Theorem of the Day
List of pending theorems with web resources
At some time in the future I hope there will be a page for each of the following theorems. To help make this possible, when I learn of relevant web resources I record them here. Links to resources which I want to study in the hopes of improving existing theorem pages are also kept here, in which case the theorem names link to their entry in my main theorem list.
Please email me and with either theorems you would like to see added to the list or weblinks which you think will be valuable to me.
Apéry's Theorem-
www.quantamagazine.org/rational-or-not-this-basic-math-question-took-decades-to-answer-20250108/
The Baire Category Theorem- "The Baire Category [theorem] is a profound triviality which condenses the folk wisdom of a generation of ingenious mathematicians into a single statement." Tom Körner
www.dpmms.cam.ac.uk/~twk/ See under Course Notes, Notes for the Part II course Linear Analysis (the quote is in Section 6
Bell's Theorem- www.quantamagazine.org/a-new-theorem-maps-out-the-limits-of-quantum-physics-20201203/
kpknudson.com/my-favorite-theorem/2023/5/2/episode-84-the-students-of-tcu
arxiv.org/abs/quant-ph/0402001
Bertrand's Postulate- web.williams.edu/Mathematics/lg5/Chebyshev.pdf Proof
arxiv.org/abs/2010.15882 Using BP to give an irrational prime generating constant and www.johndcook.com/blog/2025/02/18/the-buenos-aires-constant/
arxiv.org/abs/2501.18048 Progress on Quadratic (Landau) version of BP
oeis.org/A295262
lab.rockefeller.edu/cohenje/assets/file/464Generalizations%20of%20Bertrand%20s%20Postulate%20to%20Sums%20of%20Any%20Number%20of%20PrimesMathMag2023.pdf Generalisation of BP for large n
Bezout's Theorem- sections.maa.org/florida/proceedings/2001/fitchett.pdf
The Billiards Circuit Theorem- www.tandfonline.com/doi/full/10.1080/00029890.2023.2230860
www.quantamagazine.org/the-mysterious-math-of-billiards-tables-20240215/
Blackwell's Bet- www.futilitycloset.com/2016/06/28/blackwells-bet/
projecteuclid.org/journals/annals-of-mathematical-statistics/volume-22/issue-3
Bloom's Egyptian Fraction Theorem- arxiv.org/abs/2305.02689
Busy Beaver Theorem- scottaaronson.blog/?p=7388
mathoverflow.net/questions/461550/using-busy-beavers-to-prove-conjectures/461568?s=03#461568
www.sligocki.com//2024/05/22/bb-3-4-a14.html
www.quantamagazine.org/amateur-mathematicians-find-fifth-busy-beaver-turing-machine-20240702/
Cantor–Bernstein–Schröder- arxiv.org/abs/math/0605779
Cauchy's Integral Formula- robjlow.blogspot.com/2019/04/cauchy-and-other-mean-value-theorem.html
Cauchy's Residue Theorem- en.wikipedia.org/wiki/Residue_theorem
Cauchy–Schwarz Inequality- www.johndcook.com/blog/2021/06/21/reversed-cauchy-schwarz-inequality/
The Circle Packing Theorem- tcsmath.wordpress.com/2008/10/08/lecture-4-conformal-mappings-circle-packings-and-spectral-geometry/
people.maths.bris.ac.uk/~maetc/prospects.pdf
amslaurea.unibo.it/id/eprint/18804/1/tesi_georgian_sarghi.pdf
The Class Equation for Finite Groups- www.davcollegekanpur.ac.in/assets/ebooks/Maths/Herstein-%20Abstract%20Algebra%203rd%20Ed.pdf (Chapter 2, section 11)
dylan-c-beck.github.io/algebraqualstudygroup_Groups,%20Group%20Actions,%20and%20the%20Class%20Equation%20(Solutions).pdf
Classification of the finite simple groups- math.mit.edu/research/highschool/primes/circle/documents/2022/Gracie.pdf
Combinatorial Nullstellensatz- en.wikipedia.org/wiki/Restricted_sumset
The Continuum Hypothesis- jdh.hamkins.org/the-continuum-hypothesis-could-have-been-a-fundamental-axiom-oslo-june-2024/
Cook's Theorem on NP-completeness- quorasessionwithscottaaronson.quora.com/Do-you-expect-that-P-NP-is-decidable-in-ZFC-and-what-about-BQP-or-RP-1111
Cotes' Harmonic Means Theorem in Geometry- John D Cook's ref for Newton's Diameter Theorem
Salmon's classic text
Maker's of mathematics, Hollingdale
fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Cotes_(moyenne_harmonique)
The Crossing Lemma- www.cs.umd.edu/~gasarch/TOPICS/erdos_dist/szekely.pdf
Decidability of Tiling- arxiv.org/abs/1602.05738
terrytao.wordpress.com/2022/11/29/a-counterexample-to-the-periodic-tiling-conjecture-2/
www.quantamagazine.org/mit-math-students-continue-aristotles-tetrahedra-tiling-20210209/
www.quantamagazine.org/computer-search-settles-90-year-old-math-problem-20200819/
Desargues' Theorem- kpknudson.com/my-favorite-theorem/2020/8/12/episode-57-annalisa-crannell
kpknudson.com/my-favorite-theorem/2022/8/11/episode-78-daina-taimina
Dunn and Pretty's Triangle-Halving Deltoid- ems.press/content/serial-article-files/4415
link.springer.com/article/10.1007/s00022-005-0006-0
dl.acm.org/doi/abs/10.1016/j.aam.2022.102342
msp.org/involve/2015/8-2/involve-v8-n2-p11-p.pdf
The Erdos conjecture on arithmetic progressions- www.quantamagazine.org/landmark-math-proof-clears-hurdle-in-top-erdos-conjecture-20200803/
Euclid Book 1 Proposition 1- cgm.cs.mcgill.ca/~godfried/teaching/dm-reading-assignments/Euclid-Second-Proposition-Math-Intell.pdf
Evans' conjecture- www.math.uci.edu/~brusso/CompletionPartialLatSq18pp.pdf
www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1a1
The Feit–Thompson Theorem- www.quora.com/Why-is-the-Feit-Thompson-theorem-so-important-and-what-makes-its-proof-so-complicated-and-lengthy
terrytao.wordpress.com/2013/04/12/the-theorems-of-frobenius-and-suzuki-on-finite-groups/
The Friendship Paradox- www.science.org/doi/10.1126/science.adi5147
Frobenius' theorem on real division algebras- danshved.wordpress.com/2013/05/09/frobenius-theorem-on-real-division-algebras/
The Fundamental Theorem of Arithmetic- www.sciencedirect.com/science/article/pii/031508609290075M
The Fundamental Theorem of Calculus-
math.vanderbilt.edu/schectex/ccc/gauge/letter/
The Fundamental Lemma of the Calculus of Variations- www.ucl.ac.uk/~ucahmto/latex_html/chapter2_latex2html/node6.html
The Fundamental Theorem of Galois Theory- www.quantamagazine.org/how-galois-groups-used-polynomial-symmetries-to-reshape-math-20210803/
www.quora.com/What-was-the-ingenious-insight-of-Galois-that-laid-the-foundation-for-such-an-extensive-mathematical-framework
The Gauss Class Number Problem- en.wikipedia.org/wiki/Harold_Stark
www.scribd.com/document/330541021/01-Birch
www.quora.com/What-problems-did-Gauss-contemplate-and-leave-unsolved
Gauss's Linking formula- www.quantamagazine.org/how-tadayuki-watanabe-solved-a-topological-mystery-about-spheres-20211026/
www.maths.ed.ac.uk/~v1ranick/papers/ricca.pdf
Germain's Theorem- “Voici ce que j’ai trouvé:” Sophie Germain’s grand plan to prove Fermat’s Last Theorem
Sophie Germain childrens' book
A new criterion for the first case of Germain's Theorem
Granville - first case of Germain's Theorem is true up to...
Germain and Her Fearless Attempt to Prove Fermat's Last Theorem
Sophie Germain and Special Cases of Fermat’s Last Theorem
fermatslasttheorem.blogspot.com/2005/08/sophie-germain.html
Goldbach's Summation Identity- www.researchgate.net/publication/262857022_Series_Involving_the_Zeta_Functions_and_a_Family_of_Generalized_Goldbach-Euler_Series
www.maa.org/sites/default/files/pdf/upload_library/22/Ford/bibiloni206.pdf
The Hahn–Banach Theorem- mathshistory.st-andrews.ac.uk/Biographies/Helly/
abstrusegoose.com/124
The Hall–Paige Conjecture- cameroncounts.wordpress.com/2018/12/18/the-hall-paige-conjecture/
cameroncounts.wordpress.com/2026/08/27/complete-mappings-of-semigroups/
download.arxiv.org/pdf/2003.01798v2.pdf
Anthony Evans' book on orthogonal latin squares
The Ham Sandwich Theorem- Breuer, F. Uneven Splitting of Ham Sandwiches. Discrete Comput Geom 43, 876–892 (2010). doi.org/10.1007/s00454-009-9161-7
- www.johndcook.com/blog/2009/01/23/splitting-a-convex-set-through-its-center/
kpknudson.com/my-favorite-theorem/2022/6/9/episode-76-math-students-of-csula
Helly's Theorem- homepages.math.uic.edu/~suk/helly.pdf
Hensel's Lemma- math.uchicago.edu/~may/REU2018/REUPapers/Chen,Yuchen.pdf
www.quantamagazine.org/how-the-towering-p-adic-numbers-work-20201019/
The Hopf Umlaufsatz- people.math.harvard.edu/~knill/teaching/math22a2018/handouts/lecture15.pdf
www2.math.upenn.edu/~pjmcgrat/research/umlaufsatz.pdf
mathematik.com/Hopf/
pjmcgrat.math.ncsu.edu/research/umlaufsatz.pdf
www.fstg-marrakech.ac.ma/IGSseminar/archives/RachadBentbibPFELSTmiasi2017.pdf
L'Hospital's Rule- web.math.ucsb.edu/~agboola/teaching/2021/winter/122A/rudin.pdf
The Impossibilities- www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/Suzuki.pdf
www.jstor.org/stable/30037438
www.britannica.com/topic/Trisecting-the-Angle-Archimedes-Method-724632#ref1074301
Insolvability of the Quintic- meangreenmath.com/2022/11/25/an-elementary-proof-of-the-insolvability-of-the-quintic/
The Isoperimetric Inequality- projecteuclid.org/journals/journal-of-differential-geometry/volume-6/issue-2/A-generalization-of-the-isoperimetric-inequality/10.4310/jdg/1214430403.fulll
www.siam.org/publications/siam-news/articles/the-importance-of-mathematics-in-political-decision-making/
kpknudson.com/my-favorite-theorem/2024/4/3/episode-91-karen-saxe
prove-me-wrong.com/2021/10/27/the-isoperimetric-inequality/
The Jordan Curve Theorem- www.quora.com/Why-do-such-simple-mathematical-theorems-have-roundabout-proofs
The Kakeya Set Conjecture- www.quantamagazine.org/new-proof-threads-the-needle-on-a-sticky-geometry-problem-20230711/
terrytao.wordpress.com/2025/02/25/the-three-dimensional-kakeya-conjecture-after-wang-and-zahl/
terrytao.wordpress.com/2026/08/25/rotating-needles-in-space-the-road-to-the-kakeya-conjecture-and-why-it-matters/
The Kissing Problem- www.quantamagazine.org/mathematicians-discover-new-way-for-spheres-to-kiss-20250115/
Koebe's Theorem- www2.math.ethz.ch/t3/fileadmin/math/ndb/00238/07403/Lovasz_eth_koebe.pdf
The Krull-Schmidt Theorem- scilogs.spektrum.de/hlf/an-introduction-to-representation-theory/
math.mit.edu/~etingof/replect.pdf
Laplace's Rule of Succession- jonathanweisberg.org/post/inductive-logic-2/
Legendre's Theorem on Factorial Factors- www.cut-the-knot.org/blue/LegendresTheorem.shtml
The Lonely Runner Conjecture- terrytao.wordpress.com/2017/01/10/some-remarks-on-the-lonely-runner-conjecture/
Magnus’s Freiheitssatz- arxiv.org/abs/2501.18306
en.wikipedia.org/wiki/Freiheitssatz
The Mandelbrot set is connected- www.quantamagazine.org/the-quest-to-decode-the-mandelbrot-set-maths-famed-fractal-20240126/
www.sciencefocus.com/science/the-two-numbers-that-could-solve-maths-biggest-mysteries
Mazur's Torsion Theorem- en.wikipedia.org/wiki/Torsion_conjecture#Elliptic_curves
web.math.pmf.unizg.hr/~duje/tors/rankhist.html
The McKay conjecture- www.quantamagazine.org/after-20-years-math-couple-solves-major-group-theory-problem-20250219/
The Mean-value theorem- abesenyei.web.elte.hu/publications/meanvalue.pdf
Menelaus' and Ceva's theorems (and triple product rule)- www.mathpages.com/home/kmath442/kmath442.htm
Non-Euclidean geometry- thonyc.wordpress.com/2013/09/05/the-problem-with-parallels/
The Odd Order Theorem- www.quora.com/What-do-mathematicians-find-the-most-unforgettable-theorem-conjecture-or-proof-of-all-time
Pappus's theorem- A comment on Twitter from @ArthurNewlands reminds me to upgrade this page! "I love this theorem for many reasons. Because it is a special case of Pascal's Hexagrammum Mysticum, which is, in turn, a special case of Cayley–Bacharach; because of its dual form; because it helps characterise a projective plane over a field; because implies Desargues' thm etc."
Penrose's eight-conic theorem- arxiv.org/abs/2409.17150
The Pentagonal Tiling Theorem- www.cnrs.fr/en/press/there-are-only-15-possible-pentagonal-tiles
The PCP Theorem- www.theguardian.com/science/2024/mar/03/can-you-solve-it-the-word-game-at-the-cutting-edge-of-computer-science
The Razborov–Rudich Natural Proofs Theorem- www.quora.com/Do-mathematicians-ever-publish-research-papers-on-what-might-result-if-an-unproven-theorem-is-true
The Riemann Mapping Theorem- kpknudson.com/my-favorite-theorem/2021/4/7/episode-65-howard-masur
terrytao.wordpress.com/tag/riemann-mapping-theorem/
arxiv.org/abs/1604.04071
The Road-colouring Theorem- en.wikipedia.org/wiki/Road_coloring_theorem
Rollé's Theorem- www.quora.com/How-do-you-state-and-prove-Rolles-theorem
Shannon's Theorem- www.cs.princeton.edu/courses/archive/fall16/cos521/Lectures/lec18.pdf
arxiv.org/abs/1405.1003 (quote on p. 11 about why Shannon called it 'entropy')
arxiv.org/abs/1802.05968
Shimizu's region crossing change theorem- projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-66/issue-3/Region-crossing-change-is-an-unknotting-operation/10.2969/jmsj/06630693.full
kpknudson.com/my-favorite-theorem/2023/11/12/episode-89-allison-henrich
Shrikhande's theorem- www.ias.ac.in/describe/article/reso/020/10/0903-0918
Sierpinski's inequality- www.johndcook.com/blog/2024/01/05/sierpinskis-inequality/
mathematicaloddsandends.wordpress.com/2024/01/06/a-quick-proof-of-sierpinskis-inequality/
The Square Peg (Inscribed Square) Problem- www.ams.org/notices/201404/rnoti-p346.pdf
en.wikipedia.org/wiki/Inscribed_square_problem
Steiner–Lehmus Theorem- www.quora.com/Given-a-triangle-with-two-equal-angle-bisectors-must-it-be-an-isosceles-triangle-What-is-the-proof
Steiner's Porism- en.wikipedia.org/wiki/Steiner_chain
Tarski's high school algebra problem- en.wikipedia.org/wiki/Tarski's_high_school_algebra_problem
arxiv.org/abs/2608.08421
The Tarski–Seidenberg theorem- en.wikipedia.org/wiki/Tarski%E2%80%93Seidenberg_theorem
Thurston's Geometrization Conjecture- arxiv.org/abs/math/9404236
The Toeplitz Square Peg Problem- terrytao.wordpress.com/2016/11/22/an-integration-approach-to-the-toeplitz-square-peg-problem/
Tunnell's Theorem- www.math.rug.nl/~top/Chandrasekar.pdf
The Universal Coefficient Theorem in Homology- digitalcommons.colby.edu/cgi/viewcontent.cgi?article=1883&context=honorstheses
Van der Waerden’s Theorem on Arithmetic Progressions- pointatinfinityblog.wordpress.com/tag/van-der-waerden/
www.cs.umd.edu/~gasarch/TOPICS/vdw/GKPbook.pdf
www.sciencedirect.com/science/article/pii/S0097316508000137
arxiv.org/pdf/2102.01543.pdf
Viète's Formulae- en.wikipedia.org/wiki/Vieta%27s_formulas
Yenoda's Lemma- terrytao.wordpress.com/2023/08/25/yonedas-lemma-as-an-identification-of-form-and-function-the-case-study-of-polynomials/
Zermelo's Theorem in Game Theory- en.wikipedia.org/wiki/Zermelo%27s_theorem_(game_theory)