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.

 

Main List of Theorems
     
Theorem of the Day Home Page

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

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/

Combinatorial Nullstellensatz
en.wikipedia.org/wiki/Restricted_sumset

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

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/

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

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

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

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)
Main List of Theorems
     
Theorem of the Day Home Page