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 and, as I come across them, record web resources to help me. Links to pageswhich I want to study in the hopes of improving existing theorems 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.
https://scilogs.spektrum.de/hlf/an-introduction-to-representation-theory/
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 Körnerwww.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 Bertrand's Postulate web.williams.edu/Mathematics/lg5/Chebyshev.pdf Proofarxiv.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 BPoeis.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 nBezout'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/ 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 en.wikipedia.org/wiki/Circle_packing_theorem 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 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 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 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 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/ The Gauss Class Number Problem en.wikipedia.org/wiki/Harold_Stark 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 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/ download.arxiv.org/pdf/2003.01798v2.pdf 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/ 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 The Isoperimetric Inequality projecteuclid.org/journals/journal-of-differential-geometry/volume-6/issue-2/A-generalization-of-the-isoperimetric-inequality/10.4310/jdg/1214430403.full lwww.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 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/ 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.shtm lThe 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 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."
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 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/ Tarski's high school algebra problem en.wikipedia.org/wiki/Tarski's_high_school_algebra_problem 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)
