Theorem of the Day
Many organisations and people have contributed to this site. I have generally preferred to acknowledge this contribution here rather than clutter up individual theorems.
The value of any mathematical resource is predicated on its accuracy. I owe a great debt to all the people (some of whom are mentioned again further down this page) who have confirmed or improved the mathematical fitness-for-purpose of any part of these pages (and to whom no residual errors may be attributed): Colin Adams, Louigi Addario-Berry, Fatima Affif Chaouche, Reza Akhtar, Emmanuel Amiot, Ian Anderson, George E. Andrews, David L. Applegate, Krzysztof Apt, John Arhin, Shamil Asgarli, Dirk Augustin, Eric Bach, Jenny Baglivo, R.A. Bailey, Robert F. Bailey, June Barrow-Green, Kathleeen Bednar, Lowell Beineke, Pat Ballew, Arthur T. Benjamin, Curtis D. Bennett, Dave Benson, Valérie Berthé, Joan Birman, Justin Bledin, Alexander Bogomolny, Ethan D. Bolker, Adrian Bondy, Cliff Bott, Petter Brändén, Lev Bregman, Richard Brent, David M. Bressoud, Graham Brightwell, David Broadhurst, Ian Bruce, Derek Bruff, Edward B. Burger, Piers Bursill-Hall, Martin Bush, Steve Butler, Georgina Bygott, Peter Cameron, David Cariolaro, William A. Casselman, J.W.S. Cassels, Melody Chan, Alan Chang, Martin Chaplin, Philippe Chassaing, Amanda Chetwynd, Fan Chung, Vaek Chvátal, James Colliander, Bill Cook, Colin Cooper, Henry Crapo, Andrew I. Dale, Patrick Dehornoy, Satyan Devadoss, John D. Dixon, Dragomir Ž. Djoković, Peter Doyle, John Drost, Ioana Dumitriu, Mathieu Dutour Sikirić, Freeman Dyson, Kimmo Eriksson, Stewart N. Ethier, Anthony B. Evans, Lianghuo Fan, Włodzimierz Fechner, Rachel Fewster, Steven R. Finch, Kevin Ford, Jacob Fox, Tony Forbes, John Friedlander, Alan Freize, Stephen M. Gaglola III, Frank G. Garvan, Julian Gilbey, Edray Herber Goins, Leo Goldmakher, Dan Goldston, Len Goodman, Ronald Graham, Mike Grannell, Jack E. Graver, Sheila Greibach, Michael Griffiths, Terry Griggs, Charles M. Grinstead, Xiaoyang Gu, Ernst Hairer, Thomas Hales, Jonathan I. Hall, Martin Hall, Liam Halpenny, John H. Halton, Denis Hanson, Michael Hardy, Edmund Harriss, Sarah Hart, Jo Heath, Roger Heath-Brown, Stephan Hell, Ted Hill, Anthony Hilton, Dennis Paul Himes, Nigel Hitchen, Susan Holmes, Fred Holroyd, Dragos Hrimiuc, Stephen Huggett, Allyn Jackson, Bill Jackson, Graham Jameson, Nils o. Janus, Kevin Jardine, Oliver Jenkinson, Mark Jerrum, Peter Jones, Tibor Jordán, Stasys Jukna, Dan Kalman, Daniel M. Kane, Samuel R. Kaplan, Gyula O.H. Katona, Louis Kauffman, Donald Keedwell, Michelle Kendall, Eddie Kent, Clark Kimberling, Tinne Hoff Kjeldsen, Tony Knapp, Ron Knott, Donald E. Knuth, Cheng Yeaw Ku, Jeffrey Lagarias, Hong-Jian Lai, Tsit-Yuen Lam, Joseph M. Landsberg, David W. Lewis, Marie Lhuissier, Elliott H. Lieb, Alexei Lisitsa, Allan Siu Lun Lo, Robert Lockhart, Mark de Longueville, Heather Lonsdale, Jack Lutz, Ben Lynn, Matthew Macauley, Joseph Malkevitch, Claudia Malvenuto, Paolo Mancosu, Jan Marthedal Rasmussen, Jiří Matoušek, Thomas Mattman, Elvira Mayordomo, William McCune, Annabelle McIver, Mark McKinzie, Lin McMullin, Shawn McMurran, Francesca Merola, Stephan Mertens, Tauno Metsänkylä, Silvio Micali, Daniele Micciancio, Gérard Michon, Cole Miller, Jennifer Miller, G. Eric Moorhouse, Frank Morgan, Piers Myers, Paul J. Nahin, Peter M. Neumann, Olavi Nevanlinna, Duane Nykamp, John O'Conner, Ken Ono, Joseph O'Rourke, James Oxley, Karen Parshall, Lynn Pearson, David Penman, Florian Pfender, Nathan Pflueger, Mike Piff, János Pintz, Tony Phillips, Lionel Pournin, Cheryl Praeger, Vaughan Pratt, Andrew N. Pressley, Jean-Jacques Quisquater, Shreevatsa R, Andrew Ranicki, Clifford A. Reiter, Zinovy Reichstein, Marc Renault, Bruce Reznick, Andrew Rich, Larry Riddle, Peter J. Roquette, Jeremy Rouse, Piotr Rudnicki, Dave Rusin, Carrie Rutherford, Sy Samet, Carla D. Savage, Werner Schachinger, Winfried Scharlau, René Schoof, Gary Seitz, Jon Selig, Blagovest Sendov, Jeffrey O. Shallit, David Singerman, David Singmaster, Daniel Sleator, José Soto, Kannan Soundararajan, Joel Spencer, Richard P. Stanley, Katie Steckles, J. Michael Steele, Jacqueline A. Stedall, Ken Stephenson, Shaun Stevens, Jan Šťovíček, Volker Strassen, Francis Edward Su, Stanisław Swierczkowski, Serge Tabachnikov, John Talbot, Robert Tarjan, Jim Tattersall, Gregory Taylor, Garry J. Tee, Zach Teitler, Morwen Thistlethwaite, Andrew Thomason, Helge Tverberg, Rémi Vaillancourt, Glen Van Brummelen, Emil Vaughan, Petr Vojtěchovský, Walter Wallis, Ian Wanless, Benjamin Wardhaugh, Matthew Watkins, Bert Wehrfritz, Benjamin Weiss, Dominic Welsh, David L. Wilson, Robin J. Wilson, Ruth Williams, Douglas Woodall, Larry Wos, Taoyang Wu, Catherine Yan, Cem Yıldırım, Qiaochu Yuan, Doron Zeilberger.
Spreadsheet screen shots are from OpenOffice Calc, except where individually acknowledged.
Some pages have photo images whose ownership I have been unable to trace. Apologies for any infringements and please let me know if you are due any acknowledgement or know of some person or organisation who may be.
Gauss' Law of Quadratic Reciprocity (image of Gauss)
The Merton College Theorem (photos of Merton and Oriel Colleges)
The Pythagorean Theorem (photo of Edsger Dijkstra)
Ramsey's Theorem (photo of Paul Erdös)
Particular thanks to the following organisations...
The Australian Mathematical Society for permission to host a digitised copy of Garry J. Tee's article "Russian Peasant Multiplication and Egyptian Division in Zeckendorf Arithmetic", Australian Mathematical Society Gazette, vol. 30, no. 5, 2003, 267–276, in connection with Zeckendorf's Theorem.
London South Bank University for temporarily hosting pdf files of the early theorem descriptions belonging to the site.
The images of the 2007 Presidential $1 Coin used in the illustration of The Diaconis–Holmes–Montgomery Coin Tossing Theorem are from the United States Mint.
...and to the following individuals:
Adrian Bondy for permission to use (in default of it not being actually by somebody else!) the picture of Peter Cameron which appears in the illustration to The Girard–Newton Identities; and for drawing my attention to his 'book' proof of Ore's Theorem in Graph Theory and for helping me correct my French translation (residual errors are mine!)
Peter Cameron for drawing my attention to Babai's 1989 proof of Dixon's conjecture about Netto's conjecture; and to R.A. Bailey's 1984 paper which provided the material for Bailey's Theorem on Latin Squares; for making available the pdf of his short article on parking functions for The Parking Function Formula; and for helpful comments on several other theorems.
Mike Child for giving me the java for automatically updating the 'today's theorem' links and for help getting automated theorem display working.
Ugo Crépin for suggesting Thales' Theorem.
Tony Forbes for permission to use his jpeg image of consecutive primes in arithmetic progression to illustrate the Green-Tao Theorem; for providing me with the data necessary to create the image in the Beardwood–Halton–Hammersley Theorem; for the subtle suggestion to feature the Lucas–Lehmer Test as theorem no. 127; for showing me change of variables trick that evaluates the double integral of exp(-x^2), which illustrates The Change of Variables Theorem; for pointing out to me that Dottie's number, illustrating The Lindemann–Weierstrass Theorem, is not only irrational but transcendental; for advising me that Littlewood's famous remark about Ramanujan (see the Ramanujan Partition Congruences) was not necessarily Littlewood's; and for helpful feedback on many other theorems.
Len Goodman for alerting me to several technical problems with this website.
Michelle Kendall, for showing the example of key distribution in cryptography in which pairwise independent events do not form an independent collection, which I adapted for the The Lovász Local Lemma.
Anthony Knapp has been a generous source of advice on everything from the Cauchy Problem to cohomology (and seems to have written award-winning textbooks or monographs on most of it!) The Cameos page here was his idea too.
Robert Lockhart has made detailed and invaluable comments on many of the theorems.
William McCune for alerting me to the amazingly short single-axiom specification of Boolean Algebra, which appears in the description of The Robbins Problem; and for permission to reproduce part of EQP's solution to that problem.
Justin Mullins for permission to use his art interpretation of Euler's Identity, which for a long time acted as a fitting icon on the home page of this site.
Andrew Rich for helping me get a correct formal definition of the Sharkovsky ordering in Sharkovsky's Theorem; pointing out a non-sequitur in my original description of the Three Distance Theorem; and fixing various other little problems!
Piotr Rudnicki, for the interesting comments regarding Polish nomenclature which I have incorporated into the Remainder Theorem. (He has an interesting paper giving a `formalization' of the closely related Factor Theorem.)
Carrie Rutherford for insightful suggestions concerning too many of these theorems to mention individually.
Stanisław Swierczkowski for providing me with background historical information on the Three Distance Theorem.
Garry J. Tee for bringing my attention to his very nice article "Up with Determinants" (Image, vol. 30) in connection with the Matrix Tree Theorem; for pointing out to me that Daniel Bernoulli was also amoung those who proved Binet's Formula before him; for sending me a scan of his 1977 Math Chronicle article on Kovalevskaya to link from The Cauchy–Kovalevskaya Theorem (courtesy New Zealand Mathematical Society); and a scan of his 2003 Australian Mathematical Society Gazette article on Zeckendor arithmetic to link from Zeckendorf's Theorem (courtesy Australian Mathematical Society).
Emil Vaughan for providing the php scripts for picking today's theorem. Also for the excellent vector graphics for the Theorem on Apollonian Circle Packings; and for suggesting Vaughan Pratt's Theorem (and many others which I will surely get round to eventually!) and for telling me about the progress of Kalkowski et al on Graph Colouring by Vertex Degree.
Petr Vojtěchovský for drawing my attention to Gagola III and Hall's independent proof of the Lagrange Property for Moufang Loops.
Jon Woodring for permission to use his ripples graphic to illustrate the First Isomorphism Theorem.