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.
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/
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/
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
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/
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
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/
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
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/
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 l
Laplace's Rule of Succession
jonathanweisberg.org/post/inductive-logic-2/ l
Legendre's Theorem on Factorial Factors
www.cut-the-knot.org/blue/LegendresTheorem.shtm l
The Lonely Runner Conjecture
terrytao.wordpress.com/2017/01/10/some-remarks-on-the-lonely-runner-conjecture/
The Mandelbrot set is connected
www.quantamagazine.org/the-quest-to-decode-the-mandelbrot-set-maths-famed-fractal-20240126/
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 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
The Shoelace formula
gdz.sub.uni-goettingen.de/id/PPN352846038_0001
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 theoremm
en.wikipedia.org/wiki/Tarski%E2%80%93Seidenberg_theorem
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)