommended book, Yvette Kosmann-Schwarzbach (transl. Bertram E. Schwarzbach), The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century, Springer, New York, 2011, is superior. The French original "Les Théorèmes de Noether: Invariance et lois de conservation au XXe siècle" provides a French translation of course.
  2. Some other excellent accounts of Noether's theorem can be found here at the Physics Mill blog and here at the Perimeter Institute.
  3. A valuable source on the background and impact of Noether's theorems is Raphaël Leone, "On the wonderfulness of Noether's theorems, 100 years later, and Routh reduction"; preprint.
  4. A 2018 centenary conference for Noether's theorems was held at Notre Dame university. Only the abstracts are online at the conference website but you can get a good overview of current interest in Noether's achievements.

Theorem no. 87: Lamé's Theorem
  1. Original source for this theorem: G. Lamé, "Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers", Comptes rendus des séances du l'Académie des Sciences, 19 (1844), pp. 867–870. I don't find this issue on Gallica. However, there is a wonderful analysis of Lamé's theorem and its precursors in Jeffrey Shallit, "Origins of the analysis of the Euclidean algorithm", Historia Mathematica, Vol. 21, Issue 4, 1994, pp. 401–419; online
  2. Lamé's theorem comes in many different guises, all essentially saying that Euclid's algorithm takes longest (relative to input size) when applied to consecutive Fibonacci numbers. An excellent survey is provided by cut-the-knot.
  3. Unsurprisingly The Fibonacci Quarterly has a number of interesting papers on Lamé and the Fibonacci numbers. See, for example, J. L. Brown, Jr. and R. L. Duncan, "The Least Remainder Algorithm", The Fibonacci Quarterly, Vol. 9, No. 4, 1971, pp. 347–350, 401; online, and follow back from the references.
  4. Another charming connection between Fibonacci and Euclid lies in the fact that the gcd of two Fibonacci numbers is the Fibonacci number of their gcd: \(\gcd(F_m,F_n)=F_{\gcd(m,n)}.\) A proof may be found here on cut-the-knot. Thanks to Joshua Zelinsky for telling me this.
  5. Also a good corrective to golden ratio hype is this by Chris Budd.

Theorem no. 88: The Cauchy–Kovalevskaya Theorem
  1. Original sources for this theorem:
    1. Cauchy, Augustin, "Mémoire sur l'emploi du calcul des limites dans l'intégration des équations aux dérivées partielles", Comptes rendus hebdomadaire des séances de l'academie des sciences, tome 15, July 1842; in Œuvres completes, première série, tome 7, extrait 170, pp. 17–33; online (facsimile)
    2. von Kowalevsky, Sophie, "Zur Theorie der partiellen Differentialgleichung", Journal für die reine und angewandte Mathematik, 80, 1875, pp. 1–32; online (paywall; facsimile)
  2. Putting Kovalevskaya's work in a modern context is this HAL archives article: Elemer Elad Rosinger, "Can there be a general nonlinear PDE theory for existence of solutions?"
  3. Garry J. Tee kindly sent me a scan of his well-known article about Kovalevskaya which appeared in the Mathematical Chronicle in 1977. The Mathematical Chronicle Committee in turn kindly gave me permission to host a copy here. I have made pdf (about 3.7MB) and Powerpoint (about 9MB) versions. (The Mathematical Chronicle Committee is arranging for digitisation of Mathematical Chronicle).

Theorem no. 89: The Abel-Hurwitz Binomial Theorem
  1. Original sources for this theorem:
    1. Abel, N. H., "Beweis eines Ausdrucks, von welchem die Binomial-Formel ein einzelner Fall ist", J. reine angew. Math., 1, 1826, pp. 159–160; online (paywall; facsimile from Göttinger Digitaisierungszentrum).
    2. A. Hurwitz, "Uber Abel’s Verallgemeinerung der binomischen Formel", Acta Mathematica, 26, 1902, pp. 199–203; online.
  2. Further generalising Abel and Hurwitz is the subject of Alexander Kelmans and Alexander Postnikov, "Generalizations of Abel’s and Hurwitz’s identities", European Journal of Combinatorics, Vol. 29, Issue 7, 2008, pp. 1535–1543; online.
  3. There is a beguiling entry in Gil Kalai's blog Combinatorics and More on Abel sums. This is followed by a more substantial post about joint work on Abel sums by Kalai and Doron Zeilberger.

Theorem no. 90: Cardano's Cubic Formula
  1. A more explicit version of this formula (comparable in format to the quadratic formula) can be found here. This version is made fun of in this blog post.
  2. The case where a cubic equation has three real roots which may only be expressed in radical form using complex numbers is known as casus irreducibilis. It was determined by Pierre Wantzel in 1843.
  3. Thony Christie offers a very detailed discussion of the skirmishes around ownership of solutions to the cubic here. A posting here by Plus magazine is also illuminating.
  4. Kellie Gutman has made an English translation of Tartaglia's verse-form solution of the cubic, brought to us by
  5. David Benjamin has this nice portrait for plus magazine of Cardano and his family, while Quanta offer an excellent account of cubic solving by David S. Richeson.

Theorem no. 91: Khinchin's on Continued Fractions
  1. Original source for this theorem: Khintchine, A., "Metrische Kettenbruchprobleme", Compositio Math., Tome 1, 1935, pp. 361–382; online
  2. A closely related result of Khinchin says that that for almost all real numbers, the \(n\)-th root of the denominator of the \(n\)-th convergent of the continued fraction expansion tends in the limit to a fixed constant, known as Lévy's constant.
  3. Among real numbers which fail to give convergence to Khinchin's and Levy's constants are those related to various kinds of prime numbers, see this by Marek Wolf (thanks to Leo C. Stein for telling me about this).

Theorem no. 92: The Quadratic Formula
  1. A valuable list of derivations of the quadratic formula is provided by Cut-The-Knot.
  2. As a footnote to Rob J. Low's advocacy of the quadratic equation expressed as \(ax^2+2bx +c=0\), an analysis is given by Tony Forbes in issue 204 of M500 magazine, pp. 22–23, of the probability of the solutions being real. In contrast to the 'standard' \(ax^2+bx +c=0\) where the probability is \((41+\ln 64)/72\), the Rob J. Low form has a rational probability: \(7/9\). This is the tip of a deep iceberg, however, as indicated in this blog post in which John D. Cook summarises Alan Edelman and Eric Kostlan, "How many zeros of a random polynomial are real?", Bull. Amer. Math. Soc., Vol. 32, 1995, pp. 1–37; online. Worth a scan also this exchange on twitter (if it stays accessible).
  3. There is an interesting exchange on Twitter (24 April, 2019) between @robinhouston and @MathPrinceps on the fact that Gauss and Lagrange disagreed on whether \(bx\) or \(2bx\) is preferable, with Gauss preferring the latter. Because Twitter may be inaccessible by the time you read this I have stored a screenshot here.
  4. This Quanta magazine article by Patrick Honner is illuminating on what makes the quadratic formula tick (and what fails to tick for cubics!).

Theorem no. 93: The Generalised Hexachord Theorem
  1. Original source for this theorem: M. Babbitt, "Some aspects of twelve-tone composition", The Score, 12, 1955, pp. 53–61; not online, I think, see here for details of the journal.
  2. The application of Fourier analysis pioneered by Emmanuel Amiot and others, referenced on this theorem page, has evolved into a whole monograph by Amiot: Music Through Fourier Space: Discrete Fourier Transform in Music Theory, Springer, 2016.
  3. For readers of French this article for Images des mathématiques by Corentin Bayette is excellent.
  4. The music example in this theorem description was typeset using the CERL Sound Group LIME music notation software.

Theorem no. 94: Cayley's Formula
  1. Original source for this theorem: Cayley, A., "A theorem on trees", Quart. J. Pure Appl. Math., Vol. 23, 1889, pp. 376–378; online (facsimile).
  2. The second proof of Cayley in Stanley's book, by André Joyal, has a nice application to automata theory by Peter Cameron, described here.
  3. Also in Cameron's blog: another lovely enumeration yielding \(n^{n-2}\). A follow-up is this arxiv post with Liam Stott.

Theorem no. 95: The Convolution Theorem
  1. Original source for this theorem: Cooley, James W. and Tukey, John W., "An algorithm for the machine calculation of complex Fourier series", Math. Comput., 19 (90), 1965, pp. 297–301; online. The relevant Wikipedia page is a rich source of historical and other sources, many open access.
  2. For Gauss and FFT see Michael T. Heideman, Don H. Johnson and c. Sidney Burrus, "Gauss and the History of the Fast Fourier Transform", IEEE ASSP Magazine, October 1984, 14–21. Online here. Lanczos's contribution is mentioned in his MacTutor entry. On the connection to quadratic reciprocity see Note (4) to Theorem 29.
  3. The continuouse convolution operator and related theorems date back to Euler and Laplace, as described by Alejandro Dominguez, "A History of the Convolution Operation", IEEE Pulse, January/February 2015; online.
  4. A lovely gentle introduction to Fourier transforms by (who also provide the recommended web link on the theorem page. The recommendation used to be a very nice pdf download from Berkeley but Firefox warns me the link is risky. At your own risk it is here.
  5. Aled Walker contributes this entry (with a fine A3 poster version) on the Fourier transform to the Oxford Mathematics Alphabet.

Theorem no. 96: The Rule of Sarrus
  1. Original source for this theorem is an 1833 article "Nouvelles méthodes pour la résolution des équations" published by Sarrus in Strasbourg. It is listed in the collection Statistique des lettres et des sciences en France, edited by François Fortuné Guyot de Fère in 1834, which is free to read from google, the entry appearing on page 235 (but the online facsimile scrollbar gives it in Roman as dcci).
  2. There is apparently a 4×4 version of the Rule of Sarrus called the 'Rule of Villalobos' due to the Mexican mathematician Gustavo Villalobos Hernández. See comment (4) here which links to a Spanish wikipedia entry which unfortunately appears to have been deleted.

Theorem no. 97: Nevanlinna's Five-Value Theorem
  1. The source for this theorem is R. Nevanlinna, "Eindentig keitssätze in der theorie der meromorphen funktionen," Acta. Math., 48 (1926), pp. 367–391; online.
  2. The tribute of Lee Rubel reads "... my favorite theorem in all of mathematics is a theorem of R. Nevanlinna that two functions, meromorphic in the whole complex plane, that share five values must be identical. For real functions, there is nothing that even remotely corresponds to this." It is from the introduction to Entire and Meromorphic Functions, by Lee A. Rubel with James E. Colliander, Springer-Verlag New York, 1996. Herman Weyl described Nevanlinna's 1925 paper on meromorphic functions as "one of the few great mathematical events in our century." (Not having access to Weyl's book Meromorphic Functions and Analytic Curves, Princeton University Press, 1944, I am relying on the review of the book by Herbert Busemann and the Wikipedia article on Nevanlinna theory.

Theorem no. 98: Cartwright's Theorem
  1. Original source for this theorem: "On analytic functions regular in the unit circle II", Quart. J. Math. Oxford Ser. (2) 6 (1935) 94–105; online. In his obituary of Cartwright (Bull. London Math. Soc. 34 (2002) 91–107) Walter K. Hayman comments "With this paper the author essentially created a new field. It was almost the only paper quoted by Littlewood in his book [Lectures on the Theory of Functions, OUP, 1944] and led me to ask Mary Cartwright to become my research supervisor." The citation appears on p. 232 on Littlewood's book. He writes "The proofs, due to Cartwright, are difficult, and depend on ideas unlike any we have been considering". The book is online here.

Theorem no. 99: The Happy Ending Problem
  1. Original source for this theorem: Erdős, P. and Szekeres, G., "A combinatorial problem in geometry", Compositio Mathematica, Tome 2 (1935), pp. 463–470; online. They wrote about the problem again in 1960: P. Erdős and G. Szekeres, "On some extremum problems in elementary geometry", Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 3–4 (1960/1961), pp. 53–62; online (1.8MB pdf).
  2. The attribution to Turán of the solution to n = 5 for this problem I found in Section 10 of Imre Bárány, "Discrete and convex geometry", in János Horváth (ed.), A Panorama of Hungarian Mathematics in the Twentieth Century I, Springer, 2005, pp. 427–454; pdf reprint (July 2024). More details are given by Szekeres and Peters in their article proving n = 6: Szekeres, G., Peters, L., "Computer solution to the 17-point Erdős–Szekeres problem", ANZIAM J., vol. 48(2), 2006, pp. 151–164; online.
  3. Suk's asymptotic solution is Suk, A., "On the Erdős-Szekeres convex polygon problem", J. Amer. Math. Soc., Vol. 30, Number 4, 2017, pp. 1047–1053; online (paywall; arxiv). There is a very nice description of the Happy Ending problem and Suk's contribution here, by Kevin Hartnett for Quanta magazine.
  4. John Golden has made a beguiling Geogebra animation inspired by Esther Klein's original Happy Ending question.
  5. There is a close connectiong with Ramsey theory which is explored by Sara Freyland here.

Theorem no. 100: The Design of the Century
  1. The original source for this theorem is Forbes, A.D., Grannell, M.J. & Griggs, T.S. "The design of the century", Math. Slovaca, Vol. 57, No. 5, 2007, pp. 495–499; online.
  2. There is an interesting related entry on math.stackexchange.

Theorem no. 101: Kepler's Conjecture
  1. Original source for this theorem is: Thomas C. Hales, "A proof of the Kepler conjecture", Annals of Mathematics, Vol. 162, Issue 3, 2005, pp. 1065–1185; online. But note the extended proof process undertaken by Hales, alluded to below.
  2. Theorem under Construction icon This was a 'theorem under construction' pending Hales' Flyspeck project to reinforce his original proof with a machine-automated one. This completed in 2014 but the proof was already made bullet-proof with his 2012 publication of Dense Sphere Packings: A Blueprint for Formal Proofs. The formal proof is also described by Hales and twenty-one other authors in "A formal proof of the Kepler conjecture", Forum of Mathematics, Pi (2017), Vol. 5, e2; online.
  3. The story of Hales' original proof and publication of Kepler is well told here. An glimpse of the story lies in the 2005 Annals paper footnote (which most publishing mathematicians will empathise with!) "Received 1998, revised, 2003". Note that his main referee at Annals, namely Gábor Fejes Tóth, is the son of László Fejes Tóth who made the original breakthrough in the conjecture's resolution. By the way, a statement by the Editorial Board at Annals explicitly invites "computer-assisted proofs of exceptionally important mathematical theorems", an invitation which I understand to pre-date the Kepler submission.
  4. An earlier proof of Kepler, by Wu-Yi Hsiang, is generally regarded as incomplete. Details are given by Hales in section 4.4 of this preprint which is, generally, a first-rate introduction to the conjecture and his proof.
  5. There are some remarks about Flyspeck in Hales's contribution to an AMS Notices special issue on formal proof. The Flyspeck project is now at GitHub.

Theorem no. 102: Viète's Formula
  1. Original source for this theorem: François Viète, Variorum de rebus mathematicis, responsorum liber VIII, 1593; online (facsimile). There is a helpful response on math.stackexchange which gives the exact location (note to Corollary to Proposition 2 of Chapter 18, which is here in the above facsimile).
  2. Tom Ostler has a more detailed account of the connections between Viète's formula and ruler and compass constructions in "Geometric constructions approximating pi", Mathematical Spectrum, Vol. 40, No. 3 (May 2008), 106–108; complete issue download (600KB).
  3. For French readers this derivation of Viète (pdf download) by is good.

Theorem no. 103: The Parking Function Formula
  1. Original sources for this theorem:
    1. Ronald Pyke, "The supremum and infimum of the Poisson process", Ann. Math. Statist. 30 (1959) 568–576; online.
    2. A. G. Konheim and B. Weiss, "An occupancy discipline and applications", SIAM J. Appl. Math., 14 (6), 1966, pp. 1266–1274; online (paywall).
    3. The bijection illustrated on the theorem page is from Philippe Chassaing and Jean-François Marckert, "Parking functions, empirical processes, and the width of rooted labeled trees, Electronic J. Combinatorics., Vol. 8, Issue 1, 2001, article R14; online.
  2. The bijection to labelled trees establishes the formula but the `book' proof is due to Henry O. Pollack. An account is given in this wonderful presentation by Richard Stanley
  3. The note by Peter Cameron which is the recommended weblink for this theorem was written for the Queen Mary Maths dept student newsletter. Its topic is very nicely explored in this presentation by Thomas Prellberg.

Theorem no. 104: The Goins–Maddox–Rusin Theorem for Heron Triangles
  1. Original sources for this theorem (the latter two are the weblinks from the theorem page):
    1. N. J. Fine, "On rational triangles", Amer. Math. Monthly, Vol. 83, No. 7, 1976, 517–521; online (paywall).
    2. David J. Rusin, "Rational triangles with equal area," New York Journal of Mathematics, Vol. 4, 1998, 1–16; online
    3. Edray Herber Goins, Davin Maddox, "Heron triangles via elliptic curves", Rocky Mountain J. Math., Vol. 36, No. 5, 2006, pp. 1511–1524; online
  2. The triangle-to-curve conversion rule given in the text (from the Goins–Maddox paper) guarantees to produce an elliptic curve and a rational point on that curve. The actual curve and point depend on the order in which sides \(a,b\) and \(c\) are chosen. For example, the triangle with sides \(3, \frac{10}{3},\frac{17}{3}\), with area \(n=4\), gives \(\rho=1/4\) or \(2\) or \(2/9\), depending on which sides are called \(a,b\) and \(c\). And for the curve depicted, \(\rho=1/4\), neither ordering of the sides recovers the point \(P=(-8,24)\) on this curve (which gives the same triangle but with some negative sides) --- instead points \((2,6)\) and \((18,102)\) are discovered.
  3. The version of Heron's formula I have used is one of several given in the wikipedia entry. It is useful for showing the invariance of the rule under sign changes.
  4. There is a nice account here of searching for congruent numbers (integer areas of rational right triangles). The associated sequence at oeis is A003273.
  5. "Elliptic Curves in Recreational Number Theory" by Allan MacLeod offers an accessible introduction to Elliptic Curves and has extensive coverage of triangle problems, as well as much else.

Theorem no. 105: The Pumping Lemma
  1. Original sources for this theorem (according to its Wiki page):
    1. Rabin, Michael and Scott, Dana, "Finite automata and their decision problems", IBM J. Res. Dev., Vol. 3, Issue 2, 1959, pp. 114–125; online (paywall; pdf download's are easy to find online, here, for example, April 2024).
    2. Bar-Hillel, Y., Perles, M. and Shamir, E., "On formal properties of simple phrase structure grammars", Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung, Vol. 14, Issue 2, 1961, pp. 143–172; online (paywall). Seems harder to find downloads. There is an interesting contribution here on cstheory.stackexchange.
  2. A nice entry at Computational Complexity by Bill Gasarch poses a challenge "Find a non-reg lang that is not easily proven non-reg." Meaning, the pumping lemma plus some other basic tools always seems to be sufficient.
  3. I feel the need to mention Michael A. Harrison, Introduction to Formal Language Theory, Addison Wesley Longman Publishing Co, 1978 because, besides it being an excellent book by a pioneer in the subject, it has a wonderful pumping lemma joke on the cover.

Theorem no. 106: Babbit's Theorem
  1. I am unsure of the source of this theorem. It may well be mentioned in Babbit's 1995 paper in The Score on interval analysis (see note (1) to Theorem 93).
  2. The music example in this theorem description was typeset using the CERL Sound Group LIME music notation software.
  3. The suggestion on the theorem page that Bach may have been deliberately maximising overlap between his fugal subjects and responses is frivolous, but 'Bach the mathematician' is an enduring trope see this, for example.

Theorem no. 107: The Tverberg Partition Theorem
  1. Original sources for this theorem:
    1. Radon, J., "Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten", Mathematische Annalen, Vol. 83 (1–2), 1921, pp. 113–115; online (paywall; facsimile).
    2. H. Tverberg, "A generalization of Radon’s theorem", J. London Math. Soc., Vol. s1-41, Issue 1, 1966, pp. 123–128; online (paywall).
  2. The original recommended web link for this page was Stephen Hell's magnificent dissertation "Tverberg-type Theorems and the Fractional Helly Property". This became unavailable as a direct download but is now located at and is highly recommended.
  3. Tverberg's Theorem is a special case of the so-called Topological Tverberg Conjecture. The conjecture is true when \(r\), the number of partition parts, is a prime power but was proved false in general in 2015. It is all very well-described by Gil Kalai: follow the links back from here (you will also find a proof of Tverberg's Theorem itself).
  4. Also by Gil Kalai, a good overview, marking the birthday of Imre Bárány, of various 'Tverberg' results and conjectures. In fact he has subsequently posted a round-up of new things and previous posts, including a new short proof of Tverberg by Bárány. "Tverberg’s theorem is among my favorite mathematical theorems" he says.
  5. There is support for Gerard Sierksma's conjecture (the Dutch Cheese Problem) in Boris Bukh, Po-Shen Loh and Gabriel Nivasch, "Classifying unavoidable Tverberg partitions", J. Comput. Geom., Vol. 8, No. 1, 2017, pp. 174–205; online.

Theorem no. 108: The Analyst's Travelling Salesman Theorem
  1. Original sources for this theorem:
    1. Peter W. Jones, "Rectifiable sets and the traveling salesman problem", Inventiones Mathematicae, Vol. 102, Issue 1, 1990, page 1–16; online (paywall; facsimile)
    2. Kate Okikiolu, "Characterization of subsets of rectifiable curves in Rn", J. London Math. Soc., s2-46 (2), 1992, pp. 336–348; online (paywall).
  2. There is a good wiki page on ATST.
  3. Raanan Schul's Analyst's Traveling Salesman Theorems, a Survey is an excellent source of more detailed and more recent information.
  4. The Koch snowflake image in the illustration of this theorem was copied from a website called which appears no longer to exist.

Theorem no. 109: The Beardwood–Halton–Hammersley Theorem
  1. Original source for this theorem: Beardwood, J., Halton, J. H. and Hammersley, J. M., "The shortest path through many points", Proc. Cambridge Philos. Soc., 55, pp. 299–327; online (paywall).
  2. Entry A073008 at is "Decimal expansion of the Traveling Salesman constant" which appears to correspond to \(\beta_2\) on our page. The digits are conjectural, being the expansion of \((4/153)(1 + 2\sqrt{2})\sqrt{51}\). It is not clear whose conjecture this is; it is not in the cited article Stefan Steinerberger, "New bounds for the traveling salesman constant", Adv. in Appl. Probab., Vol. 47 (1), 2015, pp. 27–36; online (paywall; arxiv), which seems authoritative and relatively recent.
  3. An interesting counterexample of Alessandro Arlotto and J. Michael Steele puts a limit on how far the hypothesis on the variables \(X_1, X_2, \ldots\) forming the TSP tour may be relaxed.

Theorem no. 110: The Robbins Problem
  1. Original source for this theorem: W. McCune, "Solution of the Robbins Problem", J. Automated Reasoning, 19(3), 1997, pp. 263–276; online (paywall).
  2. John Baez has a nice explanation here of a single-axiom definition of a lattice. I first saw this as a Baez google+ post accompanied by his usual crop of quality comments, a couple of which I quote:
    • David Tweed: "Related, there's a claim to a more human proof that the Robbins axioms describe boolean algebra at . Frustratingly it doesn't say if this was derived from unintuitive computer proofs or independently. So a fact first proved by computer may later acquire a human proof."
    • Robert Rothenberg: It reminds me of work people have done to come up with single-axiom versions of various logics, I think Harris & Rezus. See A. Rezus, "On a Theorem of Tarski", Libertas Math., 2, pp. 63-97. Discussed in .

Theorem no. 111: De Morgan's Laws
  1. Original source for this theorem: Augustus De Morgan, Formal Logic: Or, The Calculus of Inference, Necessary and Probable, 1847 (which has a nice entry in MAA's Mathematical Treasure series). But I found the following on Peter Cameron's mathematical quotations page (under logic):
           " ... the contradictory opposite of a copulative proposition is a disjunctive proposition composed of the contradictory opposites of its parts.
    ... the contradictory opposite of a disjunctive proposition is a copulative proposition composed of the contradictories of the parts of the disjunctive proposition."
    William of Ockham (Occam), Summa totius logicae, 14th century (transl. Philotheus Boehner 1955)
    Note: This is Ockham's formulation of De Morgan's Laws, more than five hundred years before De Morgan. It is just as clear in his Latin text.
    and there is more on the prehistory of De Morgan's laws on their Wiki page.

Theorem no. 112: Bregman's Theorem
  1. Original source for this theorem: Bregman L., "Some Properties of Nonnegative Matrices and their Permanents", Dokl. Akad. Nauk SSSR, v. 211, No. 1, 1973, pp. 27–30; online. (English translation: Soviet Math. Dokl., v. 14, 1973, pp. 945–949.) The original conjecture of Minc appears in H. Minc, "Upper bounds for permanents of (0, 1)-matrices", Bull. Amer. Math. Soc., Vol. 69, Issue 6, 1963, pp. 789–791; online.
  2. A. Schrijver attributes to Brouwer (presumably L.E.J. Brouwer who was still alive when Minc originally conjectured what became Bregman's Theorem) the observation that an upper bound for permanents of arbitrary nonnegative matrices may be derived directly from the bound for (0,1)-matrices. Thus, if \(v = (b_1, b_2, \ldots, b_n)\) is a vector in descending order, and letting \(b_{n+1}=0\), define $$g(v)=\sum_{k=1}^n(b_k-b_{k+1})(k!)^{1/k}.$$ Then the Minc bound is replaced by the product \(\prod_ig(v_i)\) over the rows \(v_i\) of the matrix. (A. Schijver, "A short proof of Minc's conjecture", J. Combin. Theory, A, vol. 25, no. 1, 1978, 80–31; online.)
  3. Although computing \(\mbox{per}(M)\), the permanent of an arbitrary matrix \(M\), is hard (specifically, #P-hard) even if \(M\) is \(0\mbox{-}1\), in the case where \(M\) has nonnegative entries Jerum, Sinclair and Vigoda have given a polynomial algorithm (pdf download, May 2024) which gives a value arbitrarily close to \(\mbox{per}(M)\) with high probability (FPRAS).
  4. There is a very good blog post by Scott Aaronson on deep work by Avi Widgerson on the permanent.
  5. My illustration for this theorem ignores the fact that races can have dead heats. Indeed in Jean-Marie De Koninck's Those Fascinating Numbers, the number of outcomes of an \(n\)-horse race, taking account of dead heats, is called the \(n\)-th horse number. They are the subject of an entertaining post to the Computational Complexity blog by Bill Gasarch.

Theorem no. 113: The van der Waerden Conjecture
  1. Original sources for this theorem:
    1. M. Marcus and M. Newman, "On the minimum of the permanent of a doubly stochastic matrix", Duke Math. J., Vol.26, No. 1, 1959, pp. 61–72; online (paywall).
    2. G.P. Egorychev, "The solution of van der Waerden's problem for permanents", Advances in Mathematics, Vol. 42, Issue 3, 1981, pp. 299–305; online. (Egorychev's proof had previously appeared in Russian in G.P. Egorychev, "A solution of the Van der Waerden's permanent problem", Preprint IFSO-L3 M, Academy of Sciences SSSR, Krasnoyarks, 1980.)
    3. D.I. Falikman, "Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix", Matematicheskie Zametki, Vol. 29, No. 6, 1981, pp. 931–938; online (in Russian).
    4. Béla Gyires, "Elementary proof for a Van der Waerden's conjecture and related theorems", Computers & Mathematics with Applications, Vol. 31, Issue 10, 1996, pp. 7–21; online. Shows the equivalence of van der Waerden with Gyires's beautiful 1977 inequality $$p^2\sqrt{\mbox{Per}(AA^T)}+q^2\sqrt{\mbox{Per}(A^TA)}+2pq\mbox{Per}(A)\geq n!/n^n,$$ for \(p,q\geq 0,p+q=1\), with equality if and only if \(A\) has all entries \(1/n\). Equal credit for proving the conjecture is given to Gyires in the relevant Wiki entry and, for example, in entry 221 in this list of theorems (1.5MB pdf) by Oliver Knill. However Gyires explicitly cites Egorychev and Falikman as the first complete proofs in his "Contribution to van der Waerden's conjecture", Computers & Mathematics with Applications, Vol. 42, Issues 10–11, 2001, pp. 1431–1437; online.
  2. The Brualdi quote on the origin of van der Waerden's conjecture comes from Brualdi, Richard A., review of Permanents by Henryk Minc, Bull. Amer. Math. Soc., new series Vol. 1, No. 6, 1979, pp. 965–973; online. On this subject, there is an interesting entry at with some good links to source material.
  3. The probability calculations in this theorem description may be justified explicitly by observing that column sums add to unity, so that for, say, System 1, $$\left(\frac12+\frac14+\frac14\right)\times\left(\frac13+\frac58+\frac{1}{24}\right)\times\left(\frac16+\frac18+\frac{17}{24}\right)=1,$$ which, expanding the brackets, equals the sum of the probabilities of every possible product of output failures over the three computers.
  4. Closely related are questions about permanents of matrices having all row and column sums equal, counting perfect matchings in regular bipartite graphs of equal-size parts. Lower bounds have recieved much attention, notably the Schrijver–Valient Conjecture. See, for example, Péter Csikvári, "Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems", Journal of the European Mathematical Society, Vol. 19, 2017, pp. 1811–1844; online (paywall; arxiv).
  5. There is another van der Waerden Conjecture, concerning Galois groups of integer polynomials. A proof was announced by Manjul Bhargava in November 2021. See also this Quanta article by Leila Sloman

Theorem no. 114: The Lagrange Property for Moufang Loops
  1. The sources for this theorem are Alexander N. Grishkov and Andrei V. Zavarnitsine, "Lagrange's theorem for Moufang loops", Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 139, Issue 1, July 2005 , pp. 41–57; online (paywall); preprint; and S.M. Gagola III and J.I. Hall, "Lagrange's theorem for Moufang loops", Acta Sci. Math. (Szeged), 71 (2005), pp. 45–64; online. G. Eric Moorhouse has told me that his proof remained unpublished, having been the victim of unfortunate timing. It is cited in S.M. Gagola III and J.I. Hall as 'private communication'.
  2. There are many related questions about group-like properties of Moufang loops. Although not up-to-date, S. M. Gagola III, "How and why Moufang loops behave like groups", Quasigroups and Related Systems, Vol. 19, No.1, 2011, pp. 1–22; online, is admirably clear and authoritative.
  3. The original weblink for this theorem page was Chein, O., Kinyon, M.K., Rajah, A. and Vojtěchovský, "Loops and the Lagrange Property", Results. Math., 43, (2003), pp. 74–78; online (paywall); preprint. While just predating the proof of the theorem it nevertheless remains an excellent introduction.

Theorem no. 115: The Hardy–Ramanujan Asymptotic Partition Formula
  1. Hardy and Ramanujan published their asymptotic formula in 1918 in "Asymptotic Formulae in Combinatory Analysis", Proc. London Math. Soc., (2) 17, pp 75–115. But they had published preliminary versions as early as 1916 and the abstract to the 1918 paper appeared in the records of LMS Proceedings for March 1st, 1917 (Hardy himself, as vice-president, taking the chair). The papers can be viewed online here.
  2. In section 14.7 of Tom Apostol's Introduction to Analytic Number Theory there is an elementary derivation of a bound on the partition number \(p(n)\): $$p(n)<\exp(K\sqrt{n}),$$ where \(K=\tau/\sqrt{6}\) (thanks to Joshua Zelinksy for pointing this out to me).

Theorem no. 116: The Polynomial Coprimality Theorem
  1. Original source for this theorem: S. Corteel, C. Savage, H. Wilf, D. Zeilberger, "A Pentagonal number sieve", Journal of Combinatorial Theory, Series A, vol. 82, no. 2, 1998, 186–192; online. The constructive proof of Arthur T. Benjamin and Curtis D. Bennett is "The probability of relatively prime polynomials", Mathematics Magazine, vol. 80, no. 3 (2007), pp. 196–202; online (paywall; free pdf download, June 2024).
  2. The row and column labels in the illustration of this theorem have been supressed for \(n=3\) to save space. They are, in order, \(x^3, x^3+1, x^3+x, x^3+x+1, x^3+x^2, x^3+x^2+1, x^3+x^2+x, x^3+x^2+x+1\), the 4th and 6th being irreducible (irreducible polynomials up to degree 5 over GF(2) are listed here).
  3. Thomas Hagedorn and Jeffrey Hatley have generalised this result to consider polynomials over the ring \(\mathbb{Z}_{p^k},\ p\) prime: "The probability of relatively coprime polynomials in \(\mathbb{Z}_{p^k}[x]\)", Involve, vol. 3, no. 2, 2010, pages 223–232; online. Their paper reviews several other generalisations.

Theorem no. 117: A Theorem of Erdős and Wilson on Edge Colourings)
  1. Original source for this theorem: P. Erdős and Robin J. Wilson, "On the chromatic index of almost all graphs", Journal of Combinatorial Theory, Series B, Vol. 23, Issues 2–3, 1977, pp. 255–257; online (which is the paper linked from the theorem page, but this is the official version whereas the theorem page links to The Erdős Project at the Rényi Institute, which actually does not yet offer Erdős's complete output, currently running up to end of 1989.

Theorem no. 118: Catalan's Conjecture (Mihăilescu's Theorem)
  1. Original sources for this theorem:
    1. V.A. Lebesgue, "Sur l'impossibilité en nombres entiers de l'équation xm = y2 + 1", Nouv. Ann. Math., 9, 1850, pp. 178–181; online.
    2. J.W.S. Cassels, "On the equation ax – by = 1, II", Proc. Cambridge Philos. Soc., Vol. 56, Issue 2, 1960, pp. 97–103; online (paywall).
    3. Chao Ko, "On the Diophantine equation x2 = yn + 1, xy ≠ 0", Sci.Sinica (Notes), 14, 1964, pp. 457–460.
    4. Preda Mihăilescu, "Primary cyclotomic units and a proof of Catalan’s conjecture", Journal für die reine und angewandte Mathematik, Vol. 2004, Issue 572, pp. 167–195; online (paywall).
  2. There is more on Pillai's conjecture and on the history of Catalan's conjecture here by Michel Waldschmidt.

Theorem no. 119: Kneser's Conjecture
  1. Kneser's conjecture was proposed in Kneser, M., "Aufgabe 360", Jahresber. Deutsch. Math.-Verein, Vol. 58, 1956, p. 27 (not, as is often stated, Aufgabe 300, and in 1955); online (facsimile). Lovász's proof of Kneser's conjecture appears in "Kneser's conjecture, chromatic number, and homotopy", J. Combin. Theory A, Vol. 25, issue 3, 1978, 319–324; online. In the same issue there is a one-paragraph proof, inspired by Lovász's and also topological in nature, by Imre Bárány, "A short proof of Kneser's conjecture", J. Combin. Theory A, Vol. 25, issue 3, 1978, 325–326; online. The elementary proof of Jiří Matoušek appears in "A combinatorial proof of Kneser’s conjecture", Combinatorica, Vol. 24, Issue 1, 2004, 163–170; online (paywall; online preprint, February 2024, see under M).
  2. Bárány's short proof relies on a theorem of David Gale on the distribution of points on the sphere. An equally short proof which doesn't rely on Gale has been given in Joshua Greene, "A new short proof of Kneser's conjecture", American Math. Monthly, Vol. 109, No. 10, 2002, pp. 918–919; online (paywall; pdf download, February 2024).
  3. As well as their colourability, the question as to the Hamiltonicity of Kneser graphs has also been answered: the only non-Hamiltonian case is \(K (5,2)\) (which is the Petersen graph ). See this blog entry from Gil Kalai.

Theorem no. 120: The Lovász Local Lemma
  1. Original sources for this theorem:
    1. The original Local Lemma appears in Erdős, P. and Lovász, L., "Problems and results on 3-chromatic hypergraphs and some related questions", in A. Hajnal; R. Rado; V. T. Sós (eds.), Infinite and Finite Sets (to Paul Erdős on his 60th birthday). II., North-Holland, 1975, pp. 609–627; online.
    2. The paper of Carsten Thomassen is "The even cycle problem for directed graphs", J. Amer. Math. Soc., Vol. 5, Number 2, 1992, pp. 217–229; online: the result on hypergraph colouring appears as Theorem 5.1. (Not stated exactly as found on my theorem page; you can find more details and advances in Colin McDiarmid, "Hypergraph colouring and the Lovász Local Lemma", Discrete Mathematics, Vol. 167/168 167/168, 1997, pp. 481–486; online.)
  2. An earlier version of this theorem description may be found here. It uses an artificial number theory example to show non-independent sets which are nevertheless pairwise independent. But the conclusion of the Local Lemma is still true even though its hypotheses are false. The example replacing it gives us a false conclusion and is a bit less artificial (in fact it is based on a scenario from cryptography, described to me by Michelle Kendall, which is not artificial at all).
  3. The current example concerns an event Aij that two multisets of size 2, each chosen with repetition from the set {1,...,n}, have nonempty intersection. The number of ways that such a pair of intersecting multisets can be chosen is $${n+1\choose 2}^2-\left({n\choose 2}{n-1\choose 2}+n{n\choose 2}\right)=\frac12n\left(2n^2-n+1\right).$$ This is A081436 in The probability of \(A_{ij}\) is thus \(\frac12n(2n^2-n+1)/{n+1\choose 2}^2\).
    The probability of the triple event \(A_{ij}\cap A_{ik}\cap A_{jk}\) may be calculated as \(\frac12n(2n^3+2n^2-7n+5)/{n+1\choose 2}^3\). When \(n = 3\), this has value \(7/18\), much greater than \(8/27\), the cube of the probability of the single event.
  4. Anwer Khurshid and Haredo Sahai, "On mutual and pairwise independence: some counterexamples", Pi Mu Epsilon Journal, Vol. 9, No. 9, 1993, 563–570, looks promising as a source of further false applications of the Local Lemma. Online here (10.3MB pdf).
  5. A constructive proof of the Local Lemma is given in Robin A. Moser and Gábor Tardos, "A constructive proof of the Lovasz Local Lemma", Journal of the ACM, Vol. 57, Issue 2, 2010; online (paywall; arxiv).

Theorem no. 121: Lambert's Formula
  1. How this formula arose out of Lambert's attempts to prove Euclid's 5th postulate is beautifully described here by Evelyn J. Lamb...
  2. ...who also describes and links to a beguiling interactive tool for tiling the Poincaré disk, part of a fine non-Euclidean geometry resource by  Malin Christersson.

Theorem no. 122: The Borsuk–Ulam Theorem
  1. The original source for this theorem is Karol Borsuk, "Drei Sätze über die n-dimensionale euklidische Sphäre", Fundamenta Mathematicae, 20 (1933), pp. 177–190; online. H. Steinlein, "Spheres and symmetry: Borsuk's antipodal theorem", Topol. Methods Nonlinear Anal., Vol. 1, Number 1 (1993), pp. 15–33; online; cites two original papers by Borsuk, an earlier having appeared in 1932.
  2. In one dimension the theorem is an easy corollary of the Intermediate Value Theorem, see this from Plus magazine, for example.

Theorem no. 123: The Wedderburn–Artin Theorem
  1. Original sources:
    1. J.H.M. Wedderburn, "On hypercomplex numbers", Proc. London Math. Soc., s2-6, Issue 1, 1908, pp. 77–118; online (paywall)
    2. E. Artin, "Zur Theorie der hyperkomplexen Zahlen", Abh. Math. Sem. Univ. Hamburg, Vol. 5, No. 1, 1927, pp. 251–260; online (paywall)
  2. A superb account of the origins and consequences of this theorem, written shortly after Wedderburn's death in 1948, is Emil Artin, "The influence of J. H. M. Wedderburn on the development of modern algebra", Bull. Amer. Math. Soc., 56(1.P1), 1950, pp. 65–72; online
  3. An interesting article on the significance of Wedderburn-Artin is provided by Quora (with companion entries on significance of several other theorems)

Theorem no. 124: The Lagrange Interpolation Formula
  1. The link between Lagrange interpolation and the Chinese Remainder Theorem is discussed here.
  2. Of course linear Lagrange interpolation for sets of points in \(\mathbb{R}^2\) extends in various directions. A pretty derivation for linear interpolation in \(\mathbb{R}^n\) by Kamron Saniee is given in "A Simple Expression for Multivariate Lagrange Interpolation", SIAM Undergraduate Research Online (SIURO), Vol. 1, Issue 1, 2008; online. These lectures notes by Kostas Kokkotas do an excellent job of giving the wider context for interpolation (see chapter 3).
  3. You can find interpolation calculators on the web. This, from dCode, a French ciphers and cryptogram site, for example; or wolfram alpha which will respond to something like "interpolate (0,1), (1,4),(2,5)".
  4. Polynomial interpolation as a way of approximating a curve can be sensative to the points chosen. Equidistant points, for example, can cause distortions which, counter-intuitively, increase with the number of points sampled: Runga's phenomenon.

Theorem no. 125: The Skolem–Noether Theorem
  1. Original sources for this theorem:
    1. Skolem, Thoralf, Zur Theorie der assoziativen Zahlensysteme, Skrifter, Oslo (12), 1927 (a 50-page monograph - I don't think it is available digitally).
    2. Noether, E., "Nichtkommutative Algebra", Math. Z., 37, 1933, pp. 514–541; online.
  2. For a short (but not self-contained) proof of Skolem–Noether see this from the stacks project; an elementary proof is given by Jenő Szigeti and Leon van Wyk, "A Constructive Elementary Proof of the Skolem-Noether Theorem for Matrix Algebras", American Mathematical Monthly, vol. 124, no. 10, 2017, pp. 966–968; preprint.

Theorem no. 126: The Asymptotic (Half) Liar Formula
  1. Original sources for this theorem:
    1. Joel Spencer, "Ulam’s searching problem with a fixed number of lies", Theoret. Comput. Sci., Vol. 95, Issue 2, 1992, pp. 307–321; online.
    2. Ioana Dumitriu and Joel Spencer, "A halfliar's game", Theoret. Comput. Sci., Vol. 313, Issue 3, 2004, pp. 353–369; online.
  2. The exponential nature of the Dumitriu–Spencer asymptotic can be appreciated by noting that, while \(U_1(7)\geq 15\) (the example illustrating the theorem), the value of \(U_1(25)\) exceeds \(10^6\), i.e. asking 25 questions is guaranteed to find a value between 1 and a million, in the face of at most one lie. A nice account of this result by Deryk Osthus and Rachel Watkinson is here.
  3. The exact value of \(U_1(7)\) is \(16\); \(U_1(25)=1290554\). With such rapidly growing values it is more convenient to ask the liar question the other way round: given a value for \(n\) what is the least number, \(q_k(n)\), of questions which will guarantee a number in the range \(1,\ldots,n\) is determined, in the face of \(k\) lies? This is the version discussed by Osthus and Watkinson. Thus \(U_1(7)=16\) because \(q_1(16)=7\) but \(q_1(17)=8\).
  4. A comprehensive and very readable survey, "Searching games with errors—fifty years of coping with liars" by Andrzej Pelc, Theoretical Computer Science, Vol. 270, Issues 1–2, 2002, pp. 71–109; online.
  5. A tangential but intriguing link regarding the Fano plane, which generates Peter Cameron's trick illustrating this theorem, is this on using the Fano plane to generate poetry.
  6. Although I attribute the delphic oracle image used in my illustration of this theorem to the Staatliche Museen Berlin I think this attribution is based merely on a google image search. I have elsewhere seen it attributed to the "Collection of Joan Cadden".

Theorem no. 127: The Lucas–Lehmer Test
  1. Original sources for this theorem are:
    1. É. Lucas, "Théorie des fonctions numériques simplement périodiques", American Journal of Mathematics, 1(1878), pp. 184–240, 289–321; online, also at (1.1MB pdf).
    2. D. H. Lehmer, "An extended theory of Lucas’ functions", Annals of Mathematics, 2nd Ser.,31 (1930), pp. 419–448; online (paywall).
  2. I like this short blog entry, written by a Mersenne prime hunter, on using the Lucas–Lehmer test. There is a good, detailed description, with Python code, by Connor Krill for The Aperiodical. On the subject of hunting for primes, the Smithsonian's National Museum of American History holds a set of Factor Stencils by Derrick N. Lehmer which he used in the factoring of large numbers. Terence Tao has an excellent blog article on Lucas–Lehmer.
  3. There is a good entry here by 3010tangents about how to prove Lucas–Lehmer.
  4. For an elementary proof of Lucas–Lehmer see J. W. Bruce, "A really trivial proof of the Lucas–Lehmer test", Amer. Math. Monthly, 100 (1993), 370–371; online.

Theorem no. 128: The Euclid–Euler Theorem
  1. The MacTutor archive offers a good page on the history of perfect numbers. Fermat's contribution is discussed in Colin R. Fletcher, "A reconstruction of the Frenicle-Fermat correspondence of 1640", Historia Mathematica, Vol. 18, Issue 4, 1991, pp. 344–351; online.
  2. I found this wonderfully open-ended exploration of perfect numbers (1.8MB pdf) by Oliver Knill (one of many he has posted). Patrick Honner, for Quanta magazine, has this.
  3. Article no. 1 at John Voight's site is a nice exploration of odd perfect number properties.The current lower bound on odd perfect numbers appears to be Pascal Ochem and Michaël Rao,"Odd perfect numbers are greater than \(10^{1500}\)", Mathematics of Computation, Vol. 81, Issue 279, 2012, 1869–1877; online. They have raised this subsequently to \(10^{2000}\) according to this Quanta article by Steve Nadis, which gives a good account of recent (September 2020) research.
  4. A charming fact about perfect numbers is that they are harmonic divisor numbers, a fact proved by Øystein Ore in 1948. Indeed, Ore conjectured that all such numbers are even which would imply the non-existence of odd perfect numbers. However, number theorist Joshu Zelinsky offered this advice (quoted from Twitter, August 2023):

    "Unfortunately, we have almost no good ideas about how to prove there are no odd perfect numbers, we have even fewer good ideas about odd harmonic divisor problem. Many of the bounds we can prove about OPNs seem to break down completely for harmonics. E.g. For OPNs, we can at least prove Ochem-Rao bounds (linear inequalities relating # of distinct prime factors of n to total number of prime factors). No non-trivial bounds known of this sort for odd harmonic numbers."

    Zelinsky's website contains links to several articles from the front-line of OPN research.
  5. An elegant blog entry by Mike Spivey explains why no even perfect number can be the hypotenuse of a Pythagorean triple whereas any odd perfect number must be. More details may be found in the response of Roman Andronov to this Quora question.
  6. For French readers this from the Université Laval is very good (300KB pdf, May 2024).

Theorem no. 129: The Ollerenshaw–Brée Formula
  1. The entry for Ollerenshaw at Agnes Scott is good on technical details of her work on magic squares and cites her main publications in the area (starting 1986). The definitive source for this theorem is her book with David Brée, which is the futher reading link on the theorem page.
  2. Evelyn Lamb wrote this very nice tribute to Ollerenshaw on the occasion of her 100th birthday. Another tribute to Ollerenshaw's work on magic squares was carried by the Royal Society Publishing Blog
  3. A depiction of a most-perfect magic square dating from the 10th century is found in the Jain temple Parshvanatha at Khajuraho in Madhya Pradesh.
  4. A good source of information on magic squares generally is the website of Francis Gaspalou.
  5. Although the number of all magic squares of order 6 is unknown there is a serious attempt documented - see the link at the relevant OEIS entry.

Theorem no. 130: A Theorem on Apollonian Circle Packings
  1. The papers that inspired this theorem description can be located on Ron Graham's website dated 2003–2006. The theorem as given is from R. L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks, and C.H. Yan, "Apollonian circle packings: number theory", J. Number Theory, Vol. 100, no. 1, 2003, 1–45; online.
  2. Jerzy Kocik gives another treatment of specifying all integral Apollonian circle packings in this preprint "On a Diophantine equation that generates all integral Apollonian Gaskets".
  3. The numbers of circles in an Apollonian packing up to a given curvature is an active subject of investigation. See Alex Kontorovich and Hee Oh, "Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds", J. Amer. Math. Soc., Vol. 24, No. 3, 2011, pp. 603–648; online (paywall; arxiv preprint). (This work has been taken further by Alex Kontorovich and Christopher Lutsko.)
  4. There is a very far-reaching account of Apollonian circle packings in the PhD dissertation of Elena Fuchs which can be found here.
  5. A deep conjecture by Graham et al proposes that, subject to 'local' conditions restricting curvatures to certain congruence classes mod 24, a primitive Apollonian circle packing will be 'global' in the sense that it will exhibit any sufficiently large integer as a curvature. This so-called Local-Global conjecture seemed quite robust: indeed, Jean Bourgain and Alex Kontorovich proved in 2014 that the number of missing curvatures modulo any permitted congruency class, if not finite, was at least tightly constrained. However in July 2023 Summer Haag, Clyde Kertzer, James Rickards, Katherine E. Stange posted a comprehensive refutation of the conjecture. A fine write-up for Quanta by Max G. Levy swiftly followed.

Theorem no. 131: The Existence Theorem for Orthogonal Diagonal Latin Square
  1. Original source for this theorem (as cited in its illustration) is John Wesley Brown, Fred Cherry, Lee Most, Mel Most, E.T. Parker and W.D. Wallis, "Complete of the spectrum of orthogonal diagonal latin squares" in Rolf S. Rees (ed.), Graphs, Matrices, and Designs, Routledge, 1992, pp. 43–49.
  2. The story of Shrikhande,Bose and Parker disproving Euler's conjecture on orthogonal Latin squares is told by Nithyanand Rao in this tribute to Shrikhande for The Wire.

Theorem no. 132: Theaetetus' Theorem on the Platonic Solids
  1. A nice account of the proof of this theorem using Euler's Polyhedral Formula is given by John D. Cook here.
  2. I recommend this intriguing discussion by Pat Ballew of which of the Platonic solids is 'most spherical'. Rather in the same vein, a picture on twitter (14 Jan 2021) from @KangarooPhysics shows the solids wearing circular belts around their 'waists'.
  3. The artist Conrad Shawcross has produced a series of ten scultures based on the Platonic solids: see Perimeter Studies at (thanks to Angela Mihai for this; she has a photograph offering comments by the artist a pdf copy of which I found here).
  4. This theorem is the choice of Justin Curry in Episode 8 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.

Theorem no. 133: The Total Probability Theorem
  1. In The Doctrine of Chances: Probabilistic Aspects of Gambling, Springer-Verlag, 2010, Stewart N. Ethier writes (p. 68) "The conditioning law (... often called the law of total probability) was used without comment by Montmort and De Moivre. It was formalized in the derivation of Bayes's law."
  2. An excellent and much more thorough application of probability theory to the deuce rule of tennis is provided here by Chalkdust magazine.

Theorem no. 134: The Change of Variables Theorem
  1. The Victor Katz article referred to on the theorem page is Victor J. Katz, "Change of variables in multiple integrals: Euler to Cartan", Mathematics Magazine, Vol. 55, Issue 1, 1982, pp. 3– 11; online (paywall).
  2. The Lord Kelvin integral in our illustration is the subject of a very nice contribution by Nate Eldredge to a MathOverflow query "What do named "tricks" share?".

Theorem no. 135: Praeger's Theorem on Bounded Movement
  1. Original source for this theorem: Cheryl E. Praeger, "On Permutation Groups with Bounded Movement", Journal of Algebra, Vol. 144, Issue 2, 1991, pp. 436–442; online.
  2. The idea of movement is put into context as a 'Cayley metric' in this Cameron blog entry (note he talks of the movement of individual permutations, as opposed to the movement of the group action).

Theorem no. 136: Theorems of Euler and Rényi on e
  1. Original sources for this theorem:
    1. Euler published his result in "Calcul de la probabilité dans le jeu de rencontre", Mémoires de l'Académie des Sciences de Berlin, année 1751, 7 (1753) pp. 255–270; online. It is unfair to call it a 'theorem of Euler' since De Montmort had already derived it in 1713, although without proof. The rich history of the problem of derangements is admirably charted in Lajos Takács, "The problem of coincidences", Archive for History of Exact Sciences, Vol. 21, Issue 3, 1980, pp. 229–244; online (paywall; there was a copy here in November 2023).
    2. A. Rényi, "Some remarks on the theory of trees", Publ. Math. Inst. Hungar. Acad. Sri., 4 (1959), pp. 73–85. I cannot find an online copy but there are more details including a proof of Rényi's formula in Lajos Takács, "On Cayley's formula for counting forests", Journal of Combinatorial Theory, Series A, Vol. 53, Issue 2, 1990, pp. 321–323; online.
  2. The theory of permutation groups is naturally a rich source of results on derangements, beginning perhaps with Jordan's 1873 theorem that a transitive group always has one. Peter Cameron's blog has much on the topic (type 'derang' into the search box).

Theorem no. 137: Wallis's Product
  1. There are two superb articles on Wallis's work by Jacqueline A. Stedall in the September 2000 issue of the Royal Society's Notes and Records. The quote on the theorem page is from the first of these. They are paywalled but seem to be made open-access from time to time, so worth checking.
  2. There are closely related product formulae for other constants such as \(e\). The most elegant is perhaps that of Nicholas Pippenger, "An infinite product for e", American Mathematical Monthly, vol. 87, no. 5, 1980, p. 391; online (paywall). See also this preprint by Jonathan Sondow and Huang Yi. By the way there is a valuable memorial tribute page to Jonathan Sondow which has much else regarding formulae for constants.

Theorem no. 138: Vaughan Pratt's Theorem
  1. Original source for this theorem: Vaughan R. Pratt, "Every prime has a succinct certificate", SIAM J. Comput., Vo. 4, No. 3, 1975, pp. 214–220; online (paywall; a scanned copy was here, December 2023).
  2. Since polynomial-time solvable problems are automatically in NP this theorem is, since 2002, a corollary of the well-known algorithm of Agrawal, Kayal and Saxena (see also the web-link recommended on the theorem page).
  3. There is a good account of Pratt certification, with some Python code, by John D. Cook here.

Theorem no. 139: Strassen's Matrix Theorem
  1. Original source for this theorem: Strassen, Volker, "Gaussian elimination is not optimal", Numer. Math., 13, 1969, pp. 354–356; online (paywall; a facsimile is available from Göttinger Digitaisierungszentrumdf).
  2. Virginia Williams' account of her reduction in ω can be found in overview and technical versions at her website. There is a very good account of her work at Gödel's Lost Letter by Richard Lipton. Until October 2020, the race for the bottom for ω was between Williams and François Le Gall, who also has an article with Florent Urrutia on non-square matrix multiplication; in October 2020 a short lead was taken by Williams, as well-described by Kevin Hartnett for Quanta Magazine. There is a short discussion about the problem at cstheory.stackexchange.
  3. An interesting discussion by Bill Gasarch on limitations of current approaches to getting ω = 2 + ε can be found at Lance Fortnow' & Bill Gasarch's comutational complexity blog.
  4. The complexity of matrix multiplication is inimately connected with some conjectures in extremal set theory as discussed by Gil Kalai.
  5. Conventional wisdom says that Strassen is only advantageous for large matrices but this is challenged in Jianyu Huang, Tyler M. Smith, Greg M. Henry and Robert A. van de Geijn, "Strassen's Algorithm Reloaded", Proc. The International Conference for High Performance Computing, Networking, Storage and Analysis (SC16), Salt Lake City, UT, November 2016. A preprint is to be found on Huang's webpage.
  6. Paolo D’Alberto has a whole blog devoted to technical issues of fast matrix multiplication.
  7. Veit Elser has this intruiging machine learning preprint (2016): "A network that learns Strassen multiplication". Subsequently AI company Deepmind discovered some new Strassen-type algorithms. Ben Brubaker wrote this good explanation for Quanta.

Theorem no. 140: A Theorem on Rectangular Tensegrities
  1. Original sources for this theorem:
    1. Bolker, E. D. and Crapo, H., "How to brace a one-story building", Environment and Planning B: Planning and Design, Vol 4, Issue 2, 1977, pp. 125–152; online (paywall).
    2. Jenny A. Baglivo and Jack E. Graver, Incidence and Symmetry in Design and Architecture, Cambridge University Press, 1983, Chapter 3, Section 1.
  2. The weblink from the theorem page is to a chapter contributed by Bob Connelly to the delightful book Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, edited by Marjorie Senechal.
  3. This post by Dave Richeson gives a nice introduction to the subject of graph theory and rigidity. He posted a more 'glamorous' (lots of 3D printing!) version as part of Aperiodical's The Big Internet Math-Off 2024.
  4. Joseph Malkevitch has a valuable bibliography (up to 2001) of rigidity publications in which the work cited in this theorem description can be located.

Theorem no. 141: The Piff–Welsh Theorem
  1. Original source for this theorem: M. J. Piff and D. J. A. Welsh, "On the vector representation of matroids", J. London Math.Soc., Vol. 2, no. 2, 1970, pp. 284–288; online (paywall).
  2. A fast algorithm for constructing representations of transversal matroids is given in Rekab-Eslami, M., Esmaeili, M. & Gulliver, T.A., "A fast algorithm to construct a representation for transversal matroids", Japan J. Indust. Appl. Math., 33, 2016, pp. 207–226; online (paywall).
  3. An excellent online introduction to matroid theory is provided by Joseph Malkevitch here.

Theorem no. 142: Sylvester's Law of Inertia
  1. Original source for this theorem: J.J. Sylvester, "A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares", Philosophical Magazine IV, 1852, pp. 138–142; online (facsimile from the Hathi Trust digital library, there is a pdf via the recommended webpage for this theorem:

Theorem no. 143: The Robinson–Schensted–Knuth Correspondence
  1. Original sources for this theorem:
    1. G. de B. Robinson, "On the representations of the symmetric group", Amer. J. Math., 60 (3), 1938; online (paywall; there was a copy here, December 2023). Robinson adds a part II and a Part III to this paper, in vol. 69 (2), 1947, and in vol. 70 (2), 1948, but I think only part I is relevant here, although Knuth's investigations (see below) overlapped with part II.
    2. Schensted, C., "Longest increasing and decreasing subsequences", Canadian Journal of Mathematics, 13, 1961, pp. 179–191; online
    3. D.E. Knuth, "Permutations, matrices, and generalized Young tableaux", Pacific J. Math., Vol. 34, Number 3 (1970), 709–727; online.

Theorem no. 144: Lieb's Square Ice Theorem
  1. Elliott Lieb's original paper is "Residual Entropy of Square Ice", The Physical Review, vol. 162, no. 1, 1967, pp 162–172. Even after 50 years you still have to pay to read it online but the first two pages are displayed here. (For Russian readers it is translated online here.) This paper, by the way, figures in the citation for Lieb's 2023 Kyoto prize.
  2. I originally gave as a web link from the theorem page a very nice but technical article by Stefan Felsner, Florian Zickfeld: "On the number of planar orientations with prescribed degrees", Electronic Journal of Combinatorics, vol. 15, 2008; online. This deals with orientations of planar graphs in much more generality — Lieb's result appears in section 2.2.
  3. It has been discovered that water can form into square ice at room temperature by confining it using layers of graphene.

Theorem no. 145: The Contraction Mapping Theorem
  1. Original sources for this theorem:
    1. Banach, Stefan, "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales", Fundamenta Mathematicae, Vol. 3, Issue 1, 1922, pp. 133–181; online
    2. The Wiki page for this theorem says it was discovered independently by Renato Caccioppoli. However, I can't find any corroboration of this (the Wiki citation is to Caccioppoli's MacTutor Archive entry which doesn't mention this theorem).
  2. has a valuable collection of counterexamples showing that all the hypotheses of this theorem are needed.
  3. This theorem is the choice of Vidit Nanda in Episode 24 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast.
  4. John D. Cook has an intriguing blog entry on Kepler's use of Banach's theorem three centuries before it was discovered!

Theorem no. 146: The Panarboreal Formula
  1. Original sources for this theorem: the theorem as given is from F. R. K. Chung and R. L. Graham, "On universal graphs for spanning trees", J. Lond. Math. Soc., Vol. s2-27, Issue 2, 1983, pp. 203–211; online (paywall; 280KB pdf, April 2021) but is first mentioned in their 1979 paper "On universal graphs", Annals of the New York Academy of Sciences, 319 (1979), 136–140; online (paywall; preprint, February 2024).The \(\frac12n\log n\) lower bound on the size of a 'universal graph' is proved as Theorem 1 in F.R.K. Chung and R.L. Graham, "On graphs which contain all small trees", J. Combinat. Theory B, vol. 24, issue 1, 1978, pp 14–23; online.
  2. Some more work on panarboreal graphs and related issues is given in section 6.8 of "Spanning trees – A survey" by Kenta Ozeki and Tomoki Yamashita, 2010. A good source for recent research on universal graphs is Daniel Johannsen, Michael Krivelevich, Wojciech Samotij, "Expanders are universal for the class of all spanning trees", Combinatorics, Probability and Computing, Vol. 22, Issue 2, 2013, pp. 253–281; online (paywall; arxiv).
  3. The sequence of sizes of panarboreal graphs starts 0, 1, 2, 4, 6, 8, 11, 13, 16, 18, and is OEIS sequence A004401. This is the number of edges a graph needs to have in order to contain all \(n\)-vertex trees. This is possibly smaller than for the question asked in our presentation of Chung and Graham's theorem since they ask for the number of edges when we insist the graph must also have \(n\) vertices. The values are perhaps the same, as pointed out in the OEIS entry.

Theorem no. 147: The Sophomore's Dream
  1. Original source for this theorem: Johannis Bernoulli, "Demonstratio methodi analyticæ", 1679, in Opera Omnia, Vol. 3, pp. 376–381 (in 1742 edition); online.
  2. The decimal expansion of \(\int_0^1 x^x dx\) and related material can be found at; \(\int_0^1 x^{-x} dx\) is sequence A073009.

Theorem no. 148: A Theorem of Schur on Real-Rootedness
  1. Original source for this theorem: J. Schur, "Zwei Sätze über algebraische Gleichungen mit lauter reellen Wurzeln", J. Reine Angew. Math., 144, 1914, pp. 75–88; online (paywall; a facsimile is provieded at Göttinger Digitaisierungszentrum). Schur's use of the initial "J" is commented upon in his wiki entry. The result of Ernest Malo is in "Note sur équations algébriques dont toutes les racines sont réelles", Journal de Mathématiques spéciales, (ser.4), t. 4, 1895, p. 7–10 (I don't find this online, even for ready money).

Theorem no. 149: Euclid's Triangular Prism
  1. The original weblink for this theorem page was to Richard Fitzpatrick's site which links to a dual-language complete Elements, nearly 5MB but a truly definitive web resource. This has been replaced on the page, just because it is more immediately accessible, by a link to David E. Joyce's page for Euclid 12.7.

Theorem no. 150: Woodall's Hopping Lemma
  1. Original source for this theorem is: D.R. Woodall, "The binding number of a graph and its Anderson number", J. Combinatorial Theory, Series B, Vol. 15, Issue 3, December 1973, pp. 225–255; online. The theorem in question is Lemma 12.3.
  2. There is a good chapter on the Hopping Lemma in this 1995 LSE PhD dissertation of Sarah Jane Goodall.
  3. This arxiv preprint of Jan Kessler and Jens M. Schmidt seems of interest, offering "a polyhedral relative of Woodall's Hopping Lemma that allows cycle extensions through common neighbors of cycle vertex pairs even when none of these pairs have distance two in C". (See also this overview).

Theorem no. 151: The Small Prime Gaps Theorem
  1. Original source for this theorem: Daniel A. Goldston, János Pintz, Cem Yalçıl Yıldırım, "Primes in tuples, I", Annals of Mathematics, Vol. 170, Issue 2, 2009, pp. 819–86; online. Three further papers in the series extract further implications from the same methods: "Primes in tuples, II", Acta Mathematica, Vol. 204, Issue 1, 2010, pp. 1–47; online. "Primes in tuples III: On the difference {pn+?-pn}", Funct. Approx. Comment. Math., 35, 2006, pp. 79–89; online; "Primes in tuples IV: Density of small gaps between consecutive primes", Acta Arithmetica, 160 (1), 2013, pp. 37–53; online.
  2. For the sake of contrasting this theorem with subsequent proofs that \(p_{n+1}-p_n\leq c\) infinitely often for a constant \(c\), its conclusion may be replaced by \(p_{n+1}-p_n\leq (\log p_n)^{1/2+\epsilon}\). The progress towards current knowledge is beautifully described by Terence Tao in this youtube lecture and, for a more general audience, this lecture by Vicky Neale.
  3. The background to Zhang's proof of \(p_{n+1}-p_n\leq c\) and subsequent improvements by Maynard and Tao are described by John Friedlander in "Prime Numbers: A Much Needed Gap Is Finally Found", Notices of the AMS, Vol. 62, No. 6, June/July 2015, 660–664, online here. A more technical overview is given by Andrew Granville, "Primes in intervals of bounded length", Bull. Amer. Math. Soc., 52 (2015), 171–222, online here.

Theorem no. 152: De Moivre's Theorem
  1. The historical context of De Moivre's theorem is described in David R. Bellhouse and Christian Genest, "Maty’s Biography of Abraham De Moivre,Translated, Annotated and Augmented", Statistical Science, Vol. 22, No. 1, 2007, pp. 109–136; online. See, in particular, footnote 46 on p. 118.
  2. Here is a cute demonstration, using De Moivre, that the series \(\cos(1),\cos(2),\cos(3),\ldots\) does not converge.

Theorem no. 153: Euler's Partition Identity
  1. Original source for this theorem: Leonhard Euler, Introductio in Analysin Infinitorum, Vol. 1, 1748, Chapter 16, Section 326. A translation into English of the whole work (and Vol. 2) has been provided by Ian Bruce at his website, where the original Latin may also be found, in pdf. The 'official' source for the work is here at the Euler Archive.
  2. D. H. Lehmer has provided a valuable explanation of two generalisations of Euler's identity, Glaisher's Theorem and Roger's Theorem in "Two nonexistence theorems on partitions", Bull. Amer. Math. Soc., Vol. 52, Number 6,1946, pp. 538–544; online.

Theorem no. 154: The Remainder Theorem
  1. Colin Beveridge has a good intuitive introduction to the remainder and factor theorems at FlyingColoursMaths.
  2. Not sure how generally accessible this but a very nice twitter thread by Patrick Honner on discovering tangent lines via polynomial division.

Theorem no. 155: A Tripartite Turán Theorem
  1. The source for this theorem is Adrian Bondy, Jian Shen, Stéphan Thomassé and Carsten Thomassen, "Density Conditions For Triangles In Multipartite Graphs", Combinatorica, Vol. 26, Issue 2, 2006, pp 121–131; online (paywall); preprint.

Theorem no. 156: The Lecture Hall Partition Theorem
  1. The source for this theorem is Bousquet-Mélou, M., Eriksson, K., Lecture Hall Partitions. The Ramanujan Journal 1, 101–111 (1997); online (paywall; there is a dvi file at Bousquet-Mélou's website).

Theorem no. 157: The Transversal Matroid Theorem
  1. Original ources for this theorem:
    1. Edmonds, J; Fulkerson, D.R., "Transversals and matroid partition", Journal of Research of the National Bureau of Standards, Section B, vol 69, issue 3, 1965, pp. 147–153; online;
    2. Mirsky, L. and Perfect, H., "Applications of the notion of independence to problems of combinatorial analysis", J. Combinatorial Theory, vol. 2, issue 3, 1967, pp. 327–357; online.

Theorem no. 158: The Albert–Brauer–Hasse–Noether Main Theorem
  1. Original source for this theorem: Brauer, R., Hasse, H. and Noether, E., "Beweis eines Hauptsatzes in der Theorie der Algebren", J. Reine Angew. Math., Vol. 167, 1932, pp. 399–404; online (paywall; facsimile). Albert's contributions, and those of Käte Hey, are discussed extensively in Peter Roquette's article "The Brauer-Hasse-Noether theorem in historical perspective" (subsequently published as a monograph of the same name) which can be found online here (April 2024).
  2. The restriction of the Main theorem to number fields is essential: not every finite-dimensional division algebra is a cyclic algebra. A counter-example, due to A. Adrian Albert, is described on page 57 of Lewis's article (the weblink on the theorem page).

Theorem no. 159: The McIver–Neumann Half-n Bound
  1. Original source for this theorem: A. McIver and P. M. Neumann, "Enumerating finite groups", Quart. J. Math. Vol. 38, Issue 4, 1987, pp. 473–488; online (paywall). Some background is given via Peter Cameron's blog.
  2. The Frobenius groups, of which \(F_{20}\) is used to illustrate this theorem page, has a nice description by Emmanuel Amiot here. \(F_{20}\) makes another apparence illustrating the third isomorphism theorem.

Theorem no. 160: The Classification of Archimedean 4-Polytopes
  1. Tony Phillips' review (currently only available via an 18MB full-issue download here) of Tony Robbin's Shadows of Reality, Yale University Press, 2006, contains much fascinating material on 4-dimensional solids. Also recommended is Snezana Lawrence, "Life, architecture, mathematics, and the fourth dimension", Nexus Network Journal, 17, 587–604 (2015); online.
  2. The Historia Mathematica article by Irene Polo-Blanco is chapter 5 of her excellent 2007 University of Groningen thesis Theory and History of Geometric Models which may be found online here.
  3. If only Alicia Boole Stott could have tried the virtual reality game Hypernom!

Theorem no. 161: Quadratic Nonresidue is Zero-Knowledge Provable
  1. Original source for this theorem: Goldwasser, S., Micali, S. and Rackoff, C., "The knowledge complexity of interactive proof systems", STOC '85: Proceedings of the seventeenth annual ACM symposium on Theory of computing, December 1985, pp. 291–304; online (paywall). This is an extended abstract; the full paper appeared, under the same title, in SIAM Journal on Computing, Vol. 18, No. 1, 1989, pp. 186–208; online (paywall). Downloadable pdf versions can be found online, e.g. via the Wiki page on Zero-knowledge proofs, which is good on the origins of the paradigm.
  2. There is at least one real-life illustration of zero-knowledge proofs in the field of nuclear disarmament. A follow up.
  3. Jeremy Kun has a fine series of blog posts on zero-knowledge proofs. Follow from here.
  4. There is a well-known presentation of the zero-knowledge paradigm: Quisquater JJ. et al. (1990) "How to explain zero-knowledge protocols to your children. In: Brassard G. (ed.) Advances in Cryptology — CRYPTO’ 89 Proceedings. CRYPTO 1989. Lecture Notes in Computer Science, vol 435. Springer, New York, NY; online (paywall); facsimile. The 'et al.' in the citation represents "Myriam Quisquater, Muriel Quisquater, Michaël Quisquater, Louis Guillou, Marie Annick Guillou, Gaïd Guillou, Anna Guillou, Gwenolé Guillou, Soazig Guillou" who I presume include the 'children' (of Jean-Jacques Quisquater and Louis Guillou). Tom Berson is credited with the English version.

Theorem no. 162: Heath’s Finitely Discontinuous Function Theorem
  1. Original source for this theorem is Jo Heath, "k-to-1 functions between graphs with finitely many discontinuities", Proc. AMS, vol. 103, no. 2, June 1988, pp. 661–666; online. The term 'wiggle' for limit constructions of continuous functions appears in print in, for example, John Baptist Gauci, Anthony J. W. Hilton and Dudley Stark, "Wiggles and finitely discontinuous k-to-1 functions between graphs", J. Graph Theory, Vol. 74, Issue 3, 2013, pp. 275–308; online (paywall); but Anthony Hilton's use of the term dates at least as far back as 2008 when he talked about wiggles to the Combinatorics Study Group at Queen Mary University of London (see 12 December).

Theorem no. 163: The Friendship Theorem
  1. Original source for this theorem: Erdős, Paul, Rényi, Alfréd; Sós, Vera T., "On a problem of graph theory", Studia Sci. Math. Hungar., Vol. 1, 1966, pp. 215–235; online (1.4MB pdf download).
  2. A useful little entry at The Futility Closet links to an elementary (purely graph-theoretic) proof of this theorem by Judith Longyear and Torrence Parsons.
  3. The infinite graph constructed on the theorem page in which each pair of distinct vertices has a unique common neighbour is an example of a friendship graph. The finite friendship graphs are the windmill graphs shown in the illustration on the theorem page. Infinite friendship graphs are well-studied, dating back at least to V. Chvátal and A. Kotzig, "On countable friendship graphs", Publications du CRM-415, May 1974. I don't find this online but a follow-up paper is: Václav Chvátal, Anton Kotzig, Ivo G. Rosenberg and Roy O. Davies, "There are \(2^{\aleph_{\alpha}}\) friendship graphs of cardinality \({\aleph_{\alpha}}\)", Canad. Math. Bull., Vol. 19, No. 4, 1976, pp. 431–433; online. The iterative construction shown on the theorem page is from this paper (and originates, I believe, in Chvátal and A. Kotzig, 1974). I collected some more information in a presentation here (1.7MB pdf)
  4. Further insights into friendship graphs are given in A. Kotzig, "Degrees of Vertices in a Friendship Graph", Canad. Math. Bull., Vol. 18, No. 5, 1975, pp. 691–693; online. In particular, we learn that they are uniquely decomposable into triangles (clearly the case for windmill graphs); that every vertex and its neighbourhood induces a (possibly infinite) collection of triangles; and that in an infinite friendship graph has infinite degree.

Theorem no. 164: The Diaconis–Holmes–Montgomery Coin-Tossing Theorem
  1. Original source for this theorem is Persi Diaconis, Susan Holmes, and Richard Montgomery, "Dynamical bias in the coin toss", SIAM Review, 2007, Vol. 49, No. 2 : pp. 211–235; online (paywall). A preprint is available here (5MB pdf file); my illustration of the theorem is partly based on figures 3 and 4 of this preprint.
  2. A large scale human trial confirming the coin tossing bias has been carried out by Frantisek Bartos, preprint here.

Theorem no. 165: Lin McMullin's Theorem
  1. Original sources for this theorem are L. McMullin, A. Weeks, "The golden ratio and fourth degree polynomials, On-Math, Winter 2004-05, Vol., Number 2; and McMullin, L., "How I found the golden ratio on my CAS", The North Carolina Association of Advanced Placement Mathematics Teachers Newsletter, 13 (1) (Winter 2005) pp. 6–7. Neither article appears easy to track down now. The theorem appears to have been discovered before, by Herman Theodor Rendtorff Aude of Colgate University: HTR Aude, "Notes on Quartic curves", The American Mathematical Monthly, Vol. 56, Issue 3, 1949, pp. 165–170; online (paywall); see also, Reinert A. Rinvold, "Fourth degree polynomials and the golden ratio", The Mathematical Gazette, Vol. 93, Issue 527, July 2009, pp. 292–295; online (paywall).

Theorem no. 166: Haken's Unknot Theorem
  1. Original source for this theorem: Wolfgang Haken, "Theorie der Normalflächen: Ein Isotopiekriterium für den Kreisknoten", Acta Math., Vol. 105, Number 3-4 (1961), pp. 245–375; online.
  2. For more on the complexity of determining unknottedness there is a wiki page on the subject. Notably, a quasi-polynomial-time (i.e. \(n^{O(\log n))}\)) algorithm was announced in February 2021 by Marc Lackenby. There is more (May 2021) from Gil Kalai here. Check this entry (posted August 2021) for updates.
  3. For details of the implementation of recognition algorithms for the unknot, notably that of Joan Birman and Michael Hirsch, see Joan S. Birman, Marta Rampichini, Paolo Boldi and Sebastiano Vigna, "Towards an implementation of the B-H algorithm for recognizing the unknot", J. Knot Theory and its Ramifications, vol. 11, no. 4, 2002, pp.601–645 ; online (paywall; reprint ,June 2024). For more on Birman's work in low-dimensional topology generally see her MacTutor entry.
  4. Thistlethwaite's example unknot illustrating this theorem page features in an interesting collection of 'hard' unknots in this preprint by Benjamin A. Burton et al, which also provides a good introduction to unknotting.
  5. Knot theory deals with embeddings of the 1-sphere \(S^1\) (a closed curve) into 1 + 2 = 3 dimensions. More generally we can embed the n-sphere \(S^n\)into n + 2 dimensions. Thus \(S^2\), a hollow sphere, is embedded into 4 dimensional space. And we can ask about the decision problem: is our embedding the 'unknot'? For \(n\geq 3\) the answer is that the question is undecidable: this was proved in 1996 by Alexander Nabutovsky and Shmuel Weinberger. However, decidability in the case \(n=2\) remains an open problem. A wonderful discussion of these issues is given here by Bjorn Poonen.

Theorem no. 167: The Lindemann–Weierstrass Theorem
  1. Original sources for this theorem:
    1. C. Hermite, "Sur la fonction exponentielle", C. R. Acad. Sci. Paris, tome 77, 1873, pp. 18–24; 74–79; 226–233; 285–293; online.
    2. F. Lindemann, "Über die Zahl π", Math. Ann., Vol. 20, 1882, pp. 213–225; online.
    3. Weierstrass, K., "Zu Lindemann's Abhandlung. "Über die Ludolph'sche Zahl".", Sitzungsberichte der Königlich Preussischen Akademie der Wissen-schaften zu Berlin, 5, 1885, pp. 1067–1085; online.
  2. The proof of the transcendence of \(e\), Charles Hermite's breakthrough 1873 result, is very clearly described here.
  3. The Dottie number has its own Wikipedia page.

Theorem no. 168: The Max-Flow Min-Cut Theorem
  1. Original sources for this theorem:
    1. Ford, L. R. and Fulkerson, D. R., "Maximal flow through a network", Canadian Journal of Mathematics, Vol. 8, 1956, pp. 399–404; online; previously appeared as RAND report P-605; online (paywall)
    2. P. Elias, A. Feinstein and C. Shannon, "A note on the maximum flow through a network", IRE Transactions on Information Theory, Vol. 2, Issue 4, 1956, pp. 117–119; online (paywall). A facsimile is available via semanticscholar.
    3. The Elias et al paper cites a third source: G. B. Dantzig and D. R. Fulkerson, "On the Max-Flow Min-Cut Theorem of Networks", in "Linear Inequalities", Ann. Math. Studies, no. 38, Princeton, New Jersey, 1956, which I presume to be a reincarnation of a RAND report of the same name: P-826, 1955; onine (paywall); the purport seems to be that Dantzig-Fulkerson is the first 'constructive' (algorithmic) proof of the Max-flow Min-cut theorem.
  2. There is more on the history of this theorem at Whitty, R. W. "Some comments on multiple discovery in mathematics", Journal of Humanistic Mathematics, Volume 7 Issue 1(January 2017), pp. 172–188; online (see top of page 179).
  3. In "Some comments" there is an attribution to Anton Kotzig, which was repeated in earlier versions of this theorem page: "restricted as in our example to integer capacities, by A. Kotzig". I can no longer locate the source for this attribution; it is specific enough that I must have read it somewhere but I would expect such research to be mentioned in such an authoritative tribute as Jaromír Abrham, Alexander Rosa, Gert Sabidussi and Jean M. Turgeon, "Anton Kotzig 1919–1991", Mathematica Slovaca, Vol. 42, No. 3, 1992, pp. 381–383; online.
  4. There are probabilistic algorithms for Max-flow which are much faster than Ford-Fulkerson. See this comprehensive Quanta article by Erica Klarreich on this arxiv post by Li Chen et al and its antecedents. There is some interesting discussion in the comments following this Scott Aaronson post.

Theorem no. 169: Sokal's Theorem on Chromatic Roots
  1. The original source for this theorem is Alan D. Sokal, "Chromatic roots are dense in the whole complex plane", Combinatorics, Probability and Computing, Vol. 13, Issue 2, March 2004 , pp. 221–261; online (paywalled); preprint. Some interesting subsequent work by Adam Bohn is reported in "A dense set of chromatic roots which is closed under multiplication by positive integers", Discrete Mathematics, Vol. 321, 28 April 2014, Pages 45–52; online.
  2. Roots of chromatic polynomials on the real line have received much attention, the famous result of Jackson being that there are none in the interval \((1,32/27]\), this being tight in the sense that graphs with roots arbitrarily close to \(32/27\) may be constructed. Bill Jackson, "A zero-free interval for chromatic polynomials of graphs", Combinatorics, Probability and Computing, Vol. 2, Issue 3, 2008, pp. 325–336; online (paywall). Thomas Perrett has shown that the interval can be extended to \(\approx 1.290\) for a certain class of graphs, again tight: "A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees", Discrete Mathematics, Vol. 339, Issue 11, 2016, pp. 2706–2714; online.

Theorem no. 170: Machin's Formula
  1. Machin's Formula has an alternative statement viz  \(\tau/8=4\cot^{-1}5-\cot^{-1}239\), thanks to the relationship between \(\cot^{-1} x\) and \(\tan^{-1}(1/x)\). This version is somewhat neater (although the inverse cotangent function is not directly available on calculators or in spreadsheets) and is preferred by some writers c.f. Pat's Blog.
  2. Another entry at Pat's Blog offers an interesting synopsis of the history of Machin's series.

Theorem no. 171: The BEST Theorem
  1. Original sources for this theorem:
    1. Tutte, W. T. and Smith, C. A. B., "On unicursal paths in a network of degree 4", American Mathematical Monthly, Vol. 48, Issue 4, 1941, pp. 233–237; online (paywall).
    2. van Aardenne-Ehrenfest, T. and de Bruijn, N. G., "Circuits and trees in oriented linear graphs", Simon Stevin: Wis- en Natuurkundig Tijdschrift, 28, 1951, pp. 203–217; online.
  2. The BEST theorem finds an application in probability theory in an interesting contribution to exchangeability of random variables: Ivan Bardet, Cécilia Lancien, Ion Nechita, "de Finetti reductions for partially exchangeable probability distributions"; online preprint. The application has its origins in a paper from 1984 of Arif Zaman: "Urn models of Markov exchangeability", Ann. Probab., Vol. 12, No. 1 (1984), 223–229; online. Thanks to Ion Nechita for alerting me to this.
  3. A further rich source of connections to knot theory and graph polynomials is Richard Arratia, Béla Bollobás and Gregory B.Sorkin, "The interlace polynomial of a graph", Journal of Combinatorial Theory, Series B, Vol. 92, Issue 2, 2004, pp. 199–233; online.

Theorem no. 172: The Dyson–Andrews–Garvan Crank
  1. Original sources for this theorem:
    1. F.J. Dyson, "Some guesses in the theory of partitions", Eureka, Vol. 8, 1944, pp. 10–15; online (the same edition has a elegant proof by Dyson of the Fundamental Theorem of Algebra).
    2. George E. Andrews and F. G. Garvan, "Dyson's crank of a partition", Bull. Amer. Math. Soc. (N.S.), Vol. 18, No. 2 (1988), 167–171; online.
  2. The table illustrating this theorem page now seems to me rather impenetrable! The colour blocks (in row-major order) correspond to partitions of 17 whose \(M\) count (number of 1s) is \(17,16,15,\ldots\, 1,0\). The lengths of the blocks are given by OEIS sequence A002865. The entries (crank values) are just \(-M \!\!\mod 11\) until the 3rd entry in row 2, which records the partition consisting of 8 1s and a 9. Here \(N=1\), so the crank is \(1-8 \!\!\mod 11=4\). The final 66 table entries correspond to partitions of 17 having zero 1s, with the crank value being the value of \(\lambda\). Partitions are arranged lexicographically, with the final few partitions being \((5\,6\,6),\, (5\,12),\,(6\,11),\,(7\,10),\,(8\,9),\,(17)\).
  3. There are several detailed references to Dyson's work on partitions in the AMS memorial tribute to him in the August 2021 issue of Notices; online.

Theorem no. 173: The Ramanujan Partition Congruences
  1. Original sources for this theorem:
    1. Ramanujan, S., "Some properties of p(n), the number of partitions of n", Proc. Cambridge Philosophical Society, 19, 1919, pp. 207–210; online (transcription at
    2. Ramanujan, S., "Congruence properties of partitions", Mathematische Zeitschrift, 9 (1–2), 1921, pp. 147–153;online (paywall; a transcription at (prepared from Ramanujan's manuscripts after his death by G.H. Hardy)
    3. Ono K., "Distribution of the partition function modulo m", Annals of Mathematics, Vol. 151, Issue 1, 2000, pp. 293–307; online (paywall; a pdf is here, December 2023).
  2. There is a very nice account of the \(p=5\) congruence ('Ramanujan's most beautiful identity') by Christian Krattenthaler in the June 2017 issue of the European Mathematical Society Newsletter. A direct link to the pdf file is here (2.8MB), with Krattenthaler's article beginning on p. 41 and the Ramanujan part beginning on p. 47.
  3. A good discussion "Congruence properties of the partition function" by Tony Forbes is here (pdf, 0.5MB download, 100 pages long but the last 90 pages form an appendix listing computer-generated identities and can be ignored by most readers, I imagine).

Theorem no. 174: The Cameron–Fon-Der-Flaass IBIS Theorem
  1. Original source for this theorem: P.J. Cameron and D.G. Fon-Der-Flaass, "Bases for permutation groups and matroids", European J. Comb., Vol. 16, Issue 6, 1995, pp. 537–544; online.
  2. Peter Cameron has a nice tribute post on his blog which describes the IBIS theorem. A further entry answers a question about sizes of irredundant bases.
  3. The original weblink from this theorem page was a fine article "Quantifying symmetry" by Jonathan A. Cohen, The Australian Mathematical Society Gazette, Vol. 32, Number 2, May 2005; online (whole issue download, 5MB pdf - issues may be browsed here). It offers a very good introduction to bases of permutation groups but doesn't mention IBIS groups which is why eventually I preferred to link to a talk, by Cameron, which does.

Theorem no. 175: The Bungers–Lehmer Theorem on Cyclotomic Coefficients
  1. Original sources for this theorem:
    1. A. Migotti, "Zur Theorie der Kreisteilungsgleichung", Sitzber. Math.-Naturwiss. Classe der Kaiser. Akad. der Wiss., 87, 1883, pp. 7–14; I don't find this source online. The result appears to have been discovered indendently by A. S. Bang, "Om Ligningen φn(x) = 0", Nyt tidsskrift for matematik, Vol. 6, Afdeling B, 1895, pp. 6–12; online (paywall). The paper is in Danish which I don't read; the attributed is in Marion Beiter, "The midterm coefficient of the cyclotomic polynomial Fpq(x)", The American Mathematical Monthly, Vol. 71, No. 7, 1964, pp. 769–770; online (paywall; 300KB pdf download, March 2021).
    2. Emma Lehmer, "On the magnitude of the coefficients of the cyclotomic polynomial", Bull. Amer. Math. Soc., 42 (6), 1936, pp. 389–392; online.
    3. Rolf Bungers proof conditional on the infinitude of twin primes is cited by Lehmer as appearing in his 1934 Göttingen dissertation, which I think is "Über die Koeffizienten von Kreisteilungspolynomen", which shows up, minimally, on google books.
    4. Jiro Susuki's proof that any integer is a cyclotomic coefficient is in "On coefficients of cyclotomic polynomials", Proc. Japan Acad. Ser. A Math. Sci., 63(7), 1987, pp. 279–280; online. The result is strengthened in Chun-Gang Ji and Wei-Ping Li, "Values of coefficients of cyclotomic polynomials, Discrete Mathematics, Vol. 308, Issue 23, 2008, pp. 5860–5863; online.
    5. A tight superpolynomial bound on the growth of maximum absolute values of coefficients was obtained in 1949 by Paul Erdős (lower bound) and Paul T. Bateman (upper bound). See Erdős, P., "On the growth of the cyclotomic polynomial in the interval (0,1)", Glasgow Math. J., Vol. 3, Issue 2, 1957, pp. 102–104; online.
  2. The famous proof of Wedderburn's Little Theorem by Ernst Witt is based on cyclotomic polynomials and is a great tangent to follow. See the recommended weblink on that page.
  3. An exciting new development in the study of cyclotomic coefficients is the discovery by Gregg Musiker and Victor Reiner of a topolocial interpretation. See this (250KB pdf), for example.

Theorem no. 176: The Existence Theorem for Bachelor Latin Squares
  1. Original sources for this theorem: Evans, A.B., "Latin squares without orthogonal mates", Designs, Codes and Cryptography, Vol. 40, 2006, pp. 121–130; online (paywall); and Wanless, I.M. and Webb, B.S., "The existence of latin squares without orthogonal mates", Designs, Codes and Cryptography, Vol. 40, 2006, pp. 131–135; online (paywall); reprint (February 2024).
  2. The weblink from the theorem page is to the arxiv posting of Ian Wanless's transversals survey, published as "Transversals in latin squares", Quasigroups and Related Systems, Vol. 15, No. 1, 2007, pp. 169–190; online. Although the arxiv posting is dated 2009 it appears to be identical to the published paper. Wanless gave an invited talk of the same name at the 23rd British Combinatorial Conference in 2011 and this extends and updates the 2007 publication: Surveys in Combinatorics 2011, Cambridge University Press, 2011, pp. 403–437; online (paywall; preprint, February 2024).
  3. Ryser's conjecture, that odd-order Latin squares have a complete transversal, has been resolved for large orders by Richard Montgomery. See this blog entry by Peter Cameron for a round-up of related conjectures.

Theorem no. 177: The Heine–Borel Theorem
  1. Original source for this theorem: Borel, Émile, "Sur quelques points de la théorie des fonctions", Annales Scientifiques de l'École Normale Supérieure, 3, 12, 1895, pp. 9–55; online. Nicole R. Andre, Susannah M. Engdahl and Adam E. Parker give a wonderful early history of this result in "An Analysis of the First Proofs of the Heine–Borel Theorem", Convergence, Vol. 9, 2012, online here. The Wiki page on compactness is also very good.
  2. The notion of compactness is given a very good intuitive treatment by Evelyn Lamb here; and Manya Raman-Sundstrom has A pedagogical history .

Theorem no. 178: Sendov's Conjecture
  1. Theorem under Construction iconThis is a 'theorem under construction': I hope to chart exciting developments here towards an eventual final version, which may or may not confirm Sendov's conjecture for polynomials of arbitrary degree.
  2. The proof of Sendov for degree 8 is: Johnny E.Brown and Guangping Xiang, "Proof of The Sendov Conjecture for Polynomials of Degree at Most Eight", Journal of Mathematical Analysis and Applications, Vol. 232, Issue 2, 1999, 272–292; online.
  3. Jérôme Dégot's proof of Sendov for high degree appeared in 2014 as "Sendov conjecture for high degree polynomials", Proc. AMS, vol. 142 (2014), 1337–1349. It is pay-to-view online but a preprint is here.
  4. There is a nice snapshot of Dégot's result at about p. 100 of this cornucopia (1.1MB pdf) by Pamela Gorkin.
  5. A paper by Zaizhao Meng on the arxiv claims a proof of Sendov for polynomials of degree 9. A paper by Dinesh Sharma Bhattarai claims a proof of Sendov for polynomials of degree 10. Also this and this claiming to prove the conjecture outright, but posting errors.
  6. Progress has been made by Robert Dalmasso for the case where the zeros of the polynomial are simple.
  7. Terence Tao has posted an unconditional proof of Sendov for high degree: "Sendov’s conjecture for sufficiently high degree polynomials", Acta Mathematica, Vol. 229, No. 2, 2022, pp. 347–392; online. The introduction has a more complete review than the above of recent work on the conjecture.
  8. The Gauss–Lucas Theorem, that the convex hull of the roots of a polynomial encloses the roots of its derivative is given a fascinating physics proof here on John Carlos Baez's blog.

Theorem no. 179: The Descartes Circle Theorem
  1. Some original sources:
    1. J. Steiner, "Einige geometrische Betrachtungen", J. reine Angew. Math., Vol. 1, 1826, pp. 161–184 continues 252–288; online (paywall; facsimile)
    2. R. Lachlan, "On systems of circles and spheres", Phil. Trans. Roy. Soc. London, Ser., Vol. A177, 1886, pp. 481–625; online
    3. T. Gossett, "The Hexlet", Nature, Vol. 139, 1937, pp. 251–252; online (paywall)
    4. The Wiki article on the theorem also gives a rediscovery in 1842 by Philip Beecroft.
  2. A proof 'from the book' of this theorem is given in Levrie, Paul, "A straightforward proof of Descartes's circle theorem", The Mathematical Intelligencer, 41:3 (2019), pp. 24–27; online.

Theorem no. 180: The Greibach Normal Form Theorem
  1. Original source for this theorem: Greibach, Sheila, "A new normal-form theorem for context-rree phrase structure grammars". Journal of the ACM., vol. 12 (1), 1965, pp. 42–52; online (paywall).
  2. The algorithm converting \(G\) to Greibach Normal Form on \(O(|G|^4)\) symbols is given in Norbert Blum and Robert Koch, "Greibach Normal Form transformation revisited", Information and Computation, Vol. 150, Issue 1,1999, pages 112–118; online.
  3. Prof. Greibach was kind enough to send me a few comments on this theorem which I quote below:

    Although my original definition of GNF is as you describe, in my class notes I now permit S -> emptystring if S does not appear on the right hand side of any production and similarly for CNF (Chomsky Normal Form) as in the notes you attach, so that all context-free languages are covered.

    As far as I know, GNF is not used in any grammar-pda transformations directly; it is essential to conversion to a pda without epsilon-rules, i.e. to a nondeterministic pda which must read a new input each unit of time (I usually call this quasi-realtime). Indeed, the fact that GNF suffices for context-free languages is equivalent to the fact that nondeterministic pda are equivalent in power to quasi-realtime pda.

    My normal form was proven in 1962 and appears in my 1963 thesis but, as you note, the first full publication was in 1965.


Theorem no. 181: Archimedes’ Equiareal Map Theorem
  1. Bradley Carroll has a very nice series of pages on Archimedes' achievements, where we find (see this page) that Archimedes himself regarded his results on areas and volumes of curved bodies to be his finest work. The sphere-cylinder surface area ratio is quoted there as 2/3 whereas our version says the surface areas are equal; this is merely because our cylinder has no top or bottom. To be specific, the ratio for a sphere of radius \(r\) and a cyclinder of radius \(r\) and height \(h\), is \(2\tau r^2/(\tau r h+\tau r^2)\); we have \(h=2r=2\) and omit the second term in the denominator.
  2. I could not resist invoking E.T. Bell's celebrated triumvirate in connection with this theorem. That is shameless popularism, though, since I entirely subscribe to Thony Christie's injunction "Context is everything".
  3. The attribution to Gauss-via-Newton of differential geometry is even more shameless. In Dirk Jan Struik's classic two-part "Outline of a History of Differential Geometry", the whole of part 1 is pre-Gauss (Isis, vol. 19, no. 1, 1933, pp. 92–120; online (paywall)) with the key modern players being Clairaut, Euler and Monge. However, Gauss's work in the 1820's pervades a large proportion of part 2 (Isis, vol. 20, no. 1, 1933, pp. 161–191; online (paywall)), and the first and second fundamental forms belong to intrinsic differential geometry which was intrinsically Gauss.

Theorem no. 182: The Girard–Newton Identities
  1. A short survey of elementary proofs of these identities, together with an elegant new proof using matrix algebra, is given in Dan Kalman, "A Matrix Proof of Newton's Identities", Mathematics Magazine, Vol. 73, Number 4, October 2000, pp. 313–315; online (paywall; a copy here, scroll to 8/16/99, Jan. 2024).
  2. The photo of Peter Cameron is a cropped version of one I found on his 60th birthday conference website, maintained by Robert F. Bailey. It is attributed to Adrian Bondy by Peter Cameron himself. He told me in an email that "the picture was taken by Adrian Bondy ... at the Victoria Arms in Oxford (at Dominic Welsh's retirement conference in 2005 (I think)." From a follow-up email from Bondy "I don't recall having taken the photo, but it's possible." Since Bondy's photography is art (see his gallery website) I take the issue seriously!

Theorem no. 183: Theorema Egregium
  1. Original source for this theorem: Karl Friedrich Gauss, "Disquisitiones generales circa superficies curvas auctore Carolo Friderico Gauss. Societati regiæ oblatæ D. 8. Octob. 1827", Commentationes societatis regiæ scientiarum Gottingensis recentiores, Commentationes classis mathematicæ. Tom. VI. (ad a. 1823–1827), Gottingæ, 1828, pp. 99–14. An English translation with introduction is offered as a pdf (1MB) download by Project Gutenberg.
  2. An excellent and beautifully illustrated technical source on this theorem is Nigel Hitchin's notes on Geometry of Surfaces, under Teaching here.

Theorem no. 184: von Neumann's Minimax Theorem
  1. Original source for this theorem is: von Neumann, J., "Zur Theorie der Gesellschaftsspiele", Mathematische Annalen, 100 (1),1928, pp. 295–320; online (paywall; facsimile at Göttinger Digitaisierungszentrum).
  2. A superb analysis of the origins of von Neumann's theorem is Tinne Hoff Kjeldsen, "John von Neumann’s Conception of the Minimax Theorem: A Journey Through Different Mathematical Contexts", Arch. Hist. Exact Sci. 56 (2001) 39–68; online (paywall; there is reprint online here, December 2023).
  3. Regarding the 'underlying technology': George B Dantzig's classic "Reminiscences about the origins of linear programming", Operations Research Letters, Vol. 1, Issue 2, 1982, pp. 43–48; online (paywall; DTIC tech. report).

Theorem no. 185: Kőnig's Bipartite Matching Theorem
  1. This theorem is commonly referred to as the Kőnig–Egerváry theorem, having been discovered independently and simultaneously by Kőnig's compatriot Jenő Egerváry. Both papers appeared in the same volume of Matematikai és Fizikai Lapok:
    1. Egerváry, Jenő, "Matrixok kombinatorius tulajdonságairól", Matematikai és Fizikai Lapok, Vol. 38, pp. 16–28.
    2. Kőnig, D., "Gráfok és mátrixok", Matematikai és Fizikai Lapok, Vol. 38, 1931, pp. 116–119.
    The complete volume in the original Hungarian is free as a 95MB pdf download at An English translation of Kőnig's paper has been put in the public domain by Gábor Szárnyas: "Graphs and matrices: A translation of "Graphok és matrixok" by Dénes Kőnig (1931)"; arxiv.
  2. Corresponding to this theorem is an algorithm for finding a maximum matching in a bipartite graph which operates in \(O(mn)\) time, where the graph has \(m\) edges and \(n\) vertices. The best algorithm (modulo some speed-ups via rendomisation) remains the \(O((m+n)\sqrt{n})\) algorithm of Hopcroft and Karp, "A n^5/2 algorithm for maximum matchings in bipartite graphs", SIAM J. Comput., Vol. 2, No. 4, 1973, pp. 225–231; online (paywall; pdfs are easy to find on the web, and see the algorithm's Wiki page). Finding a maximum matching in general (not necessarily bipartite) graphs can, perhaps surprisingly, also be achieved in \(O(m\sqrt{n})\) time, via the Micali–Vazirani (MV) algorithm. The proof of correctness of this algorithm has a long history, nicely described in this guest post at Computational Complexity (also contains a link to a preprint of the paper giving a complete proof).

Theorem no. 186: The Insolvability of the Entscheidungsproblem
  1. Original sources for this theorem:
    1. Emil L. Post, "Finite combinatory processes — formulation 1", The Journal of Symbolic Logic, Vol. 1 , Issue 3, 1936, pp. 103–105; online (paywall; free pdf download June 2024). This paper famously only offers a glimpse of Post's grasp of computability and undecidability. See John Stillwell, "Emil Post and his anticipation of Gödel and Turing", Mathematics Magazine, Vol. 77, No. 1, 2004, pp. 3–14; online (paywall), and Liesbeth de Mol, "Closing the circle: an analysis of Emil Post's early work", Bulletin of Symbolic Logic, Vol. 12, No. 2, 2006, pp. 267–289; online (paywall).
    2. Alonzo Church, "An unsolvable problem of elementary number theory", American Journal of Mathematics, Vol. 58, No. 2, 1936, pp. 345–363; online (paywall; free pdf download June 2024).
    3. A.M. Turing, "On Computable Numbers, with an Application to the Entscheidungsproblem", Proc. London Math. Soc., Vol. s2-42, Issue 1, 1937, pp. 230–265; online.
  2. "Did Turing prove the undecidability of the halting problem?" by Joel David Hamkins and Theodor Nenu is very worthwhile reading.
  3. Some contextual information is given in a presentation (500KB pdf) I gave at Rewley House on 23 June 2012.
  4. An good source of undecidable problems is this 2012 survey (450KB pdf) by Bjorn Poonen. Related links on Diophantine undecidability are: James P. Jones paper "Diophantine Representation of the Fibonacci Numbers" (1.3MB pdf) , and this paper by Yuri Matiasevich (in which some of the characters seems to print strangely but not unreadably).
  5. Details of Jack Copeland's Essential Turing are given here; Charles Petzold's reading guide to Turing's 1936 paper is listed here. Biographies of Turing are listed here.
  6. Hilbert's 1930 "Wir müssen wissen, Wir werden wissen" radio address is online here with a transcription and an accompanying English translation.
  7. There is a poetry version of the proof of non-decidability of the Halting problem here by Geoffrey K. Pullum, as I learnt from Pat'sBlog.
  8. At the heart of Turing's result is the demonstration that not all functions from the natural numbers to the natural numbers can be computed. Joel David Hamkins has the intriguing result that any function is computable if the right model of arithmetic is chosen. This means arithmetics which satisfy the axioms of the natural numbers but which contain additional 'non-standard' numbers. A good introduction is provided by John Baez.
  9. This blog post from Gödel's Lost Letter is an excellent source on proving the unsolvability of the Halting Problem.

Theorem no. 187: Karp's Theorem
  1. Karp's original article is R.M. Karp, "Reducibility among combinatorial problems", in Complexity of Computer Computations (R.E. Miller and J.W. Thatcher, eds.), Plenum Press, 1972, pp. 85–103. It is reprinted with a nice introduction by Richard Karp in Michael Jünger, Thomas M. Liebling, Denis Naddef, George L. Nemhauser, William R. Pulleyblank, Gerhard Reinelt, Giovanni Rinaldi and Laurence A. Wolsey (eds.), 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art, Springer, 2010.

Theorem no. 188: al-Kāshi's Law of Cosines
  1. Garry J. Tee kindly provided the following amplification on the history of trigonometric functions: "Hipparchus in (c-130) invented the chord, the first trigonometric function, and he constructed a short table of values of the chord function.  (On a sphere of radius R, the chord of angle x is the distance between 2 points on the sphere subtending angle x at the centre). By the 5th century, Hindu astronomers had replaced the chord  by the more convenient sine function, with \(2R\sin x  = \mbox{chord}(2x)\). In 499, Aryabhata commenced his renowned astronomical treatise “Aryabhatiya” with a short table of sines."
  2. As is often the case with formulae in Euclidean geometry there are spherical and hyperbolic versions of this theorem. The spherical law of cosines is described here; the hyperbolic has a Wiki page.

Theorem no. 189: The Handshaking Lemma
  1. Architectural historian Dr Lynn Pearson kindly sent me the following comments in response to a query regarding the origins and attribution of the tiling pattern illustrating this theorem:
           "While investigating the 6/7 murals query, I chanced upon the QM Physics archives website; the brochure about the new Physics Building (1962) is available at;
    this details the six panels. I too made it six as there are six architectural 'bays'. But I see what you mean about the different section you have used for your theorem. Looking at the six panels, the middle 4 have the main pattern in white on a blue ground, with various small sections of coloured tiling around. But the two end panels have the blue section, and another smaller vertical section on a gold background. The one furthest from the road has the precessing orbit, plus what looks to me like ellipses/catenary curves?? – see archive pic:
    Are these two elements of the panel connected maths-wise? And the same goes for your panel, the one nearest the road – that's 'spreading of dislocations from a Frank-Read source' plus the diagram you have in your theorem. Are these things linked in some way? It seems that the designers chose to 'finish off' the series of murals with slightly more ornate ones at each end. Anyway, all these panels were by Carter's, but my notes from the Carter's photographic archive at Poole Museum show that the firm worked closely with the building's architects, Playne & Lacey, on the project. A man called R. Khosla, who worked for the architects, helped in the design of the 6 panels, but left before they were complete. As the head of design at Carter's, A. B. Read did much of the mural work; I'd think he completed the job. The tile painting would have been done by the firm's (lady) artists. I think that is as good an attribution as you will get."
    Dr Pearson also provided a link to a relevant article of hers, although for copyright reasons its images cannot be displayed. And there is a little more in the Tile Gazetteer.
  2. From plus magazine: applying the double counting argument proof of the Handshaking Lemma to the complete graph on n + 1 vertices is a neat way of proving that $$1+2+\ldots + n = \frac12n(n+1).$$
  3. Other impressive applications of the Handshaking Lemma: the proof of Sperner's Lemma in 2D; and this proof that the so-called 'Lights-Out' game is solvable for any graph in which all vertices are initially 'turned on'.

Theorem no. 190: Jackson's Theorem on Compatible Euler Tours
  1. Original source for this theorem is Jackson, Bill, "A characterisation of graphs having three pairwise compatible Euler tours", J. Combinatorial Theory, Series B, Vol. 53, Issue 1, September 1991, pp. 80–92; online. The paper conjectures that, in an Eulerian graph with minimum degree \(2k\), there are \(2k-1\) pairwise compatible Euler tours if and only if $$(2k-1)(\omega_B-1)\leq (2k-2)|B|,$$ for all sets \(B\) of bitransitions, using (a suitable generalistion of) the notation of our theorem page. Thus Jackson's theorem is the case \(k=2\). As far as I know \(k\geq 3\) is still open. If there are fewer than \(2k-1\) pairwise compatible Euler tours then Jackson has conjectured elsewhere that there are \(2k-2\). This was known to be true for \(k=2\) and Jackson's paper extended this to \(k=3\): an Eulerian graph with minimum degree 6 has at least four pairwise compatible Euler tours.
  2. The original weblink from this theorem page was to the fine set of notes (850KB pdf) posted here by Tero Harju. Still recommended, of course, but the replacement link to the Egerváry Research Group page is more directly relevant.

Theorem no. 191: L'Hospital's Rule
  1. A nice short account of the Bernoulli vs L'Hospital ownership of this theorem is given here at Life Through a Mathematician's Eyes.
  2. A nice 'double' example of L'Hospital in action is the proof that \(\ln(x)\tan(x) \rightarrow 0\) as \(x\rightarrow 0\), starting in the \(\infty/\infty\) form as \(\ln(x)/\cot(x)\) and then transferring to the \(0/0\) form.
  3. The expression \(0^0\) gets a thorough investigation by Michael Huber and V. Frederick Rickey in "What is \(0^0\)?", Convergence, Vol. 5, 2008. Online here. Other good treatments are this blog post by David A. Tanzer (with over 50 very informative reader responses) and this from askamathematician (which has over 1000 responses, which I haven't read but I suppose there must be some interesting stuff there as well!)
  4. There are some interesting insights into early writing on L'Hospital's rule (including by L'Hospital himself) in a classic book review by Underwood Dudley for MAA, which they have kindly put online here.

Theorem no. 192: The Rotation-distance Bound
  1. Original source: Daniel D. Sleator, Robert E. Tarjan and William P. Thurston, "Rotation distance, triangulations, and hyperbolic geometry", J. Amer. Math. Soc., Vol.1, No.3., 1988, pp. 647–681; online.

Theorem no. 193: Frieze's Theorem on Expected Minimum Tree Length
  1. Original source: Alan M. Frieze, "On the value of a random minimum spanning tree problem", Discrete Applied Mathematics, 10 (1985), 47–56; online.
  2. A sharper asymptotic for Frieze's result is given in Colin Cooper, Alan Frieze, Nate Ince, Svante Janson, Joel Spencer, "On the length of a random minimum spanning tree", Combinator. Probab. Comp., 25 (2015) 89–107; online (paywalled), open-access preprint.
  3. Find more wonderful properties of \(\zeta(3)\) in this preprint by David Broadhurst.

Theorem no. 194: Wilson's Theorem
  1. The (contrapositive to the) 'if' converse to the theorem follows because if \(n\) is composite then some \(d\),\(1<d<n\), divides \(n\). Then \(d\) appears as a factor of \((n-1)!\) and therefore cannot divide \((n-1)!+1\), in which case, neither can \(n\).
  2. Fredrik Johansson describes a neat trick for reducing the computation required for testing primality via Wilson's Theorem.
  3. A combinatorial argument which seems in the same spirit as P.G. Anderson et al's combinatorial lemma is found in Szilárd András, "A combinatorial generalization of Wilson’s theorem", Australasian J. Comb., Vol. 49, 2011, pp. 265–272; online. (direct pdf, 125KB)
  4. It seems convenient to record here Gauss's generalisation of Wilson's Theorem: if \(n>2\) then $$\prod_{\stackrel{k=1}{\gcd(k,n)=1}}^n\hspace{-.2in}k = \left\{\begin{array}{rcl} -1\mbox{ mod }n & & n=4, p^m, 2p^m,\\ 1\mbox{ mod } n &&\mbox{otherwise.}\end{array}\right.$$ A very nice proof is given here by Pete L. Clark (under "Expository Papers").

Theorem no. 195: The Erdős–Ko–Rado Theorem
  1. Original sources for this theorem:
    1. P. Erdős, Chao Ko and R. Rado, "Intersection theorems for systems of finite sets", Quart. J. Math., Oxford Ser. (2) 12, 1961, pp. 313–320; online. The paper was written in 1938, however, see P. Erdős, "My joint work with Richard Rado", in Surveys in combinatorics 1987, London Math. Soc. Lecture Note Ser., 123, pp. 53–80, Cambridge Univ. Press, 1987; online.
    2. The Katona proof was published in Katona, G.O.H., "A simple proof of the Erdős–Chao Ko–Rado theorem", Journal of Combinatorial Theory, Series B, Vol. 13, Issue 2, 1972, pp. 183–184; online.
  2. An interesting discussion by John Mount of proofs of Erdős–Ko–Rado can be found in this Win-Vector blog entry.
  3. The requirement that \(n\geq 2k\) is necessary since when \(k>n/2\) any pair of subsets must necessarily intersect. This makes \(n/2\) a transition point where the size of a maximum intersecting family jumps up, as illustrated below for \(n=50\). The horizontal axis is \(k\); the red dots are values of \({n-1 \choose k-1}\); the blue dots are values of \({n \choose k}\).
    max size of intersecting family

Theorem no. 196: The Cantor–Bernstein–Schröder Theorem
  1. Original source for this theorem: Dedekind R. 1887 "Ähnliche (deutliche) Abbildung und ähnliche Systeme", in Gesammelte mathematische Werke, vol. 3 (eds R Fricke, E Noether, Ö Ore), pp. 447–449. Braunschweig: Vieweg; online. The proof history of the theorem is complicated as can be inferred from the discussion (cited on the theorem page) at Gödel's Lost Letter. An in-depth analysis is provided by Wilfried Sieg, "The Cantor–Bernstein theorem: how many proofs?", Philosophical Transactions of the Royal Society, A, Vol. 377 Issue 2140, March 2019; online.
  2. The proof presented for this theorem is streamlined by appealing to the Knaster-Tarski fixed-point theorem which guarantees a fixed point for a monotone function on a complete lattice (in this case the power set lattice). More details at this MathWorld entry.
  3. CBS is, in the analysis of Williard Quine, one form of the 'law of comparability. In Set Theory and Its Logic, Harvard University Press, revised edition,1969, p. 208, he memorably says, "Accidents of definition aside, there are three distinct things here: the Axiom of Choice, the Schröder-Bernstein Theorem, and triviality."
  4. A graph-theoretic proof of this theorem generalises to one about paths, as described in Reinhard Diestel & Carsten Thomassen, "A Cantor-Bernstein theorem for paths in graphs", American Mathematical Monthly, Vol. 113, No. 2, pp. 161–165; online (a reprint is here, October 2023, under "Erdős–Menger conjecture; paths in infinite graphs").

Theorem no. 197: The Robin–Lagarias Theorem
  1. Original source for this theorem: Jeffrey C. Lagarias, "An elementary problem equivalent to the Riemann Hypothesis", Amer. Math. Monthly, Vol. 109, No. 6, 2002, pp. 534–543; online (paywall; arxiv, which is the recommended weblink on the theorem page). Guy Robin's breakthrough is Robin, Guy (1984), "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées. Neuvième Série 63 (2), 1984, pp. 187–213. I think it may not exist online but check this math.stackexchange entry for updates. The earlier sources for this theorem are well charted in Lagarias's paper.
  2. A collection of assertions equivalent to the Riemann Hypothesis is given here. A collection of proposed proofs/refutations of RH is given here. In an interesting contribution by Brian Conrey, RH is implied by the non-negativity over a small interval \([0,\epsilon]\) of the function \(f(x)=\sum_{n=1}^{\infty}\lambda(n)\sin\tau n x/n^2\), where \(\lambda(n)\) is the Liouville function.
  3. A nice Quora entry by Alan Amit discusses the implications of RH being false. In passing he points out that Lagarias's version of RH is elementary enough that a counterexample to RH would mean a disproof in Peano arithmetic, not something that can be easily deduced from finding a zero off the critical line.
  4. Out of curiosity I plotted the divisor function \(\sigma(n)\) (red) against the RHS of Laragias's inequality \(H_n+\ln(H_n)\exp(H_n)\) for the first \(10^5\) values of \(n\). I find it hard to imagine betting against RH!     LHS(red) vs RHS (blue) of Lagarias
  5. On the subject of computational data on RH: Dave Platt and Tim Trudgian, "The Riemann hypothesis is true up to 3×1012", Bulletin London Math. Soc., Vol. 53, Issue3, 2021, pp. 792–797; online (paywall; arxiv). It is a good source for related work, as are the citation links from the LMS publication page.

Theorem no. 198: The Art Gallery Theorem
  1. Original sources for this theorem:
    1. Chvátal, V., "A combinatorial theorem in plane geometry", J. Comb. Theory, Series B, Vol. 18, Issue 1, 1975, pp. 39–41; online.
    2. Fisk, S., "A short proof of Chvátal's watchman theorem", J. Comb. Theory, Series B, Vol. 24, Issue 3, 1978, p. 374; online.
  2. There is a good account of the Art Gallery Theorem by Erica Klarreich for Quanta Magazine here. This talk (1.7MB pdf, August 2022) by Hemanshu Kaul gives good overview of theorems and algorithms in the 'Art Gallery' field.
  3. The theorem as proved here guards an art gallery with 'vertex guards', who are stationed at vertices of the polygon, although it is stated in terms of 'point guards', who may be anywhere inside (or on the boundary of) the polygon (the lower bound of \(\lfloor n/3\rfloor\) applies equally in both cases). The example chosen is of an 'orthogonal' gallery, in which all walls meet at angle \(\tau/4\). In fact, for orthogonal galleries a sharper result is possible: \(\lfloor n/4 \rfloor\) vertex guards are sufficient (and a square version of the 'comb' polygon shown on the theorem page proves necessity). J. Kahn, M. Klawe and D. Kleitman, "Traditional galleries require fewer watchmen", SIAM Journal on Algebraic Discrete Methods, 1983, vol. 4, No. 2 : pp. 194–206; online paywalled.
  4. A natural extension of the problem guards an \(n\)-vertex polygon containing \(h\) disjoint polygons ('holes'). Here the tight bound for point guards is \(\lfloor (n+h)/3 \rfloor\). See I. Bjorling-Sachs & D. L. Souvaine, "An efficient algorithm for guard placement in polygons with holes", Discrete & Computational Geometry, vol. 13, pp. 77–109 (1995); online. In fact \(\lfloor (n+h)/3 \rfloor\) is conjectured to be sufficient even if only vertex guards are allowed (Shermer, 1982). Again for orthogonal galleries the numerator is conjectured to be 4, for vertex guards (Shermer again). A good survey is given by Paweł Żyliński "Placing guards in art galleries by graph coloring", chapter 13 in Marek Kubale (ed.), Graph Colorings, American Mathematical Society, 2004.

Theorem no. 199: Fermat's Two-Squares Theorem
  1. The Wiki page for this theorem gives a good account of its history. In fact, there is another Wiki page devoted to just proofs of this theorem.
  2. The representation of a prime \(p=4n+1\) as a sum of two squares is unique (up to order of summands). A nice proof using Gaussian primes, is given here.
  3. Strictly speaking, Lagrange's Lemma only goes one way: if \(p\) is congruent to 1 mod 4 then \(-1\) is a quadratic residue mod \(p\). Lagrange used his lemma in 1773 to give a simpler proof than Euler's of Fermat's theorem. There is a short discussion in chapter 6 of Stillwell's Elements of Number Theory, Springer 2003 (see sections 6.5 and 6.8 and chapter 9 for the converse). The lemma is an instance of the Law of Quadratic Reciprocity (see this, for example).
  4. There is a famous 'one-sentence' proof of this theorem by Don Zagier, "A one-sentence proof that every prime \(p = 1\ (\!\!\!\mod 4)\) is a sum of two squares", Amer. Math. Monthly, Vol. 97, No. 2, 1990, p. 144; online (paywall; pdf download here, March 2024, and there is a good explanation of the proof by James McIvor here (Lecture 6).

Theorem no. 200: Minkowski's Convex Body Theorem
  1. The original source for this theorem seems somewhat obscure. It seems to be widely cited with the date 1889. Minkowski's work in the 1880s is well-described by Jay Goldman in Chapter 22 of The Queen of Mathematics: A Historically Motivated Guide to Number Theory where he writes "Minkowski continued to work on these ideas and on November 6, 1889, he wrote to Hilbert
    Perhaps you or Hurwitz are interested in the following theorem (which I can prove in half a page): in a positive definite form of determinant \(D\) with \(n(\geq 2)\), one can always assign such values to the variables that the form is \(< nD^{1/2}\).
    This theorem was based on geometric reasoning and it revolutionized the subject. We now explain these ideas, first systematically presented in Minkowski's fundamental book." The Minkowski quote in turn is from Chapter 9 of Winfried Scharlau and Hans Opolka (transl. W.K. Bühler and G. Cornell) From Fermat to Minkowski: Lectures on the Theory of Numbers and Its Historical Development. Minkowski's 'fundamental book' is Geometrie der Zahlen, 1910, and this is the usual citation for Minkowski's theorem (e.g. in Blichfeldt's 1914 paper). So it seems reasonable to give Nov. 6, 1889, as the date of birth of the theorem. However, the French version of its Wiki page (which is very different from the English version and well worth a visit) gives 1891 as the first publication date.
  2. Blichfeldt's theorem, however, is easy to trace (and read) online: Blichfeldt, H. F., "A new principle in the geometry of numbers, with some applications", Trans. Amer. Math. Soc., Vol. 15, Issue 3, 1914, pp. 227–235; online.
  3. Our proof of Fermat's 2-Squares theorem (theorem no. 199) uses Minkowski's theorem (i.e. is a theorem in his 'geometry of numbers'). Pete L. Clark has posted a 125-page set of notes here on geometry of numbers with lots more examples of applications and a great deal of valuable contextual material.

Theorem no. 201: Jensen's Inequality
  1. Original source for this theorem: Jensen, J. L. W. V., "Sur les fonctions convexes et les inégalités entre les valeurs moyennes", Acta Mathematica, Vol. 30, 1906, pp. 175–193; online.
  2. Jim Wilson at University of Georgia has this good discussion of applying the AMGM inequality (scroll down).
  3. This theorem's illustration originally featured soup cans based on Andy Warhol's Campbell's Soup pop art. However, Campbell Soup Company declined to give me permission for this use "in part because the image you have used is not a reproduction of our famous trademarks, but rather what we consider a 'mutilation' of our marks." If you would like to view this mutilation for yourself let me know and I will smuggle you a copy!
  4. A good real-life application from the world of finance is given here, taken from Sam L. Savage, The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty, John Wiley & Sons, paperback edition, 2012.
  5. Meanwhile the issue in our illustration, that of choosing height and radius for a tin can, is given real-life treatment in this charming tweet from @mathematicsprof. I have posted a pdf screenshort here for posterity.

Theorem no. 202: The Freidlander–Iwaniec Theorem
  1. The web link for this theorem is to an overview of its proof. The actual proof is set out in a subsequent paper of almost a hundred pages: John Friedlander and Henryk Iwaniec, "The polynomial X2+Y4 captures its primes", Annals of Mathematics, Vol. 148, no. 3, 1998, pp 945–1040; online (paywall; there is an archived copy here May 2024).
  2. The question of whether there are infinitely many primes of the form \(n^2+1\) is the first of Landau's Problems, all unsolved as of May 2024. See this by János Pinz for more details.

Theorem no. 203: Euler's Continued Fraction Correspondence
  1. Original sources for this theorem:
    1. Leonhard Euler, Introductio in analysin infinitorum, Chapter 18, 1748; see note to Theorem no. 153: Euler's Partition Identity for more on this source.
    2. L.J. Lange, "An elegant continued fraction for π", American Mathematical Monthly, Vol. 106, No. 5, 1999, pp. 456–458; online (Lange's paper describes Douglas Bowman's contribution).
  2. There is a good account of Kerala Gargya Nilakanth's work in R. Roy, "The discovery of the series formula for π by Leibniz, Gregory and Nilakantha", Mathematics Magazine, Vol. 63, Issue 5, 1990, pp. 291–306; online (paywall; reprint here, May 2024, file Roy-pi.pdf).
  3. There is more on the \(\tau\) continued fraction here.Tony Foster gives a continued fraction in terms of cubes for \(\pi=\tau/2\) via a nice exploitation of Nilakantha's series in the same vein as the derivation by Douglas Bowman. Suggests a nice exercise to give the corresponding result for \(\tau\). He also has a similar derivation for the golden ratio to contrast to the simple continued fraction (all 1's) which everyone knows.·e·s

Theorem no. 204: Singmaster's Binomial Multiplicity Bound
  1. Original sources for this theorem:
    1. Singmaster, D., "Research Problems: How often does an integer occur as a binomial coefficient?", American Mathematical Monthly, 78 (4), 1971, pp. 385–386; online (paywall; there is a copy here at
    2. H. L. Abbott, P. Erdős, D. Hanson, "On the number of times an integer occurs as a binomial coefficient", American Mathematical Monthly, 81 (3), 1974, pp. 256–261; online (paywall; there is a copy here at The Erdős Project).
    3. Daniel M. Kane, "Improved bounds on the number of ways of expressing t as a binomial coefficient", Integers, Vol. 7, 2007, pp. 1–7; online.
  2. Theorem under Construction iconThis is a 'theorem under construction': I hope to chart exciting developments here towards an eventual final version, which may or may not be \(N(k)\leq 8\) for all \(k\).
  3. There is a nice entry on Singmaster's conjecture at Gödel's Lost Letter.
  4. A version of Singmaster's conjecture in terms of algebraic geometry is given in Hugo Jenkins, "Repeated binomial coefficients and high-degree curves", Integers, vol. 16, 2016.
  5. Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Terence Tao and Joni Teräväinen have announced (June 2021) a proof of this conjecture for the 'interior' of Pascal's triangle, with the result that it remains to be proved for an edge region: the number of solutions to the equation \(\binom{n}{m}=t, t\geq 2,\) is bounded by an absolute constant provided this is true in the region $$2\leq m<\exp(\log^{2/3+\epsilon}n).$$. Published subsequently as "Singmaster’s Conjecture In The Interior Of Pascal’s Triangle", The Quarterly Journal of Mathematics, Vol. 73, Issue 3, 2022, pp. 1137–1177; online.
  6. Coincidentally with Matomäki et al, a nice paper generalising Singmaster to monomials has appeared: Jean-Marie de Koninck, Nicolas Doyon and William Verreault, "Repetitions of Multinomial Coefficients and a Generalization of Singmaster's Conjecture", Integers, Vol. 21, 2021, paper A34; online.

Theorem no. 205: The Classification of the Semiregular Tilings
  1. Original sources for this theorem:
    1. Kepler's Harmonices Mundi has its own Wiki page.
    2. Paul Robin, "Carrelage illimité en polygones réguliers", La Nature, 1887 : Quinzième année, deuxième semestre : n. 731 à 756, pp. 95–96; online (facsimile).
    3. A. Andreini, "Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative", Mem. Società Italiana della Scienze, Ser.3, 14, 1905, pp. 75–129 online (facsimile via, 4.5MB pdf)
    4. D. M. Y. Sommerville, "Semi-regular networks of the plane in absolute geometry", Earth and Environmental Science Transactions of The Royal Society of Edinburgh, Vol. 41, Issue 3, 1906, pp. 725–747; online (paywall; facsimile).
  2. There is more on Kepler's investigation into plane tilings in this lovely article by plus magazine. In fact they have a whole collection of tiling articles! This Quanta Magazine article by David S. Richeson is very valuable.
  3. The general subject of plane tilings is deep and wide: for example, the decidability of whether a given single tile will tile the plane is an open question (very elegantly introduced by Chaim Goodman-Strauss in "Can't Decide? Undecide!", Notices of the AMS, Vol. 57, No. 3, 2010, 343–356, online here).

Theorem no. 206: Euler's Formula
  1. Original source for this theorem is Volume 1 of Euler's 1748 Introductio in analysin infinitorum; online. English translation at Ian Bruce's
  2. A sweet, and sweetly presented, proof of Euler's Formula, given by Math With Bad Drawings, makes a very nice example of solving differential equations by separation of variables.
  3. The appearance of a protractor (angle measurer) in the illustration of this theorem gives me an excuse to link to this delightful little History of Protractors from Life Through a Mathematician's Eyes.

Theorem no. 207: The Eratosthenes-Legendre Sieve
  1. Original source for this theorem: A. M. Legendre, Essai sur la théorie des nombres, Coucier, Paris, second edition, 1808. I owe this citation to James A. Farrugia's excellent dissertation "Brun's 1920 Theorem on Goldbach's Conjecture" Utah State University, 2018; online (see footnote, page 6).
  2. A very nice motivation for sieving in number theory is given by Terence Tao here.

Theorem no. 208: Torricelli's Trumpet
  1. In response to my query, Paolo Mancosu kindly gave me following comments on the origins of Torricelli's result and his methods:
           "I am quite sure [that] Oresme and Fermat and Roberval certainly did not anticipate Torricelli's discovery. Fermat wrote about similar solids (de infinitis hyperbolis) after Torricelli. As for Roberval, I cite that report of Mersenne (given by Torricelli) where it is reported that, according to Mersenne, Roberval had written some kind of speech claiming that Torricelli's result was impossible! Had he anticipated him, he would certainly have claimed priority rather than trying to prove that the result was impossible. You are right that Torricelli does not prove that the lateral surface is infinite. I do not know who first did that."       
  2. In his review (Notre Dame J. Formal Logic, vol. 40, no. 3, 1999, 447–454) of Mancosu's Philosophy of Mathematics & Mathematical Practice in the Seventeenth Century, Craig Fraser says (p. 448)  "Torricelli discovered the remarkable fact that the solid of revolution obtained by rotating the hyperbola \(y=1/x\) about the \(x\)-axis has finite volume and infinite surface area." I have found no other evidence that Torricelli calculated the surface area of his solid, however.
  3. Dave Richeson's well-known Division by Zero blog has provided a template for constructing a paper version ofTorricelli's Trumpet (under the pseudonym Gabriel's Horn). (Possibly the arxiv version of Richeson's paper is more of a permalink than the one given in the blog entry.)
  4. A very nice animated illustration of indivisibles, applied to circular area, is given by Matt Henderson here.
  5. Although paradoxical the finite volume vs infinite surface area can be motivated by elongating a solid of finite volume: it's volume is preserved but its surface area increases. This is well explained at Quora by Nishanth Jayram.

Theorem no. 209: The Erdős Discrepancy Problem
  1. Theorem under Construction iconThis was originally posted as a 'theorem under construction': September 2015 brought news of Terence Tao's complete resolution of the conjecture. This made it the 2nd 'declassification', after Kepler's Conjecture. The role of the Polymath in Tao's proof has been commented on helpfully by Gowers here.
  2. There is much more to the background to the conjecture, and approaches to it, at the official Polymath 5 page.
  3. The sequence of length 1160 appearing in the table in this theorem description, reproduced from Konev and Lisitsa's paper, is available in Excel 2003 here. The first 11 terms:
             −  +  +  −  +  −  −  +  +  −  +
    happen to constitute a maximum-length sequence with discrepancy C=1. The terms sum to +1 so continuing with a +1 would give a summation to 2; the even-index terms sum to −1, so continuing with a −1 would give a summation to −2. The sequence 0,11,1160, ... is; it is known (Konev and Lisitsa) that the next term exceeds 130000.
  4. There is a very good description of Konev and Lisitsa's proof of \(C=2\) by Richard Lipton and Ken Regan here.
  5. Erdős mentions Nikolai Chudakov (spelt 'Tchudakoff') in connection with this conjecture here (in Problem 49).
  6. Although Mathias' paper was published in a 1997 tribute volume for Erdős, this was actually the proceedings of Erdős' 80th birthday celebration, held in March 1993.
  7. Terence Tao has a valuable blog entry on 'near-counterexamples' to the conjecture and its unexpected relationship to the 'Elliott Conjecture'. (But see note 1!)

Theorem no. 210: The Basel Problem
  1. The proof given in the description of this theorem is called the 'Lewin argument' by Kalman and McKinzie who cite its first known appearance as Leonard Lewin's Polylogarithms and Associated Functions, Elsevier Science, 1981 (although they stress that Lewin did not claim credit). The book is an update of Lewin's earlier Dilogarithms and Associated Functions, Macdonald, 1958, in which the same material may be found (chapter 1, section 3.1). Both books are out of print, sadly. There is a valuable review of the former by Richard Askey in Bull. AMS, vol. 6, no. 2.
  2. Regarding Euler's solution to the Basel Problem, the accepted sequence of events appears to be: discovered in 1734, presented in 1735, published in 1740. The Euler Archive gives the date of presentation of Euler's paper as December 5,1735. It might appear that 'December 5, 1734' is the correct date. In "Euler and the Zeta Function" Raymond Ayoub gives 1934 as the year of "Euler's first triumph" and says "Euler communicated his result to Daniel Bernoulli and, while unfortunately this letter has been lost, the reply does exist". However, the reply is dated in the Euler Archive as 12th September 1736. Like many of Bernoulli's letters to Euler it deals with several matters giving no clue as to when they arose but it would seem more consistent with a 1735 letter from Euler than a 1734 one.

Theorem no. 211: Willans' Formula
  1. The source for this page is C. P. Willans, "On formulae for the nth prime number", The Mathematical Gazette, Vol. 48, No. 366 (Dec., 1964), pp. 413–415. Not free to view online but 1st page can be previewed here.
  2. The reduction, for composite \(k=a\times b\), of \(\sin^2(1+(k-1)!)\tau/4k)\) to \(\sin^2(\tau/4k)\) requires justification. Indeed \(1+(k-1)!\equiv 1\!\!\!\mod k\) because both \(a\) and \(b\) will divide \((k-1)!\). And the quotient \((k-1)!)/k\) will be a multiple of 4 when there are sufficient even factors in \((k-1)!\), which is when \((k-1)/2>2+\log_2 k\). This occurs for \( k >12\) (but \( k = 6, 9,10,12\) all reduce to \(\tau/4k\) since the greatest power of 2 in their factorisations is low).
  3. See note 1 for Theorem 194 regarding the computation required to implement Willans' formula.
  4. The famous 26-variable polynomial of Jones–Sato–Wada–Wiens whose positive values, over the positive integers, are precisely the prime numbers, is also based on Wilson's Theorem. There is a nice explanation here. Another nice paper of Tsangaris and Jones, by the way, describes how a 19th century summation formula for GCD, due to Mathias Jacob Hacks, can be fashioned into summation formulae for \(\pi(x)\), n-th prime number and next prime number: Panayiotis G. Tsangaris and James P. Jones, "And Old Theorem on the GCD and Its Application to Primes", Fibonacci Quarterly, Vol 30, No. 30, 1992, pp. 194–198; online. This mathstackexchange entry is good on Willans and related formulae.

Theorem no. 212: Vizing's Theorem
  1. For details of Vizing's publication of his theorem see its Wiki page. Details regarding Gupta's discovery of the theorem are supplied in the preface of Michael Stiebitz, Diego Scheide, Bjarne Toft and Lene M. Favrholdt, Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture, Wiley-Blackwell, 2012:
           "Vizing's bound was discovered independently by Ram Prakash Gupta during his Ph.D. studies, mostly at the Tata Institute of Fundamental Research in Bombay, 1965–1967, supervised by Sharadchandra Shankar Shrikhande, and stimulated by Claude Berge. Also Gupta's proof was based on a variation of the fan idea (discovered independently by Gupta), and it was extended to locally bounded infinite graphs i.e. infinite graphs with a finite maximum degree."       
    (Gupta's Wikipedia entry explains that although his PhD research was unofficially directed by Shrikhande his official supervisor was C.R. Rao, at the Indian Statistical Institute, Calcutta, and their Math Geneology entries support this. However, Rao was a statistician while Shrikhande was a combinatorialist.)
  2. The 'fan idea' is the basis of textbook proofs of Vizing's theorem (but not the proof from Schrijver which I have chosen to link to) and extends to prove the generalisation to graphs with multiple edges: \(X'(G)\leq \Delta +m\), where \(m\) is the maximum edge multiplicity of \(G\). A good account is here.

Theorem no. 213: The 6-Circles Theorem
  1. Original source for this theorem: Evelyn, C. J. A., Money-Coutts, G. B. and Tyrrell, J. A., The Seven Circles Theorem and Other New Theorems, Stacey International, 1974. The theorem in question appears as a section called "A Theorem about a triangle and six circles", pp. 49–58, but I have not seen the volume. There is a review here (paywall; but you can see the first page). The generalisation published by Tyrrel and Powell is J. Tyrrell and M. Powell, "A theorem in circle geometry", Bull. Lond. Math. Soc., Vol. 3, Issue 1, 1971, pp. 70–74; online (paywall).
  2. This theorem is sometimes referred to as the Money-Coutts Theorem although it is not clear why Money-Coutts deserves more credit than Evelyn or Tyrrell. The name 'Six Circles' is a bit ambiguous: cut-the-knot, for instance, has three quite distinct theorems which qualify for the name (located here, alphabetically).
  3. Although Tyrrell has been described as a 'professional' and Evelyn and Money-Coutts as 'amateurs', an obituary of Evelyn (by Tyrrell) appears in Bull. London Math. Soc., vol. 9, no. 3, 1977. He published professional-level work during the 1930s and then again in the 1960s.
  4. A generalisation by Serge Tabachnikov in a different direction from that discussed in this theorem description, namely from triangles to n-gons, is given in "Going in Circles: Variations on the Money-Coutts Theorem", Geometriae Dedicata, 80, 2000, 201–209, online (paywall); reprint (June 2024; scroll down under 'Papers').
  5. As well as the animation linked from the theorem page there is this posted to twitter by Andrew Peason.

Theorem no. 214: A Theorem on Maximal Sum-Free Sets in Groups
  1. Original source for this theorem (which is the weblink from the theorem page): Michael Giudici and Sarah Hart, "Small maximal sum-free sets", Electronic J. Comb., Vol. 16, Issue 1, 2009, Article R59; online.
  2. Finite groups do not necessarily have large sum-free sets: W.T. Gowers, "Quasirandom groups", Combinatorics, Probability and Computing, Vol. 17, Issue 3, 2008, pp. 363–387; online (paywall; arxiv).
  3. Extremal problems concerning sum-free sets in abelian groups are the subject of a blog entry by Terence Tao.
  4. This theorem gets a neat description in the context of Sarah Hart's other mathematical activities here at Gödel's Lost Letter.

Theorem no. 215: Wedderburn's Little Theorem
  1. Original source for this theorem: J. H. Maclagan-Wedderburn, "A theorem on finite algebras", Transactions of the American Mathematical Society, Vol. 6, Number 3, 1905, pp. 349–352; online. Regarding the original discovery and proof of Wedderburn's theorem, Karen Parshall is the authority: “In Search of the Finite Division Algebra Theorem and Beyond: Joseph H. M. Wedderburn, Leonard E. Dickson, and Oswald Veblen”, Archives Internationales d’Histoires des Sciences, vol. 35 (1983), pages 274–299 (not online, that I can find). There is a nice exploration of one aspect by Michael Adam and Birte Julia Mutschler: "On Wedderburn's theorem about finite division algebras"; paper 99 here.
  2. Multiplication in the quaternions is described in the description of Moufang's Theorem; you can check that the given multiplication table for the Dickson near-field of order 9 is identical to quaternion multiplication under the isomorphism: $$\left(\begin{array}{rrrrrrrr} 1 & a & b & c & d & e & f & g \\ 1 & -1 & i & j & -k & -i & k & -j \end{array}\right)$$ whereby the multiplication is seen to be almost commutative in the sense that the table is skew symmetric.
  3. Zinovy Reichstein has drawn my attention to an uncomfortable but unavoidable footnote: an elegant, one-page, group-theoretic proof of Wedderburn's Little Theorem was published by the so-called Unabomber, Ted Kaczynski, while a PhD student at the University of Michigan. A reference can be found in this bibliography resource of John D Bullough.
  4. One-page proofs continue to appear. E.g. John Schue, "The Wedderburn Theorem of Finite Division Rings", Amer. Math. Monthly, Vol. 95, No. 5 (May, 1988), pp. 436-437 (using properties of field extensions); Nicolas Lichiardopol,"A New Proof of Wedderburn's Theorem", Amer. Math. Monthly, Vol. 110, No. 8 (Oct., 2003), pp. 736-737 (ring theory, exploiting, like Kaczynski's proof, an initial lemma from number theory).
  5. It long remained an intriguing circumstance that Wedderburn's theorem gave an algebraic proof that Desargue's theorem implies Pappus's for finite projective planes, and that no geometric proof was known (see, e.g., Peter Cameron's Projective and Polar Spaces, chapter 2, page 23). John Bamberg and Tim Penttila have resolved the issue by providing a geometric proof of Wedderburn, "Completing Segre's proof of Wedderburn's little theorem", Bull. Lond. Math. Soc., vol. 47, no. 3, 2015, pp. 483–492; preprint. (Additionally, the paper is an excellent source on Wedderburn's theorem generally.)

Theorem no. 216: Irrationality of Circumference of Unit Circle
  1. Original sources for this theorem:
    1. Lambert, Johann Heinrich, "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques", Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin, 17, 1768, pp. 265–322; online (facsimile, it is reproduced in J. Lennart Berggren, Jonathan M. Borwein and Peter B. Borwein, PI: A Source Book, 3rd edition, Springer-Verlag, New York, 2004, followed by a translation into English of the part relating to Lambert's irrationality proof. They give the date for Lambert's first announcement of his proof as 1766, but published in 1770 in "Vorläufige Kenntnisse für die, so die Quadratur und Rectification des Circuls suchen", Beyträge zum Gebrauche der Mathematik und deren Anwendung, Berlin, 1770, 140–169; online (facsimile by Göttinger Digitaisierungszentrum).
    2. Charles Hermite, "Extrait d'une lettre de Mr. Ch. Hermite à Mr. Borchardt", Journal für die reine und angewandte Mathematik, 76, 1873, pp. 342–344; online (facsimile by DigiZeitschriften), he had laid the groundwork in an earlier article in the same volume "Extrait d'une lettre de Monsieur Ch. Hermite à Monsieur Paul Gordan", pp. 303–311.
  2. A more detailed account of Lambert's irrationality proof is given at here at math.stackexchange. Wikipedia has a page on the proof of irrationality of Pi. Niven's famous 1-page proof (linked from the theorem page) is given a very nice 'reading' by Timothy Y. Chow in "A well-motivated proof that pi is irrational".
  3. Featured in Math Scholar's thread Simple proofs of great theorems.

Theorem no. 217: Taylor's Theorem
  1. An annotated English translation of Brook Taylor's Methodus Incrementorum Directa & Inversa can be found here at Ian Bruce's invaluable
  2. An alternative justification for Hugh Worthington's Rule is given by Colin Beveridge here. The explanation illustrating theorem no. 217 is by Tony Forbes, M500 magazine, issue 260, 2014, p. 17. He observes that using degrees instead of radians allows an even better approximation: \(\displaystyle \tan^{-1}\frac{a}{b}\approx \frac{172a}{b+2c}\) where \(c=\sqrt{a^2+b^2}\).
  3. A step-by-step proof of the Lagrange remainder form of Taylor's theorem is given by Gowers here.

Theorem no. 218: The Riemann Rearrangement Theorem
  1. Riemann's habilitation thesis "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" was published posthumously in 1867. A facsimile can be found here and the text is transcribed here. An English translation is available although it may be out of print. Riemann's habilitation work is discussed in detail in Detlef Laugwitz (transl. Abe Shenitzer), Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics, Birkhauser, 2nd printing, 2008. A French translation is here (§1–§8) and here (§9–§13) (presumably the one by Darboux and Houel, c.f. these notes, although no credit is given).
  2. It seems worthwhile to give an English translation of Riemann's proof of his rearrangement theorem (from §3 of his thesis):
           "In Crelle’s Journal in January 1829 a memoir by Dirichlet appeared in which rigorous conditions were established for representing, by trigonometric series, functions which are integrable and which do not possess infinitely many maxima or minima.
            "He discovered the correct path to follow to solve this problem by consideration of the fact that infinite series fall into two classes according to whether or not they remain convergent when all their terms are made positive. In the first class, the terms may be permuted in an arbitrary manner; whereas in the second class, the value of the series depends on the ordering of the terms. Indeed, if one denotes, in a series of the second class, the positive terms by $$a_1,a_2,a_3,\ldots,$$ and the negative terms by $$-b_1,-b_2,-b_3,\ldots,$$ it is clear that \(\sum a\), and similarly \(\sum b\), must be infinite; for if both sums were finite then the series would still be convergent on giving all terms the same sign; if just one of the sums where infinite, then the series would diverge. It is now clear that the series, if its terms are placed in a suitable order, may take an arbitrary given value \(C\); for if one takes alternately the positive terms of the series until its value exceeds \(C\), and then the negative terms until the value falls below \(C\), the difference between this value and \(C\) will never exceed the value of the term immediately preceeding the most recent change of sign. Now the \(a\) values, and similarly the \(b\) values must eventually become infinitesimally small as their indices increase, and thus the differences between the series sum and \(C\) must also become infinitesimally small, as one extends the series sufficiently long, which is to say that the series converges to \(C\).
            "It is only series of the first class which are amenable to the laws governing finite sums; only they may be considered as the collection of their terms; those of the second class may not be so considered: a circumstance which was missed by mathematicians of the last century, in the main because series which extend according to ascending powers of a variable belong, generally speaking (which is to say, with the exception of certain exceptional values of that variable), to the first class. "
  3. Regarding the rearrangements of Leibniz's series given in figure A of the theorem description, it is remarkable that closed forms may be given to their sums (allowing for special functions). Thus for the highest valued rearrangement shown (approx. 0.95868) we have (thanks to Maple): $$\sum_{k=0}^{\infty}\left(\frac{1}{8k+1}+\frac{1}{8k+5}-\frac{1}{4k+3}\right)=-\frac14\gamma-\frac34\ln 2+\frac{1}{16}\tau-\frac18\Psi\left(\frac18\right)-\frac18\Psi\left(\frac58\right),$$ where \(\gamma\) is the Euler-Mascheroni constant, \(\tau\) is circumference of unit circle, and \(\Psi\) is the digamma function (the slope of the log of the gamma function).
  4. The alternating harmonic series provides an even more fascinating example of Riemann's theorem in the hands of Larry Riddle in this article which originally appeared in Kenyon Mathematics Quarterly, vol. 1, no. 2 (1990), 6–21. It is also given an animation by here by CindyJS. This Boise State University dissertation by Monica Josue Agana is a rich source of examples and related theorems.
  5. A variant of Riemann's theorem says that we may change signs of terms in a conditionally convergent series to achieve any sum. I'm not sure of the origins of this observation. It is mentioned in "Almost sure limit sets of random series" (pdf preprint) by Pete L Clark and is treated in more depth in Teresa Bermúdez and Antonio Martinón, "Changes of signs in conditionally convergent series on a small set", Applied Mathematics Letters, Vol. 24, Issue 11, 2011, pp. 1831–1834; online. A spectacular example, again involving the harmonic series, is described in William Dunham, "Euler's miracle", Euleriana, Vol. 1, Issue 2, 2021, pp. 172–180; online.
  6. In another version, Riemann's theorem tells us that a series is absolutely convergent if and only if every rearrangement converges. This becomes a test for absolute convergence, in principle, if it can be shown that a finite number of convergent rearrangements is enough. This proposed 'rearragement number' is the object of study by Andreas Blass, Will Brian, Joel David Hamkins, Michael Hardy and Paul Larson, see also this arxiv article which also has interesting comments on the history of Riemann's theorem (published subsequently: Trans. Amer. Math. Soc., Vol. 373, Number 1, 2020, 41–69; online, but the history comments are absent).

Theorem no. 219: Integration by Parts
  1. A very attractive discussion about "striking applications of integration by parts" is ongoing at stackexchange.
  2. Ian Bruce wrote to me of his experience with his valuable project 17centurymaths: "Most of the elementary calculus material can be found in Euler's Differential and Integral Calculus books, and in fact he starts Book I on Integration with integration by parts; Ch.1 of this book is Top of the Pops in my line of business, and gets first place consistently in downloads, followed by Newton's definitions & Axioms ... Euler's work is still highly readable, and more so than others of that age and before; in fact he seems to have set the standard for generations of mathematicians to come."
  3. Ernst Hairer has provided an elegant geometrical interpretation of integration by parts, which may be viewed here.
  4. There is a nice description here by Murray Bourne of an alternative to integration by parts called the Tanzalin Method which is apparently commonly used in Indonesia. As you will see, it too can lead to infinite series!

Theorem no. 220: The Pappus–Guldin Theorems
  1. According to Andrew Leahy's article, no proof by Pappus of his theorems has been discovered and Paul Guldin gave no proof, the first known proof being supplied by Giannantonio Rocca in 1644.
  2. There is a very nice discussion of the 17th century pre-calculus debate in Chapter 5 of Amir Alexander's Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, Oneworld Publications, 2014.
  3. Peter Harremoës has drawn my attention to a little irony: the surface area of the torus is usually given in terms of its major radius \(R\) and minor radius \(r\), as \(\tau^2rR\). You can instead use inner radius \(a=R-r\) and outer radius \(A=R+r\) and in this case surface area is given as \(\pi^2(A^2-a^2)\). Rather sneakily it is the latter, less standard, presentation which is used about 2.5 mins into this film debate on the π vs τ question as an argument that pi makes things simpler!

Theorem no. 221: The Inclusion-Exclusion Principle
  1. Attributions of Inclusion-Exclusion often include the name of Poincaré (e.g. 'formule du crible de Poincaré') and this seems a bit obscure. In Encyclopaedia of Mathematics, Supplement III: 3 (ed. Michiel Hazewinkel, Springer, 2002) this attribution carries a reference to Poincaré's book Calcul des probabilités, Gauthier-Villars, 1896, and this book may have been influential in making the principle widely known in France.
  2. Inclusion-Exclusion may be generalised in several ways. A good example is given here by Stewart Weiss; probably the most famous is due to Gian-Carlo Rota and is described by Peter Cameron in Lecture 9 of this course. Rota's original article can (and should!) be read here.
  3. Stewart N. Ethier has contributed the following: "the expected number of boxes needed for a full set of coupons has the nice formula \(\displaystyle n\!\left(1+\frac12+\frac13+\ldots+\frac{1}{n}\right)\), which either can be derived from [the inclusion-exclusion formula] (via Theorem 1.4.2 of my book [The Doctrine of Chances: Probabilistic Aspects of Gambling]), or can be derived directly by writing the random variable of interest as the sum of \(n\) independent geometric random variables with success probabilities \(1, (n-1)/n, (n-2)/n, \ldots, 1/n\) (and using the fact that a geometric(\(p\)) random variable has mean \(1/p\))."
  4. It should perhaps be recorded that the number of surjections from \(m\) objects onto \(n\) is directly expressable in terms of Stirling numbers of the second kind, \(S(m,n)\) being the number of ways to partition \(m\) objects into \(n\) nonempty subsets: thus \(S(m,n)\) counts all ways to choose which objects will map to the same image point, and then \(n!S(m,n)\) incorporates the order in which we choose the image points.
  5. A nice application of Inclusion-Exclusion is to vary the standard combinatorial question "How many ways to put n indistinguishable balls into k distinguished boxes" by adding "so that no box gets more than C balls". An excellent explanation of the answer is given here by Brian M. Scott.

Theorem no. 222: Faulhaber's Formula
  1. For a general survey of properties of Bernoulli numbers, Pascal Sabah and Xavier Gourdon's article "Introduction on Bernoulli's numbers" is excellent, see here under "Miscellaneous". There is a pdf version here (March 2024).
  2. There is a fascinating investigation of what Faulhaber achieved and how he achieved it in Knuth, D.E., "Johann Faulhaber and sums of powers", Math. Comp., Vol. 61, No. 203, 1993, pp. 277–294; online.
  3. Seki's discovery of the Bernoulli numbers is described in Silke Wimmer-Zagier and Don Zagier's chapter in Eberhard Knobloch, Hikosaburo Komatsu and Dun Liu (eds.), Seki, Founder of Modern Mathematics in Japan: A Commemoration on His Tercentenary, Springer, 2013. Online here (March 2024). There is some further information at the beginning of Tsuneo Arakawa, Tomoyoshi Ibukiyama and Masanobu Kaneko, Bernoulli Numbers and Zeta Functions, Springer, 2014.
  4. Faulhaber's formula can be elegantly presented in terms of the inverse of Pascal's triangle minus one, see this presentation.

Theorem no. 223: Tutte's Golden Identity
  1. Tutte's Golden Inquality appears in W.T.Tutte, "On chromatic polynomials and the golden ratio", J. Comb. Theory, Vol. 9, Issue 3, 1970, pp. 289–296; online, inspired by his investigations with Gerald Berman reported in G.Berman and W.T.Tutte, "The golden root of a chromatic polynomial", J. Comb. Theory, Vol. 6, Issue 3, 1969, pp. 301–302; online. The Golden Identity appears in W.W. Tutte, "The golden ratio in the theory of chromatic polynomials", Ann. New York Acad. Sci., Vol. 175(1), 1970, pp. 391–402; online (paywall)..
  2. There is a description of Tutte's work on graph polynomials in the excellent obituary by Arthur Hobbs and James Oxley which appeared in Notices Amer. Math. Soc., Vol. 51, No. 3, 2004, pp. 320–330; online (but this is full-issue pdf download, so it's big!)
  3. Calvin McPhail-Snyder has drawn my attention (via Twitter) to a proof of this identity from mathematical physics! "One way to prove this", he says, "involves quantum algebra! It turns out there are nontrivial connections between representation theory and chromatic polynomials". Paul Fendley and Vyacheslav Krushkal, "Tutte chromatic identities from the Temperley–Lieb algebra", Geometry & Topology, Vol. 13, Issue 2, 2009, pp. 709–741; online.

Theorem no. 224: Green's Theorem
  1. The original source for this theorem is "An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism" which Ralf Stephan has transcribed here. It was published by Green at his own expense but received little attention until William Thomson (later Lord Kelvin) rediscovered it and arranged for its publication in Crelle's Journal in the 1840s.
  2. Paul Nahin, whose Inside Interesting Integrals is the recommended further reading for this theorem, also writes interestingly about its background in Chapter 7 of An Imaginary Tale: The Story of \(\small\underline{\sqrt{-1}}\), Princeton University Press, 1998.
  3. An important exhibition commemorating Green was held at the University of Nottingham in the autumn of 2014 and the curator's blog post is very interesting. Much of historical as well as scientific interest can be found in Lawrie Challis and Fred Sheard, "The Green of Green Functions", Physics Today, 56, 12, 2003, 41–46; online (paywall; a reprint here,July 2023).

Theorem no. 225: The Spherical Law of Cosines
  1. You can find latitudes and longitudes of cities, and compute great-circle distances between them here.
  2. Pat Ballew has drawn my attention to a dual version of this theorem, relating three angles A, B, C and one side, say c: $$\cos(C)=-\cos(A)\cos(B)+\sin(A)\sin(B)\cos(c),$$ (this is referred to by Van Brumellen in Heavenly Mathematics as the "Law of Cosines for Angles").
  3. A mnemonic of Napier for spherical trigonometry (also from Van Brumellen's book, I think) has been nicely summarised by John D. Cook here.
  4. plus magazine have provided a very nice introduction to longitude and latitude.

Theorem no. 226: Wolstenholme's Theorem
  1. Original sources for this theorem:
    1. Babbage, Charles, "Demonstration of a theorem relating to prime numbers", The Edinburgh Philosophical Journal, Vol. 1, 1819, pp. 46–49; online.
    2. Wolstenholme, Joseph, "On certain properties of prime numbers", The Quarterly Journal of Pure and Applied Mathematics, Vol. 5, 1862, pp. 35–39; online.
  2. The converse of Wolstenholme's Theorem, that \({2n-1\choose n-1}\not\equiv 1 \hspace{-0.05in}\mod n^3\) for all composite values of \(n\), is a famous open question. It is known to be true for even \(n\) and for all \(n<10^9\). See for example, Vilmar Trevisan and Kenneth Weber, "Testing the converse of Wolstenholme's theorem", Matemática Contemporânea, 21 (2001), 275–286; online. Recent progress on the conjecture is described in Saud Hussein, "A note on the converse of Wolstenholme’s Theorem", Integers, vol. 18 (2018), Paper No. A94; online, where it is attributed to James P. Jones.
  3. A generalisation of Wolstenholme due to James Whitbread Lee Glaisher in 1900, says that \({kp-1\choose p-1}\equiv 1 \hspace{-0.05in}\mod p^3\) for any prime \(p\geq 5\) and any positive integer \(k\). In this case the converse does not hold. Small counterexamples exist for nonprimes \(p=4,9,25\), for example (thus \(p=4,k=33\) gives \({131\choose 3}\equiv 1 \hspace{-0.05in}\mod 64\)).
  4. A proof of Babbage's \(p^2\) prototype of the theorem is given here.
  5. More on the 'harmonic numbers' context for Wolstensholme can be found in Zhi-Wei Sun, "Arithmetic theory of harmonic numbers", Proc. Amer. Math. Soc., 140, no. 2, 2012, 415–428, online.

Theorem no. 227: Cauchy's Theorem in Group Theory
  1. A thorough analysis of the origin of Cachy's 1845 'Mémoire sur les arrangements ...', in which his theorem is asserted, has been given by Peter M. Neumann, 'On the date of Cauchy's contributions to the founding of the theory of groups', Bulletin of the Australian Mathematical Society, vol. 40, 1989, 293–302; online.
  2. James H. McKay's proof of Cauchy's theorem was published as "Another proof of Cauchy's group theorem", American Math. Monthly, Vol. 66, No. 2, 1959, p. 119; online (paywall; copies are not hard to find online, e.g. here, July 2024).
  3. Incidental to the choice of \(D_{10}\) to illustrate this theorem, is a corollary to Cauchy's theorem that any group of order twice an odd prime is either cyclic or dihedral. This is Prop. 3.34 in our recommended book, Smith and Tabachnikova's Topics in Group Theory, Springer, London, 2000.
  4. Michael Meo claims, in his article "The mathematical life of Cauchy's Group Theorem", Historia Mathematica, vol. 31, issue 2, pp. 196–221; online, that Cauchy's proof of his theorem contains an 'egregious error' and that a subsequent attempt by Dedekind in the 1850 is also incomplete. This would suggest that the first complete proof of the theorem comes as a corollary to its own generalisation (Sylow's theorems of 1872). I corresponded with Peter M. Neumann about this and he disagreed: "Ambiguity and slips aside, the fact is that Cauchy's ideas are fundamentally sound". Meo has talked about his investigations on Quora.
  5. This theorem has an extension to certain non-prime divisors of group orders: Peter J. Cameron, David Craven, Hamid Reza Dorbidi and Benjamin Sambale, "Minimal cover groups"; arxiv. Peter Cameron has an overiew on his blog.

Theorem no. 228: Fisher's Inequality
  1. Original source for this theorem: R.A. Fisher, "An examination of the different possible solutions of a problem in incomplete blocks", Annals of Eugenics, vol. 10, 1940, pp. 52–75; online. Bose's paper containing his short proof of the inequality is R. C. Bose, "A note on Fisher's inequality for balanced incomplete block designs", Ann. Math. Statist., Vol. 20, Number 4 (1949), pp. 619–620; online.

Theorem no. 229: Poncelet's Porism
  1. An elegant and (relatively) simple proof is given by Lorenz Halbeisen and Norbert Hungerbühler in "A Simple Proof of Poncelet’s Theorem (on the occasion of its bicentennial)", American Mathematical Monthly, in press, (preprint, November 2023). For a modern proof, from algebraic geometry, see this by David Speyer.
  2. A bicentennial survey of past and current research into Poncelet's theorem is given by Vladimir Dragović and Milena Radnović in "Bicentennial of the Great Poncelet Theorem (1813–2013): Current advances", Bull. Amer. Math. Soc., Vol. 51, No. 3, 2014, 373–445.
  3. The example given of a quadrilateral inscribed in and circumscribing two eillipses is a cheat! The parameters were chosen by trial and error to give an adequate illustration: outer ellipse is centered on the origin and inclined at \(\tau/8\) to \(x\) axis; major radius \(a = 9.3\), minor radius \(b = 4.1\); inner ellipse is centered at \((1.05046,1.3)\) and has no inclination; major radius \(c = 4.0448\), minor radius \(d = 3.22\).
  4. Some very good notes by Tony Forbes are available here (about 1.5MB), including detailed instructions for creating genuine examples for all combinations of conics (not just ellipses) and also giving a brief description of the link to elliptic curves.
  5. Poncelet's Porism is also known as his Closure Theorem for reasons made beautifully clear in Jonathan King, "Three Problems in Search of a Measure", The American Mathematical Monthly, Vol. 101 (1994), pp. 609–628, online here. We find in the same article that the theorem is intimately related to Gelfand's Question!
  6. There is a French version of this theorem description. If you read French the weblink from the French version is a very fine popular account of Poncelet's porism.

Theorem no. 230: Ore's Theorem in Graph Theory
  1. Original source for this theorem: Ore, Ø, "Note on Hamilton circuits", American Mathematical Monthly, 67 (1), 1960. p. 55; online (paywall).
  2. Bondy's short proof appears in "Short proofs of classical theorems", J. Graph Theory, Vol. 44, No. 3, 2003, 159–165; online (paywall; there was a pdf copy here, December 2023). The algorithmic interpretation given here is similar in spirit to an adaptation of Ore's original proof by E.M. Palmer, "The hidden algorithm of Ore's theorem on Hamiltonian cycles", Computers & Mathematics with Applications, Vol. 34, No. 11, 1997, 113–119; online.
  3. Apart from the left-most, the graphs illustrating this theorem were generated in Maple. However I manually replaced the vertices in order to get the permuted numberings (I was too lazy to work out how to get Maple to do this).

Theorem no. 231: Sophie Germain's Identity
  1. The correspondence of Sophie Germain is online at the Bibliothèque nationale de France via gallica. The letter reproduced here is located by searching for '9118' under 'Manuscripts'.
  2. Leonard Dickson's History of the Theory of Numbers, Volume I: Divisibility and Primality can be read online here courtesy of The references to Euler and Germain are on pages 381 and 382, respectively.
  3. The letter from Euler to Goldbach cited by Dickson can be read at the Euler Archive (August 28 in the 1742 correspondence). Not every letter from Euler to Goldbach of that year is online but it seems clear that this is the one which Dickson intends.

Theorem no. 232: The Riemann Explicit Formula
  1. Original source for this theorem is Riemann's 1859 paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse". The paper is so famous as to have its own Wiki page!
  2. An excellent account of the explicit formula, starting from scratch, is this at by Jørgen Veisdal.
  3. The relationship between the distribution of primes and (logarithmic) spirals has a rich history. A good example is given by Matthew Watkins here; the idea of spotting patterns in prime spirals goes back to (at least) Ulam in 1963. A nice variant by Edmund Harriss can be found here, and there is a very elegant 3D conical spiral by Dan Bach here.
  4. The weblink for this theorem by Matthew Watkins offers a very clear account of Riemann's formula in the Chebyshev \(\psi\) function version (as preferred in the Wikipedia entry for example). He has much more on the Riemann Hypothesis here; and indeed, the whole prime distribution story is the subject of his trilogy of books (with illustrator Mark Tweed) Secrets of Creation.
  5. There is a famous Bonn University inagural lecture by Don Zagier on the subject of Riemann's prime counting function which can be found in English translation here (pdf, 2.5MB).
  6. Much intriquing recent commentary on the Riemann Hypothesis, including an extended essay by Alaine Connes, can be found starting at this post from Not Even Wrong.
  7. Andrew Odlyzko provides tables of zeros of the Riemann zeta function via his home page, in case you want to experiment with Riemann's formula. (Thanks to David Bernier for this. His test using the first 10^5 zeros is on twitter here).

Theorem no. 233: The Circle Area Theorem
  1. The Archimedes proof of this theorem still qualifies as a textbook one, e.g. here, although calculus variants of the \(\int_0^{r\tau}\frac12r\mbox{dt}\) variety are presumably more respectable from a modern perspective.

Theorem no. 234: A Generalised Hlawka Inequality
  1. The original source for Hlawka's Inequality is Hans Hornich, "Eine Ungleichung für Vektorlängen", Mathematische Zeitschrift, Volume 48, Issue 1, (1942/43), 268–274. It may be viewed online here thanks to the Göttinger Digitalisierungszentrum. (Hornich says merely "For the special case m = 1, n = 2, Herr Hlawka has given me a purely algebraic proof..." so that the name Hornich–Hlawka as preferred by seems more appropriate. However 'Hlawka' seems to be the generally adopted nomenclature.)
  2. The original result of Dragomir Djoković appeared in "Generalizations of Hlawka's inequality", Glasnik Matematičko-Fizicki i Astronomski, Ser. II, vol. 18, (1963), issue 3, 169–175; online (direct 1.4MB pdf download, only Glasnik Matematički, the successor to Glasnik Matematičko-Fizicki i Astronomski appears to be fully online ). D.M. Smiley & M.F. Smiley's paper is "The polygonal inequalities", Amer. Math. Monthly, Vol. 71, No. 7 (1964), 755–760; online (paywall). In both papers something more general is proved for the sequence of \(n\) vectors: that for \(2\leq k<n\) we have $$d_k\leq {n-2\choose k-2}d_n+{n-2\choose k-1}d_1,$$ using the notation in the statement of the theorem. The inequality as stated is found by summing over \(k\). Djoković and Smiley & Smiley also gave conditions for equality.
  3. A nice derivation of Hlawka's Inequality from the Ptolomeic Inequality is given by Alice Simon and Peter Volkmann in Annales Mathematicae Silesianae, Vol. 9, 1995, 137-140. The article is online here.

Theorem no. 235: A Theorem on Modular Fibonacci Periodicity
  1. The period lengths of the modulo-reduced Fibonacci sequences continue to be the subject of intensive research. A good recent (2012) example is here. They also go by the name of Pisano periods. (after Leonardo Pisano aka Fibonacci). They are sequence A001175 at
  2. Of particular interest is the so-called 'Wall's Question': for a prime \(p\), is it possible that the period mod \(p\) and mod \(p^2\) should be equal? Such a prime is termed a Wall–Sun–Sun prime. The question has links to the Fermat's Last Theorem via Germain's Theorem. See, Klaška, J., "Criteria for testing Wall's question", Czechoslovak Mathematical Journal, vol. 58 (2008), issue 4, pp. 1241-1246, online.
  3. D.D. Wall's paper is "Fibonacci series modulo m", The American Mathematical Monthly, Vol. 67, No. 6, 1960, 525–532; online (paywall). Covering rather the same material is a roughly contemporary paper, "The Fibonacci matrix modulo m" by the Caltech physicist David W. Robinson. This was published in the 2nd ever issue of Fibonacci Quarterly and this is free online here.
  4. The papers of Morgan Ward on linear recurrences are a good source of information on modular periodicity. They appears in Transactions of the American Mathematical Society and are free online here (1931) and here (1933). The main result from 1931 is that if \(m\) has prime decomposition \(p_1^{a_1}p_2^{a_2}\cdots p_n^{a_r}\) then period length mod \(m\) is equal to the LCM of period lengths mod \(p_i^{a_i}, i=1,\ldots, r\).

Theorem no. 236: Kemeny's Constant
  1. Original source for this theorem, as indicated on the theorem page, is John G. Kemeny and J. Laurie Snell, Finite Markov Chains, Van Nostrand, Princeton, NJ, 1960, Chapter 4, section 4.10 (this will have changed in the new Springer edition found in our bibliography).
  2. The directed graph modelling Alice's casino is a finite automaton which finds the remainder of an input binary number (with \(H=1\) and \(T=0\)) mod 8. Doubling appends a zero to a binary number; adding 1 thereafter appends instead a 1, so the action of the automaton is the same as step (3) in the casino game. Another example of such an automaton illustrates The Pumping Lemma. There is a nice non-binary take on this (for mod 7, but see comments) by David Wilson guesting at Tanya Khovanova's Math Blog.
  3. If you operate a casino and would like to compete with Alice using Kemeny's constant, Tony Forbes has offered a neater and more intuitive version of her game (750KB pdf, see p. 20) in M500 magazine.
  4. There is an interesting contrast between \(K\), the expected time to reach the stationary distribution, and the probability of reaching the distribution in fewer than \(K\) steps. The latter will be greater than \(1/2\) (to compensate for the occasional long runs). So Bob will often find himself losing money to Alice but he will be seduced by the prospect of a long run, just as in any lottery you hardly ever win anything but play for the prospect of a jackpot.
  5. An interesting question from Piers Myers is: can other averages for time to stationary distribution, e.g. median, also have constant values for Markov chains? For the 8-state chain used here the answer for median values appears to be, roughly, yes, according to simulations: 2500 runs from each starting state to a target state selected u.a.r. gave median times 5,4,5,4,4,4,4,5. But Piers points out that this cannot hold in general: the chain \(\left(\begin{array}{cc} 0 & 1 \\ 1/100 & 99/100\end{array}\right)\) has stationary distribution \((1/101, 100/101)\); Kemeny's constant is \(100/101\); but median time to reach stationary distribution is 1 from state 1 and 0 from state 2.
  6. A very nice Markov chain animation provided by might be of use for 'visualising' Kemeny's constant for small chains.

Theorem no. 237: Sylvester's Catalecticant
  1. The last section of "On the Alexander-Hirschowitz Theorem" by Maria Chiara Brambilla and Giorgio Ottaviani is very good on the history of Waring's problem for forms. Also very good are the opening pages of Power Sums, Gorenstein Algebras, and Determinantal Loci, Springer 1999, by Anthony Iarrobino and Vassil Kanev. This paper by Zach Teitler and Alexander Woo gives a good picture of the current state of play.
  2. Zach Teitler provided me with much help in getting to grips with the subtleties of Sylvester's work to the point where I felt it worth quoting his comments verbatim, in the form of some additional notes.
  3. A classic paper in the theory of binary forms is Joseph P. S. Kung and Gian-Carlo Rota, "The invariant theory of binary forms", Bull. Amer. Math. Soc. (N.S.), Vol. 10, No. 1, (1984), 27–85, online here.
  4. Bruce Reznick draws my attention to the charming description of Sylvester on his work on binary forms:
           "I discovered and developed the whole theory of canonical binary forms for odd degrees, and, as far as yet made out, for even degrees too, at one evening sitting, with a decanter of port wine to sustain nature's flagging energies, in a back office in Lincoln's Inn Fields. The work was done, and well done, but at the usual cost of racking thought—a brain on fire, and feet feeling, or feelingless, as if plunged in an ice-pail. That night we slept no more."       
    (which can be found on p. xxiv of The Collected Mathematical Papers of James Joseph Sylvester: Volume 4, 1882-1897. Bruce observes, aptly I think, "If he had been known as a writer, rather than as a mathematician, this would be a famous quote!" (Of course Sylvester was proud of, although not remembered for, his poetry, see Chapter 8 of Karen Hunger Parshall's, James Joseph Sylvester: Jewish Mathematician in a Victorian World, The Johns Hopkins University Press, 2006.) By the way, an excellent slideshow by Bruce Reznick on representations of forms can be found here (1.2MB pdf).
  5. Catalecticant matrices are also known as Hankel matrices. The appropriate Wiki page gives a way in to this side of the story.

Theorem no. 238: Euler's Even Zeta Formula
  1. Euler discovered his formula in 1739 and it appeared in De seriebus quibusdam considerationes which can be read in the original Latin and in German or English translation as entry E130 at the Euler Archive. The role of the Bernoulli numbers was made explicit by Euler in his 1755 classic textbook Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, volume 1 which is entry E212.
  2. Max Woon (publishing as See Chin Woon) gave his binary tree generation of the sequence of Bernoulli numbers in "A Tree for Generating Bernoulli Numbers", Mathematics Magazine, Vol. 70, No. 1, 1997, 51–56; online (paywall). A generalisation to arbitrary complex sequences using elementary methods has been given by Petr Fuchs: "Bernoulli numbers and binary trees", Tatra Mountain Mathematical Publications, 20 (2000), 111–117, online (postscript) here.
  3. Euler's work on \(\zeta(3)\) and related series is described in William Dunham, "Euler and the cubic Basel problem", The American Mathematical Monthly, Vol. 128, Issue 4, 2021, pp. 291–301; online (paywall; winner of a 2022 Paul R. Halmos-Lester R. Ford award, so should become free to read at some point). Thanks to Arthur Newlands @ArthurNewlands for telling me about this).
  4. Although there may be no direct calculations of \(\zeta(2n+1)\) in terms of Bernoulli numbers, there are infinite series formulae. The most famous approach is probably Ramanujan's, see Section 3 of this nice overview of Ramanujan's notebooks by Bruce C. Berndt: the approach is given a thorough workout by Marc Chamberland and Patrick Lopatto in "Formulas for Odd Zeta Values and Powers of \(\pi\)", Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.5, online here. The best-known formula is a special case of Ramanujan's first discovered by Mathias Lerch in 1901: if \(n\) is odd then $$\zeta(2n+1)=\frac12\tau^{2n+1}\sum_{k=0}^{n+1}(-1)^{k+1}\frac{B_{2k}}{(2k)!}\frac{B_{2n+2-2k}}{(2n+2-2k)!}-2\sum_{k=1}^{\infty}\frac{k^{-2n-1}}{e^{k\tau}-1},$$ whereby \(\zeta(2n+1)\), for large, odd \(n\), is very close to a rational multiple of \(\tau^{2n+1}\).

Theorem no. 239: Kuratowski's 14-Set Theorem
  1. This theorem was apparently made famous when it featured as an exercise in John L. Kelley's General Topology (first published by Van Nostrand, 1955). It again appears as an exercise in James Munkres' Topology (chapter 2, section 17, exercise 21) where it is also required to find a set which generates the maximum of 14. The answers to Munkres' exercises are helpfully provided by Jesper Michael Møller here (November, 2023).
  2. The particular 14-set chosen for my illustration was generated with the help of Mark Bowron's fun interactive diagram.
  3. There is actually no need to state this as a theorem about topological spaces. P. C. Hammer, "Kuratowski’s closure theorem", Nieuw Archief voor Wiskunde, 7 (1960), 74–80, has shown that Kuratowski's theorem remains true for a more abstract closure operator defined set-theoretically. There is a nice discussion by Jeffrey Shallit and Ross Willard here.
  4. Joshua Zelinsky has brought to my attention an attractive paper of David Sherman generalising Kuratowski in terms of number of operators and number of sets: David Sherman, "Variations on Kuratowski's 14-set theorem", American Math. Monthly, Vol. 117, no. , 2010, pp. 113–123; online (paywall; there was a reprint on Sherman's home page, November, 2023).
  5. A good blog post by David Richeson. For French speakers, this blog account of the theorem by Blogdemaths is very fine.
  6. The history of the closure operation, and Kuratowski's role in it, is very well presented in this MAA Convergence article by Nicholas A. Scoville

Theorem no. 240: The Jones Knot Polynomial Theorem
  1. Strictly speaking, our presentation of this theorem uses the 'normalised bracket polynomial'. The substitution \(x=q^{1/4}\) is used in the Jones polynomial proper (as recorded in the Knot Atlas, for example). I asked Louis Kauffman about this and he explained it as " a historical accident having to do with the fact that Jones defined the invariant in a different way (via a representation of the braid group to a Temperley—Lieb algebra) than I did by using the bracket state sum. The state sum is close to physics via ideas in statistical mechanics. The Temperley—Lieb algebra is close to physics also."
  2. The definitive published resource for relationships between knot theory and physics must be Louis Kauffman's Knots and Physics, World Scientific, 4th revised edition, 2013. Kauffman's webpage is also an essential visit, with such gems as his "New Invariants in the Theory of Knots" (an Amer. Math. Monthly write-up of his 1987 breakthrough).
  3. There is apparently no convention for orienting links when calculating the writhe of a multi-component link. However, if a link has more than one component then the orientations of individual components only changes the value of the Jones polynomial by a power of its variable. See, for example, Sandy Ganzell, Janet Huffman, Leslie Mavrakis, Kaitlin Tademy and Griffin Walker, "Unoriented links and the Jones polynomials", preprint online here.
  4. One of the biggest questions in knot theory is whether the Jones polynomial distinguishes the unknot, that is, can any knot \(K\) other than the unknot have \(J(K)=1\)? In the case of links with more than one component the Jones polynomial cannot distinguish the unlink, as shown for example by Shalom Eliahoua, Louis H. Kauffman and Morwen B. Thistlethwaite in "Infinite families of links with trivial Jones polynomial", Topology, vol. 42, no. 1, 2003, 155–169, online via Elsevier Open access. It is known, Haken's Unknot Theorem, that distinguishing the unknot is decidable, and much has been discovered about the problem, algorithmically (see the notes page for Haken's theorem); but distinguishing the unknot by an invariant, even one as expensive to evaluate as the Jones polynomial, would represent a qualitative advance in understanding.
  5. Erica Klarreich has a good short article on knot invariants in Quanta magazine.

Theorem no. 241: The Large Prime Gaps Theorem
  1. Original source for this theorem: Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Chi-Shen Tao, "Long gaps between primes", J. Amer. Math. Soc., 31, no. 1, 2018, pp. 65–105; online (paywall; AMS preprint version, 500KB pdf).
  2. Theorem under Construction iconThis is a 'theorem under construction': I hope to chart exciting developments here towards an eventual final version, which may or may not mean something approaching Cramér's 1936 conjecture.
  3. The composite sequence in our example is \(m+k\) where \(m=293357\) and \(k=1,\ldots,25\). The fact that \(Y(17)=25\) does not mean that larger \(k\) values will give primes: in fact \(293357+k\) is composite until \(k=42\). By the way, online Chinese Remainder Theorem solvers generally don't appear to accept congruences of the form \(m=-a_p\!\!\mod p\) (this by is an exception) but solving with positive \(a_p\) and then negating the answer is fine, as can be seen immediately at (Theorem 5, notes(1)).
  4. Work on long prime gaps has historically used the Jacobsthal function: \(j(n)\), for positive integer \(n\), is the smallest positive integer \(m\), such that every sequence of \(m\) consecutive integers contains an integer coprime to \(n\) (alternatively, \(j(n)\) is the maximal gap between integers coprime to \(n\)).The first thirty values A048669 are \(1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 3, 2, 2, 4, 2, 4, 3, 4, 2, 4, 2, 4, 2, 4, 2, 6,\ldots\). Ford et al's paper observes that \(Y(x)=j(P(x))-1\) (with \(P(x)\) the product of primes not exceeding \(x\)).
  5. An excellent article about Cramer's model of the prime numbers and his conjecture is this by Andrew Granville, hosted at Chance News, who also have a whole series of lectures on probabilistic number theory, with part 2 focussing on Cramér's work.

Theorem no. 242: The Pólya–Redfield Enumeration Theorem
  1. Original sources for this theorem:
    1. J. Howard Redfield, "The theory of group-reduced distributions", Amer. J. Math., Vol. 49, No. 3, 1927, pp. 433–455; online (paywall).
    2. J. Howard Redfield, "Enumeration by frame group and range groups", J. Graph Theory, Vol. 8, No. 2, 1984, pp. 205–223; online (paywall). Accompanied by a modern reading of Redfield's original 1940 submission: J. I. Hall, E. M. Palmer and R. W. Robinson, "Redfield's lost paper in a modern context", J. Graph Theory, Vol. 8, No. 2, 1984, pp. 225–240; online (paywall). The E. Keith Lloyd paper providing one of the weblinks from the theorem page, is also indispensable: "Redfield's contirubtions to enumeration", MATCH Communications in Mathematical and in Computer Chemistry, Vol. 46, 2002, pp. 215–233; online.
    3. G. Pólya, "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen", Acta Math., 68, 1937, pp. 145–254; online.
  2. Space did not allow for the evaluation of the cycle index for the group action on the edges of the tetrahedron. The calculation gives \({b}^{6}+{b}^{5}r+2\,{b}^{4}{r}^{2}+4\,{b}^{3}{r}^{3}+2\,{b}^{2}{r}^{4}+b{r}^{5}+{r}^{6}\), whence the determination that there are, up to rotational symmetry, four colourings with three red and three blue edges. The total number of colourings is \(1+1+2+4+2+1+1=12\), as was already established by using the Cauchy–Frobenius Lemma.
  3. The description of this theorem makes a simplification by going straight from a set of labels \(L\) to the formal power sums \(\sum x_i^k, x_i\in L,\) substituted into the cycle index. More properly, we should associate a weight with each label and it is power sums of weights which are substituted. Thus, for example, our red-blue edge colourings of the tetrahedron each 'choose' a subset of the edges (say, the blue edges). We can think of this as having a label 'absent' and a label 'present' with weights 1 and \(x\), respectively. And we can enumerate, say, the number of different 3-sets up to tetrahedral symmetry by substituting into the cycle index the polynomials \(1+x^k, k=1,\ldots,3\). More formally still, we can define a 'figure-counting series' \(A(t)=\sum a_it^i\) in which \(a_i\) is the number of 'figures' (labels) having weight \(i\). Then what is substituted into the cycle index are the polynomials \(A(t^k)\). This allows \(A(t)\) to be an infinite sum (with constant coefficients!). In Peter Cameron, Permutation Groups, Cambridge University Press, 1999, section 5.13, this approach gives the enumeration of \(n\)-sets up to symmetry via the figure counting series \(A(t)=t^0+t^1=1+t,\) the weights being 0 and 1.
  4. There is a lovely body of theory called 'orbital combinatorics' which combines enumeration up to both symmetry and structure. It originates in Peter J. Cameron, Bill Jackson and Jason D. Rudd, "Orbit-counting polynomials for graphs and codes", Discrete Mathematics, Vol. 308, Issues 5–6, 2008, 920–930, online here. There is a more up-to-date overview on Cameron's blog and see also this presentation.

Theorem no. 243: A Theorem of Anderson, Cameron and Preece on Groups of Units
  1. Original sources for this theorem (which are the weblinks from the theorem page):
    1. D. A. Preece and Ian Anderson , "Obtaining all or half of Un as ⟨ x ⟩ ×  ⟨ x + 1 ⟩, Integers, Vol. 12, 2012, paper #A52; online.
    2. P. J. Cameron and D. A. Preece, "Three-factor decompositions of Un with the three generators in arithmetic progression", arxiv.
  2. A good presentation of this work in context by Peter Cameron, as well as much else of relevance and interest, are linked from his blog entry on the Donald Preece Memorial Day.
  3. The fact that \(-3\) is a quadratic residue mod \(p\) for an odd prime \(p\) if and only if \(p\equiv 1 \pmod 6\) is a textbook exercise, c.f. Problems 9.3, no. 5(a) in David M. Burton, Elementary Number Theory 7th edition, McGraw-Hill, 2010.

Theorem no. 244: The LYM Inequality
  1. Original sources for this theorem (which are the weblinks from the theorem page):
    1. K. Yamamoto, "Logarithmic order of free distributive lattices", J. Math. Soc. Japan, Vol. 6, Issues 3– 4, 1954, pp. 343–353; online.
    2. L.D. Meshalkin, "A generalization of Sperner’s theorem on the number of subsets of a finite set", Theor. Probability Appl., Vol. 8, Issue 2, 1963, pp. 203–204; online (paywall)
    3. D. Lubell, "A short proof of Sperner’s theorem", J. Combin. Theory, Vol. 1, Issue 2, 1966, p. 299; online.
  2. There is a classic probabilitistic proof of the LYM inequality due to Peter Frankl, "A probabilistic proof for the lym-inequality", Discrete Math., Vol. 43, Issues 2–3, 1983, p. 325; online.
  3. How LYM is situated in the study of antichains in partially ordered sets is very well explained by Dominic Yeo in this Eventually Almost Everywhere post.

Theorem no. 245: The Alternating Series Test
  1. A good source of information on the origins of the concept of series convergence is Giovanni Ferraro, "Convergence and formal manipulation in the theory of series from 1730 to 1815", Historia Mathematica, Vol. 34, Issue 1, 2007, 62–88, online here.
  2. There are some good discussions about alternating series at for example this, and this.
  3. A wonderful compilation of proofs of divergence of the harmonic series is: Steven J. Kifowit and Terra A. Stamps, "The Harmonic Series Diverges Again and Again", The AMATYC Review, Vol. 27, No.2, Spring 2006. A preprint can be found here (April 2024, scroll down to "VIII. Articles of Interest", item 2). The first proof of divergence is attributed to Nicolas Oresme in the 14th century. More on the harmonic series via Theorem 218 Notes (3,4).
  4. A well-known occurrence of the harmonic series is in creating a large overhang when stacking overlapping blocks: the stack remains stable when the overlaps are, successively, \(1/2,1/4,1/6,\ldots\), giving a total overhang of \(\frac12 H_n\), \(H_n\) being the \(n\)-th partial sum of the harmonic series. Thus an unbounded overhang is possible. But in fact much more can be achieved, as demonstrated by Paterson and Zwick in 2007. See this prepint, for example.
  5. Mathwithbaddrawings has a lovely entry about how slowly the harmonic series diverges and the curious fact that, omitting terms containing the digit 9 restores convergence (the Kempner series).
  6. A nice example of a series which defies AST is \(\sum (-1)^nn/p_n\) where \(p_n\) is the \(n\)-th prime. The monotonic increase in \(p_n\) is too inconsistent to guarantee that the ratios \(n/p_n\) are monotonic decreasing, so AST does not apply. Erdős asked whether the series nevertheless converges and this has been answered in the affirmative by Terence Tao, conditional on a conjecture of Hardy and Littlewood. (Tao remarks that Erdős showed that the variant with numerator \(n\ln n\) is divergent.)

Theorem no. 246: Euler's Product Formula for ζ(s)
  1. The convergence of Euler's formula for real \(s>1\) is sometimes attributed to Kronecker in the 1870s. However, it seems likely that convergence issues would have been resolved before that, even by the time of Riemann's investigations in the 1850s. Some good background is given here.
  2. Euler's formula is the wellspring from which emerged group representations and harmonic analysis in the masterful account by Anthony Knapp in the April 1996 issue of the AMS Notices.

Theorem no. 247: Euler's Product Formula for Sine
  1. Euler's cotangent series can also be derived from a 'half-angle' formula: \(f(x)=\frac12\left(f(x/2)+f\left((x\pm 1)/2\right)\right).\) See, e.g., Konrad Knopp (transl. Young), Theory And Application Of Infinite Series, Dover edition, 1990. The book is online here via and the relevant text can be found on page 205ff. My presentation (with thanks to Andy Rich) is essentially the same and needs to be acknowledged as what is referred to by Knopp as an "in general faulty mode of passage to the limit" (his emphasis), which he spends two pages making rigorous (even confined to the reals). The derivation of Euler's cotangent formula from the half-angle formula is attributed by Knopp to Heinrich Schröter, 1868. There is more on its history in Chapter 11 (§ 2) of Reinhold Remmert's, Theory of Complex Functions, Springer, 1991, (transl. Robert Burckel).
  2. A very interesting account of the Mittag-Leffler theorem, which is the 'ultimate' generalisation of Euler's formula, is available in this dissertation by Laura E. Turner.
  3. An explanation by Jim Belk of why convergence of infinite products is subsumed within convergence of infinite series. The convergence of Euler's formula is illustrated by John D. Cook here.
  4. Euler's formula's LHS may be replaced with the more elegant (from this website's perspective!) \(\sin\tau x\) but at the expense of elegance in the RHS, where the factors in the product become \(1-4x^2/k^2.\)

Theorem no. 248: A Theorem about Gaussian Moats
  1. Theorem under Construction iconThis is a 'theorem under construction': I hope to chart exciting developments here towards an eventual final version, which may or may not mean a proof that Gaussian moats can have unbounded width.
  2. Original source for this theorem: Ellen Gethner, Stan Wagon, and Brian Wick, A stroll through the Gaussian primes", American Mathematical Monthly, vol. 105 (1998), pp. 327–337; online (this is the recommended weblink from the theorem page).
  3. The widest Gaussian moats found thus far have width \(D=6\), found in 2004 by Nobuyuki Tsuchimura: "Computational Results for Gaussian Moat Problem", IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Volume E88-A Issue 5, May 2005, 1267–1273; online (paywalled); preprint (400KB pdf, June 2024).
  4. An arxiv preprint by Madhuparna Das asserts that there is no sequence of Gaussian primes of form \(a^2+b^2\) with bounded distance between consecutive entries. Das's arxiv account lists three or four other articles on Gaussian moats.
  5. Ellen Gethner has kindly supplied information on the origins of Gordon's Gaussian Moat problem:
           "One of the most difficult aspects of the problem was in finding out who actually posed it; the paper by Jordan and Rabung attributes that to Erdős. I had the opportunity to ask Erdős about the problem at an analytic number theory conference at the University of Illinois in 1995. I remember going to a conference party at someone’s home, and there was Erdős on a chaise lounge in the middle of an enormous back yard with no other chairs in sight and 100+ mathematicians milling around him. I had a fairly lengthy conversation with him about the Gaussian Moat problem; I learned right away that he hadn’t posed the problem and he wasn’t sure who had. I asked him if he thought the conjecture was true (i.e., that there are indeed arbitrarily wide Gaussian Moats) and his response was a pause followed by “what do YOU think?” I answered that I thought the conjecture was correct, to which he responded “so do I!”
    In the meantime, later on at the same conference, I happened to be in a car with several other mathematicians on the way to a session. One of the mathematicians was Basil Gordon; I had heard from one of his PhD students that Gordon might be able to help me find who had posed the problem. During that car ride, I asked my question, and oddly enough, Gordon turned out to be the original poser! I think (I’m a little fuzzy here) that he said that he had posed the problem during a session of one of the ICMs. In any case, all of the encounters were purely serendipitous and in looking back on the whole thing, I’m surprised that I succeeded in solving some of the mysteries."
    (the "paper by Jordan and Rabung" is J. H. Jordan and J. R. Rabung, "A conjecture of Paul Erdős concerning Gaussian primes", Math. Comp., vol. 24 (1970) 221–223. They construct a 4-moat, i.e. they show that steps of size at least \(D=4\) are required to reach infinity.)
  6. I found these notes (pdf) on Gaussian integers by Christian Wuthrich of great value.

Theorem no. 249: Bézout's Identity
  1. Original sources for this theorem:
    1. Bachet de Méziriac's original assertion of the identity is traced in its Wiki entry.
    2. Bézout, É., "Recherches sur le degré des Équations résultantes de l'évanouissement des inconnues, & sur les moyens qu'il convient d'employer pour trouver ces équations", Mémoires de l'Académie royale des sciences, 1764, publ. 1767, p. 288-338; online. The English Wiki entry cites Théorie générale des équations algébriques, Paris, 1779; online. But the account of Bézout's work in Liliane Alfonsi, "Étienne Bézout : Analyse algébrique au siècle des Lumières" places the identity firmly in the 1764 mémoire, see Section 4 of Chapter 3, pp. 35ff.
    3. Taher Elgamal, "A public-key cryptosystem and a signature scheme based on discrete logarithms", IEEE Transactions on Information Theory, Vol. 31 (4), 1985, pp. 469–472; online (paywall; pdf download, October, 2023). A little footnote to this publication from Gödel's Lost Letter.
  2. It should be noted that the discrete logarithm problem is solved by quantum computers: Peter W. Shor, "Algorithms for quantum computation: discrete logarithms and factoring", in Proc. 35nd Annual Symposium on Foundations of Computer Science (Shafi Goldwasser, ed.), IEEE Computer Society Press, 1994, 124–134; online (paywall; pdf download, October, 2023 (the last entry under 'Quantum Computation', the pages of the pdf version appear in reverse order for some reason), Shor's arxiv version is also available but this is expanded from the FOCS publication and is a bit less accessible, in my opinion.)
  3. I find that the photograph featuring Merkel and Obama which accompanied the blog entry cited on the theorem page, does so no longer.

Theorem no. 250: the Power of a Point Theorem
  1. Original sources for this theorem:
    1. Louis Gaultier, "Mémoire sur les Moyens généraux de construire graphiquement un Cercle déterminé par trois conditions, et une Sphère déterminée par quatre conditions", Journal de l'École Polytechnique, Cahier 16, 1813, pp. 124–214; online. (There are some detailed comments on this paper, although not directly relevant to power of a point, in Jemma Lorenat's fine dissertation "Die Freude an der Gestalt" : methods, figures and practices in early nineteenth century geometry; online).
    2. Jakob Steiner, "Einige geometrische Betrachtungen", Journal für die reine und angewandte Mathematik, Vol. 1, 1826, pp. 161–184; online.
  2. Michael N. Fried, the author of "Mathematics as the science of patterns", the recommended weblink from this theorem, gave me the following insight, which I find charming and valuable:
           \(h\), as you defined it, can be thought of in a slightly different way. Consider the function \(f(x,y)=|(x-a)^2+(y-b)^2-r^2|\). Its zeros are the points on a circle with center \((a,b)\) and radius \(r\); but its value at an arbitrary point \(P(x,y)\) is the power of \(P\) with respect to that circle. Students typically learn to solve equations \(f(x,y)=0\), that is, to find the curve they represent, but then ignore the values of \(f\) at other points. For example, the curve given by \(f(x,y)=|ax+by+c|=0\) (normalized so that \(a^2+b^2=1\)) is of course a straight line \(L\), while the value of \(f\) at other points is the distance between those points and \(L\) (including of course those points whose distance from \(L\) is zero!).       
    Michael Fried has, by the way, a fascinating Youtube presentation comparing the relevant bits of Euclid with Steiner's geometry.
  3. Cut-The-Knot offers a very nice application of the Intersecting Chords Theorem.

Theorem no. 251: The Hanani–Tutte Theorem
  1. Hanani's original 1934 paper, published under his Polish birth name of Chaim Chojnacki as "Über wesentlich unplättbare Kurven im dreidimensionalen Raume", Fundamenta Mathematicae, Vol. 23, Issue 1, 1934, pages 135–142, can be read online here. Bill Tutte's 1970 paper "Toward a theory of crossing numbers", Journal of Combinatorial Theory, Vol. 8, Issue 1, 1970, pages 45–53, can be read online here.
  2. The algebraic specification of planarity is independently credited to Wen-Jun Wu, "On the planar imbedding of linear graphs", Journal of Systems Science and Mathematical Sciences, 1985 Issue 4, pages 290–302, with independent work of some others preceding it. More details at the Wikipedia entry for the theorem.
  3. Although we can solve equations to turn a graph drawing into one which has only evenly crossing independent edge pairs it is not obvious how to turn the result into a drawing with no crossings at all. For this we need a direct algorithmic proof of Hanani–Tutte and this was first provided by Michael J. Pelsmajer, Marcus Schaefer and Daniel Štefankovic in "Removing even crossings", Journal of Combinatorial Theory B, Vol. 97, Issue 4, 2007, pages 489–500, online here.
  4. For small graphs, testing solvability of the Tutte–Wu equation system allows planarity testing without recourse to graph algorithms or data structures. However, for large graphs this does not compete realistically with known linear algorithms since the worst-case running time is \(O(n^6)\) where \(n\) is the number of vertices of the graph (strictly, the number of edges is involved in the running time but a non-linear number of edges guarantees non-planarity by the \(3n-6\) bound, see Kuratowski's Theorem).

Theorem no. 252: Bertrand's Ballot Theorem
  1. Original source for the Cycle Lemma is A. Dvoretzky and Th. Motzkin, "A problem of arrangements", Duke Math. J., Vol. 14, No. 2 (1947), 305–313; online (paywall). A good secondary source is Nachum Dershowitz and Shmuel Zaks, "The cycle lemma and some applications", Europ. J. Combinatorics, vol. 11, no. 1, 1990, pp. 35–40; online.
  2. As so often, the theorem is not accurately named, since it was apparently first stated by William Allen Whitworth in 1878. The general form I have given is due to neither, having evolved from the original case \(k=1\). Details may be found in Marc Renault, "Four Proofs of the Ballot Theorem", Mathematics Magazine, vol. 80, no. 5, 2007, pp 345–352; online via Renault's Ballot Problem page (which is the recommended weblink from the theorem page).
  3. Just to fill in the details, there is a claim on the theorem page that removing a sequence of \(k\) \(a\)'s and a \(b\) from a cycle with \(n\) \(b\)'s and \(m=n(k-1)+S\) \(a\)'s, "gives a new cycle in which the surplus \(S\) is reduced by exactly 1." Indeed, with \(M=m-k\) \(a\)'s and \(N=n-1\) \(b\)'s remaining, we have \(M=n(k-1)-k+S=N(k-1)+k-1-k+S=N(k-1)+(S-1).\) (The proofs of the Cycle Lemma I have seen invoke the pigeon hole principle to repeatedly remove the \(a\)-\(b\) sequences. But I find this obscure — what are the boxes? Saying 'a counting argument shows...' would seem more appropriate. And I have gone so far as to actually spell out the counting argument.)

Theorem no. 253: The Third Isomorphism Theorem
  1. Original source for this theorem: see Theorem 34.
  2. This page has been separated off from an earlier combined description of the 2nd and 3rd isomorphism theorems, see Theorem 35, notes(1).
  3. There is a temptation, when dealing with quotient groups to use shorthand group notation as in \(F_{20}/C_5\cong C_4\). It has to be kept in mind, however, that quotienting by isomorphic subgroups need not result in isomorphic quotient groups. See this by, for example.

Theorem no. 254: Kasteleyn's Theorem
  1. Original source for this theorem is: P. W. Kasteleyn, "Dimer statistics and phase transitions", J. Math. Phys., 4, 1963, pp. 287–293; online (paywall). I have not seen this paper but its abstract certainly seems to concern the general planar graph version of Kasteleyn's theorem. But the theorem seems more generally attached to the paper "Graph theory and crystal physics", in Graph Theory and Theoretical Physics, F. Harary, ed., Academic Press, London, 1967, pp. 43–110. There is a Wiki page for the theorem which calls it the FKT algorithm, giving equal billing to Fisher and Temperley. Sources for their earlier contributions can be found there.
  2. This theorem is sometimes stated with the argument of the square root being the absolute value of the determinant. This protects against an eventuality, of the determinant being negative, which in fact cannot arise, since the matrix in question is necessarily real, skew symmetric and thus has non-negative determinant.
  3. Donald Knuth gives some valuable history of the Pfaffian function in "Overlapping Pfaffians", Electronic Journal of Combinatorics, vol. 3, no. 2, 1996; online.
  4. David E. Speyer has given short topological proofs of Kasteleyn's theorem, and variants of it, in "Variations on a theme of Kasteleyn, with application to the totally nonnegative Grassmannian", Electronic Journal of Combinatorics, vol. 23, no. 2, 2016; online.

Theorem no. 255: Countability of the Rationals
  1. I am not sure if Cantor ever explicitly published the countability of the rationals as a theorem. It is mentioned in the Wiki article on Cantor's first (1874) set theory paper as arising in correspondence between Dedekind and Cantor. It follows at once from more general results such as the countability of a countable union of countable sets (but this latter result requires the axiom of choice!)
  2. The version of the Stern–Brocot tree used in my illustration is often attributed to Neil Calkin and Herbert Wilf (2000). However it has been traced back to George Raney (1973) by Alessandro De Luca and Christophe Reutenauer in "Christoffel words and the Calkin-Wilf tree", The Electronic Journal of Combinatorics, 18(2), 2011, P22; online. A very nice description of how 'Stern's diatomic sequence' may be derived from Pascal' triangle appears in Aperiodical's The Big Internet Math-Off 2024

Theorem no. 256: Moreau's Necklace Formula
  1. Original source for this theorem: Moreau, C. , "Sur les permutations circulaires distinctes", Nouvelles annales de mathématiques, Sér. 2, tome 11, 1872, pp. 309–314; online.
  2. Moreau's formula was independently derived, by the same approach, by Édouard Jablonski, "Théorie des permutations et des arrangements circulaires complets", Journal de mathématiques pures et appliquées, 4e série, tome 8, 1892, pp. 331–350; online. And again, much later, by Hazel Perfect, "Concerning Arrangements in a Circle", The Mathematical Gazette, Vol. 40, No. 331, 1956, pp. 45–46; online (paywall). (This was how I discovered the formula. I wrote an more extended account of this theorem entry for M500 magazine in which a summary is given of Perfect's solution to the necklace counting problem: Whitty, R., "Perfect's necklace formula", M500, Issue 285, December 2018, pp. 12–15; online (600KB pdf download). Perfect only went as far as writing down the equations which solve the problem. I presented the solution in terms of the Möbius function on the poset of divisors of the bead numbers. Only afterwards I did due diligence on the history of the theorem and my page now bears little resemblence to the originally posted version! Of which an echo persists, however, in the name of the pdf file, which I preferred not to disturb). I submitted to M500 an addendum (150KB pdf) to complete the derivation of Moreau from Perfect's presentation.
  3. For good measure the formula is sometimes traced to MacMahon, P.A., "Application of a theory of permutations in circular procession to the theory of numbers", Proc. Lond. Math. Soc., Vol. s1-23, 1892, pp. 305–313; online (paywall; facsimile). However, MacMahon merely states the formula, attributing it to Moreau and Jablonski, before moving on to applications.
  4. Necklace counting belongs to the general topic of combinatorics of words for which a good historical account is Jean Berstel and Dominique Perrin, "The origins of combinatorics on words", European Journal of Combinatorics, Vol. 28, Issue 3, 2007, pp. 996–1022; online. Another good source is Romeo Meštrović, "Different classes of binary necklaces and a combinatorial method for their enumerations"; arxiv.
  5. My page ended with the suggestion to list the 24 necklaces on four balls of three or fewer colours; also with the suggestion that these enumerations are best done with, say, Pólya–Redfield enumeration. Accordingly I gave the job to my computer (Maple) and got the following multinomial: $$ {b}^{4}+{b}^{3}g+{b}^{3}w+2\,{b}^{2}{g}^{2}+3\,{b}^{2}gw+2\,{b}^{2}{w} ^{2}+b{g}^{3}+3\,b{g}^{2}w+3\,bg{w}^{2}+b{w}^{3}+{g}^{4}+{g}^{3}w+2\,{ g}^{2}{w}^{2}+g{w}^{3}+{w}^{4},$$ where \(3b^2gw\), for example, means that, up to rotational symmetry, there are three necklaces with two blue balls, one green and one white.
  6. The best-known algorithm for generating necklaces for a given number of beads and colours is by Harold Fredericksen, Irving J. Kessler and James Maiorana and is known as the FKM algorithm (you can see it in action on Jason Davies' necklaces page). An interesting alternative and a good source of information is Frank Ruskey, Terry MinYih Wang and Carla D.Savage, "Generating Necklaces", Journal of Algorithms, vol. 13, no. 3, 1992, 414–43; preprint.

Theorem no. 257: Distribution of Local Maxima in Random Samples
  1. The original publication is T. Austin, R. Fagen, T. Lehrer, and W. Penney, "The Distribution of the Number of Locally Maximal Elements in a Random Sample", Annals of Mathematical Statistics, Vol. 28, Number 3 (1957), 786-790; online.
  2. T. Lehrer is apparently the Tom Lehrer famous as a satirical singer-songwriter. The website mentions the above and another article with the somewhat mysterious commentary "Unfortunately these mathematical publications did not have a lasting effect on society."
  3. At any rate, the Austin et al article provoked a reaction in the profession: M.O. Glasgow, "Note on the Factorial Moments of the Distribution of Locally Maximal Elements in a Random Sample", Ann. Math. Statist., Vol. 30, Number 2 (1959), 586–590; online.
  4. There is a nice tribute to Lehrer at Gödel's Lost Letter which links to a previous short entry on his theorem with Austin et al.

Theorem no. 258: Sylow's Theorems
  1. The original publication is L. Sylow, "Théorèmes sur les groupes de substitutions", Mathematische Annalen, Vol. 5, 1872, pp. 584–594; online (paywall); at Göttinger Digitaisierungszentrum. An annotated English translation by Robert A. Wilson is provided on his webpage.
  2. Geoff Smith, whose book is the recommended reading for this theorem, gave me following nice picture of \(A_4\) not having an order-6 subgroup:
           When discussing the non-existence of a subgroup of order 6 in \(A_4\), you do have the option to geometrize. Colour the vertices of a cube red and blue so that no vertices joined by an edge are the same colour. The group of rotations of the cube which preserve the blue vertices is a copy of \(A_4\) (label the blue vertices 1 to 4). The elements of this \(A_4\) are then rotations about grand diagonals and rotations through pi using skewers centre-face to centre-face. This enables one to reason geometrically about \(A_4\) and to "see" what is going on. This has the disadvantage that people with poor geometric intuition will melt, but the advantage of dealing with things more concrete than permutations.       
  3. The bell-ringing illustration for this theorem deserves a little amplification, which space on the page itself did not allow. The Plain Bob method starts with the 2-Sylow subgroup and is completed by switching to its cosets in \(\mbox{Sym}_4\): permutations 8 to 15 are the left coset by \((2 4 3)\); permutations 16 to 23 are the left coset by \((3 4)\). Bell ringing in general provides good examples of Lagrange's theorem in action! There is a good introductory article in this vein "Bells, Motels and Permutations Groups" by Gary McGuire.
  4. On the subject of bell ringing, kudos to, a site "built for change ringers to continue ringing with one another even when socially distanced".
  5. A couple of very valuable blog posts on Sylow appeared close on each other's heels at the end of 2020: this by Daniel Litt and this by Qiaochu Yuan.

Theorem no. 259: Schur's Commuting Matrices Bound
  1. Original sources for this theorem:
    1. Schur, J., "Zur Theorie der vertauschbaren Matrizen", Journal für die reine und angewandte Mathematik , Vol. 130, 1905, pp. 66–76; online (paywall; facsimile at Göttinger Digitaisierungszentrum). The first few paragraphs are translated into English here.
    2. Jacobson, N. "Schur's theorems on commutative matrices", Bull. Amer. Math. Soc., Vol. 50, Number 6, 1944, pp. 431–436; online.
  2. A short proof of Schur's theorem (in Jacobson's extension to arbitrary fields) is given in M. Mirzakhani, "A simple proof of a theorem of Schur", American Math. Monthly, Vol. 105, no. 3, 1998, pp. 260–262; online.
  3. In a general modern setting, this theorem is about the dimension of various kinds of subalgebra. See this arxiv post by J. Szigeti, J. van den Berg, L. van Wyk and M. Ziembowski, for example.
  4. The values of \(\lfloor n^2/4\rfloor+1\) are sequence A033638 at OEIS where many further references may be found.

Theorem no. 260: Dunn and Pretty's Triangle-Halving Deltoid
  1. Original sources for this theorem:
    1. Dunn, J.A. and Pretty, J.E., "Halving a triangle", Math. Gaz., Vol. 56, No. 396, 1972, pp. 105–108; online (paywall).
    2. The history of triangle, and tetrahedron, bisection is authoritatively traced in W. A. Beyer and Blair Swartz, "Bisectors of triangles and tetrahedra", The American Mathematical Monthly, Vol. 100, No. 7, 1993, pp. 626–640; online (paywall). To quote from their introductory remarks, "... the problems have a much older history in hydrostatics and naval architecture, as they are also connected with the orientation and stability of floating bodies." They formulate a version of the deltoid theorem which they say 'extends' various textbook entries. To give a flavour using sources available online: p. 190 of George Greenhill, A treatise on hydrostatics, MacMillan, London, 1894; online; and p. 232, e.g. 3 of Horace Lamb, Statics, including hydrostatics and the elements of the theory of elasticity, Cambridge University Press, 1912; online.
    3. Subsequent work on the bisection deltoid is recorded in Allan Berele and Stefan Catoiu, "Bisecting the perimeter of a triangle", Mathematics Magazine, Vol. 91, Issue 2, 2018, pp. 121–133; online.
  2. Variations on the triangle area bisection theme may be found on blogs and forums: math.stackexchange, Saving School Math,
  3. A description of how the hyperbolae were plotted in the illustration for this theorem page is given here and how the bisecting line of arbitrary slope was plotted is described here.

Theorem no. 261: Euclid's Pythagorean Formula
  1. Original sources for this theorem: David E. Joyce is the recommended source for an English language reading of Euclid. The relevant page is here. Richard Fitzpatrick's dual-language source allows something like the original Greek to be viewed.
  2. St Exupéry's problem appears in various places on the internet, for instance on the official Antoine de Saint-Exupéry website, whence the French version of the problem text. A variant of the problem is found here, which may actually be the original, the Egyptian story having been added by popularisers subsequently. His Pharoah problem has led to his name being given to those integers which are products over Pythagorean triples, the Saint-Exupéry numbers being entry A057096 at OEIS. It is observed that it is unknown if there can be two triples yielding the same St-Exupéry number.
  3. Antoine de Saint Exupéry was obviously at ease with elementary number theory. As a little footnote, an article in the International Herald Tribune, "350 Years Later, Math Conundrum Bites the Dust," by Gina Kolata, June 25, 1993, celebrated Andrew Wiles's announcement of a proof of Fermat's Last Theorem. In response came a letter (IHT, July 29, 1993) from Isia Leviant (who I think must be the Isreali artist of that name):
           Antoine de Saint-Exupéry, the famous French writer, was a fan of mathematics. In April 1943, a few months before being downed over the Mediterranean, he had lunch with me in an Algiers bistro. Knowing my training in math, he wanted to show me that he had solved the Fermat riddle. He started writing a series of equations on a paper napkin. Unfortunately, I had to stop him halfway: There was a mistake in his calculations. My own mistake was not to have kept the paper napkin with its precious manuscript.
  4. St Exupéry's scholarly record in mathematics is examined authoritatively by Roger Mansuy for Images des Mathématiques.
  5. The response of Roman Andronov to this Quora question explains some nice geometry of Pythagorean triples: the radius of the incircle of a right triangle with integer sides \(a,b,c,\,c\) the hypotenuse is \(r=(a+b-c)/2=n(m-n)\), where \(m\) and \(n\) are as given on the theorem page (with \(k=1\)) (see also notes (5) to Theorem 128).

Theorem no. 262: The Polygonal Number Theorem
  1. Original sources for this theorem:
    1. A.-L. Cauchy, "Démonstration du théorème général de Fermat sur les nombres polygones", Mémoires de la classe des Sciences mathématiques et physiques de l'Institut de France, 14 (1813–1815), pp. 177–220; I don't find this online. Cauchy's result generally seems to be credited to him with the date 1813. However, catalogue entries for this paper cite it as "lu à` l'académie, le 13 novembre 1815", e.g. here.
    2. Melvyn B. Nathanson, "A short proof of Cauchy's polygonal number theorem", Proc. Amer. Math. Soc., Vol. 99, No. 1, 1987, pp. 22–24; online.
  2. My presentation of Nathanson's proof risks giving an impression of circularity: locate \(b\) such that quadratic equations in \(b\) specify an interval allowing \(b\) to be located. Lack of space prevented me from clarifying: for a given \(n\) and \(m\) an interval for \(b\) may be expressed, via the quadratic formula, purely in terms of \(n\) and \(m\). Namely, the interval \([1/2+\sqrt{6(n/m)-3}, 2/3+\sqrt{8(n/m)-8}\,]\) is guaranteed to be bounded by the zeros of the quadratics and to have length at least 4 for \(n\geq 120 m\).
  3. Nathanson's proof gives a stronger version of Cauchy's theorem: any nonnegative integer may be written as a sum of \(m+2\) polygonal numbers of order \(m+2\) at most four being greater than 1. (Nathanson says \(m+1\) polygonal numbers but I don't see this for \(m=3\) when his \(0\leq r\leq m-3\) will fail to give residue zero).
  4. Polygonal numbers are a special case of figurate numbers. The weblink from the theorem page is to a presentation of Elena Deza and Michel Marie Deza's, Figurate Numbers, World Scientific, 2012.
  5. Some more context for this theorem is given in this presentation (600KB pdf).

Theorem no. 263: The Shoelace Formula
  1. Original source for this theorem: Albrecht Meister, "Generalia de genesi figurarum planarum et inde pendentibus earum affectionibus", Novi Commentarii Societatis Regiae Scientiarum Gottingensis, I, 1769/70, 1771, pp. 144–180; online (facsimile at Göttinger Digitaisierungszentrum; the figures are separated from the text here). Burkard Polster in this first-rate Mathologer video on the formula, explains that Gauss mentions it admiringly in his writings, hence the attribution to him.
  2. Writing the Shoelace formula in terms of the exterior or Grassmann algebra (as is done in its Wiki entry, for example) is over-elaborate for plane applications, where the calculation is merely a sum of cross products (treating the coordinates as position vectors). However, the exterior algebra notation is rather concise for formal calculations, as in our application to area bisection.
  3. In any case the exterior algebra is much more general and wide-reaching than what I present for the purposes of encoding the Shoelace formula. See the wiki entry, for example. Writing the formula sum as a sum of \(2\times 2\) matrix determinants, this encoding is the \(n=2\) case of the lemma that says a wedge product of \(n\) weighted sums of the algebra invariants is equal to the determinant of the \(n\times n\) matrix of the weights.
  4. The formula we give for area bisections of a triangulated polygon may in principle be extended by applying the same calulation to any vertex and an opposite edge. A careful interpretation of the components of the formula is required however. For example, in our 5-vertex polygon, we may take the triangle from vertex \(v_2\) to the 'opposite edge' \(v_3v_4\). Then \(A_{co}\) is the area created by travelling counterclockwise round the polygon from the head of triangle edge \(v_2v_3\) (which happens also to be a polygon edge) until we arrive back via this triangle edge to arrive at \(v_3\). This gives the whole polygon, which has area 8. For area \(A_{cl}\) we start by traversing the triangle edge in the other direction, \(v_3v_2\), and then continuing counterclockwise around the polygon, but this brings us immediately back to \(v_3\), giving zero area. The denominator is twice the area of the triangle but with the orientation \(v_3v_4v_2\). Because this is clockwise it is assigned a negative area: \(-4\). So we have the result \(t=(8-0)/(2\times-4)=-1\).

Theorem no. 264: Tunnell's Theorem
  1. Original source for this theorem: Jerrold B. Tunnell, "A classical Diophantine problem and modular forms of weight 3/2", Inventiones Mathematicae, Vol. 72, Nmber 2, 1983, pp. 323–334; online. The work of Coates and Wiles and of Kolyvagin is cited in the PNAS article "Congruent numbers" by John H. Coates (which is the recommended weblink from the theorem page).
  2. The restriction to square-free positive integers is necessary for the theorem as stated. For example, the congruent number 24 has 8 solutions on the LHS of Tunnell's condition but zero on the RHS, failing to qualify. Conversely the non-congruent value 12 has zero solutions on both sides of the condition and would thus qualify vacuously.
  3. Tunnell's theorem was key to the identification, in 2009, of all congruent numbers up to one trillion, described here. The accuracy of the list is, accordingly, subject to Birch and Swinnerton-Dyer.
  4. Congruent numbers are entry A003273 at OEIS, where there is much additional information. Hover over the entries under Crossrefs for primitive congruent numbers, non-congurent numbers and for numbers of solutions to the Tunnell theorem equations.
  5. I made a slide presentation on Tunnell's life and work which can be found here (1MB pdf). In French: another slide show about congruent numbers has quite a lot on their history and is here (2.5MB pdf).
  6. The congruent number problem is the choice of Matilde Lalín in Episode 43 of Kevin Knudson and Evelyn Lamb's My Favorite Theorem podcast (actually Matilde Lalín says Mordell's Theorem is her favourite but Tunnell's theorem gets talked about too).

Theorem no. 265: Bondy's Theorem on Subsets
  1. Original source for this theorem: J.A. Bondy, "Induced subsets, J. Combinatorial Theory, Series B, Vol. 12, Issue 2, 1972, pp. 201–202; online.
  2. Bondy's original proof was graph theoretic and is essentially what is given on the theorem page. The presentation in terms of n-cubes follows the recommended book Extremal Combinatorics by Stasys Jukna. An inductive proof is given in R. Crowston, G. Gutin, M. Jones, G. Muciaccia and A. Yeo, "Parameterizations of test cover with bounded test sizes", Algorithmica, Vol. 74, Issue 1, 2016, pp. 367–384; online (paywall). Thanks to Florian Foucaud for this reference. Regarding algebraic proofs see Andreas Winter, "Another algebraic proof of Bondy's theorem on induced subsets", J. Combinatorial Theory, Series A, Vol. 89, Issue 1, 2000, pp. 145–147; online.
  3. Sauer's Lemma was proved independently and more or less simultaneously in
    1. N. Sauer, "On the density of families of sets", J. Combinatorial Theory, Series A, Vol. 13, Issue 1, 1972, pp. 145–147; online; a version of the lemma, privately communicated, is stated in Bondy's paper.
    2. S. Shelah, "A combinatorial problem; stability and order for models and theories in infinitary languages", Pacific J. Math., 41, 1972, pp. 271–276; online;
    3. V. N. Vapnik and A. YA. Chervonenkis, "On the uniform convergence of relative frequencies of events to their probabilities", Theory Probab. Appl., Vol. 16, Issue 2, 1971, pp. 264–280; online (paywall; reprints can be found on the web, e.g. here, pdf 1.2MB, January 2024). This is an English translation of the Russian original. It is reproduced in Vladimir Vovk, Harris Papadopoulos and Alexander Gammerman (eds.), Measures of Complexity: Festschrift for Alexey Chervonenkis, Springer, 2015 and more details are given on the Springer page for the chapter giving, for example, the date of the first draft of the paper as 1966.
  4. For the French version of the theorem page, the recommended weblink is Aline Parreau's posting of her PhD thesis. The official Institut Fourier posting is here but is a much larger file and does not have clickable cross-links.

Theorem no. 266: Turing-completeness of Conway’s Game of Life
  1. Original source for this theorem: Conway's proof of Turing-completeness appeared in Elwyn R. Berlekamp, John H. Conway and Richard K. Guy, Winning Ways for Your Mathematical Plays, Volume 2, Academic Press, 1982 (link is to bibliography entry for 2nd edition).
  2. The Game of Life was originally presented in Martin Gardner's Mathematical Games column: "The fantastic combinations of John Conway's new solitaire game 'life'", Scientific American, Vol. 223, no. 4., 1970, pp. 120–123; online (paywall; a copy is here, Jan. 2024, but appears to be missing the final figures). I don't think Bill Gosper's glider gun has an original source — it may have been communicated directly to Martin Gardner who presented Conway's $50 challenge to find a Life configuration with unlimited growth in his 1970 article.
  3. The further reading for this theorem is Paul Rendell's book on constructing a universal Turing machine in the Game of Life. I found a nice overview by Rendell, "A Turing Machine In Conway's Game Life" here (Jan. 2024) whose provenance I am unsure of. Rendell has a site where it is not but much else is.
  4. Tangential to Turing-completeness but offering a very nice introduction to Life is Alex Stone's Quanta magazine article on the finally complete demonstration that oscillators of all periods exist in Life.
  5. The list of things which are Turing-complete is long and varied. A 2023 addition is origami ! See this elegant exposition by Jordana Cepelewicz for Quanta magazine.
  6. There is a wonderful visualisation, by Alec Singh, of Life evolution plotting against Time as a vertical axis.

Theorem no. 267: The Intermediate Value Theorem
  1. Bolzano's 1817 paper is "Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (Prague 1817)." A translation into English is given in S.B Russ, "A translation of Bolzano's paper on the intermediate value theorem", Historia Mathematica, Vol. 7, Issue 2, 1980, pp. 156–185; online.
  2. Augustin-Loius Cauchy's contribution to the development of this theorem is also seminal. Very good is Michael J. Barany, "Stuck in the Middle: Cauchy’s Intermediate Value Theorem and the History of Analytic Rigor", Notices of the AMS, Vol. 60, Number 10, 2013, pp. 1334–1338; online.
  3. For an introduction to intuitionist rejection of the theorem see, for example, section 3.4 of the relevant entry at the Stanford Encyclopedia of Philosophy.
  4. A good self-contained proof of the theorem is given here (Day 6) at Oxford College, Emory.

Theorem no. 268: Cramer's Rule
  1. Original source: Gabriel Cramer, Introduction à l'analyse des lignes courbes algébraique. The book has its own Wikipedia page (in French) with a link (under references) to a facsimile copy. Attributions to Maclaurin are also common by A.A. Kosinski, "Cramer's Rule Is Due To Cramer", Mathematics Magazine, Vol. 74, No. 4, 2001, pp. 310–312; online (360KB pdf download), seems definitive to me.
  2. Contrary to popular belief, Cramer's Rule is not necessarily inferior to Gaussian elimination in terms of efficiency and stability. See Ken Habgood and Itamar Arel, "A condensation-based application of Cramer’s rule for solving large-scale linear systems", Journal of Discrete Algorithms, Vol. 10, January 2012, pp. 98–109; online.

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