Theorem of the Day



Acknowledgements 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 fitnessforpurpose of any part of these pages (and to whom no residual errors may be attributed): Colin Adams, Louigi AddarioBerry, 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 BarrowGreen, 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, Andreas Brusinsky, Edward B. Burger, Piers BursillHall, 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, Vašek 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, Michael N. Fried, John Friedlander, Alan Freize, Stephen M. Gaglola III, Frank G. Garvan, Ellen Gethner, Julian Gilbey, Colin Glanville, 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 HeathBrown, Stephan Hell, Ted Hill, Anthony Hilton, Dennis Paul Himes, Nigel Hitchen, Susan Holmes, Fred Holroyd, Dragos Hrimiuc, Stephen Huggett, David Ireland, 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, HongJian Lai, TsitYuen 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, Keith M. Martin, 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, JeanJacques 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 for whose reproduction I have been unable to gain authorisation. 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. Euler's Product Formula for ζ(s) (screen shot of Euler Archive) Gauss' Law of Quadratic Reciprocity (image of Gauss) The Pythagorean Theorem (photo of Edsger Dijkstra) Ramsey's Theorem (photo of Paul Erdös)
Particular thanks to the following organisations... The American Mathematical Society for permission to reproduce from Paul R. Halmos' I Have a Photographic Memory the photograph of Derrick H. Lehmer used to help illustrate the Lucas–Lehmer Test. The Archimedeans for permission to reproduce two poems from Eureka to illustrate the Contraction Mapping Theorem. 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. DailyNewsPix for permission to reproduce from the New York Daily News, the picture of Roger Federer illustrating the Total Probability Theorem. IKEA for permission to use an image from their instruction booklets in the illustration for Karp's Theorem (Detail). Latimer Clarke Corporation Pty Ltd, for permission to use a map from Atlapedia to illustrate the Four Colour Theorem (in the version replaced on 07.10.12) London South Bank University for temporarily hosting pdf files of the early theorem descriptions belonging to the site. The Mathematical Chronicle Committee for permission to reproduce online Garry Tee's 1977 Math Chronicle article on Kovalevskaya to link from The Cauchy–Kovalevskaya Theorem. The Math Forum for permission to use their image of Pascal's triangle to illustrate Lucas' Theorem. Nature Publishing Group for permission to reproduce Frederick Soddy's poem in the Descartes Circle Theorem. Perseus Book Group for permission to quote the dedicatory poem from Harary's Graph Theory in Kuratowski's Theorem. 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. The World Meteorological Organization for permission to use screen shots from its weather information service to help illustrate the Borsuk–Ulam Theorem.
...and to the following individuals: Colin Adams for helping with the accuracy of some of the text in Haken's Unknot Theorem Reza Akhtar for allowing me to host his fine notes on elliptic curves, linked from The Lutz–Nagell Theorem. Emmanuel Amiot for help with the translation into French of Le Porisme de Poncelet (any errors which may have crept back in are mine, obviously). Shamil Asgarli, for providing me with an updated version of his elegant summary of three proofs of Wedderburn's Little Theorem, which provides the recommended weblink from that theorem description. Eric Bach for providing a pdf of his lecture notes, being the web link from Lamé's Theorem. Robert F. Bailey for permission to use his image of the BiggsSmith graph to illustrate Cameron's theorem on distancetransitive graphs. Valérie Berthé for sending me the .pdf version of her survey article with Pascal Alessandri, being the web link from the ThreeDistance Theorem. Petter Brändén for providing me with the historical background and modern context for a Theorem of Schur on RealRootedness. 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. Alan Camina for suggesting the Stable Marriage Theorem. John Caughman for making available a pdf version of Katie Leonard's dissertation for the web link from the Friendship Theorem. 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. Ellen Gethner, for shedding light on the origins of the Gaussian Moat Problem, as recorded in the notes to the corresponding theorem. Tony Forbes for permission to use his jpeg image of consecutive primes in arithmetic progression to illustrate the GreenTao 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. Jessica Fridrich for permission to use images of Rubik's cube from her Speed cubing site (check out the video clips of her sub15second cubing!) to illustrate Fermat's Little Theorem. Len Goodman for alerting me to several technical problems with this website. Ernst Hairer for providing me with a pretty geometrical interpretation of integration by parts to link to from Theorem 219 (notes). Thomas Hales for patiently correcting my misconceptions about Kepler's Conjecture. John H. Halton for giving me much help in preparing the Beardwood–Halton–Hammersley Theorem. The index of mathematicians here was his idea too. Stephen Hell for drawing my attention to the Dutch Cheese Problem, featured in The Tverberg Partition Theorem, which also benefited from comments by Jiří Matoušek and Tverberg himself. Dennis Paul Himes for providing me with a 2nd example of a naturally occuring K_{4} in map colouring: Paraguay, Bolivia, Brazil, Argentina, for the Four Colour Theorem. Allyn Jackson for helping me track down cameo appearances of theorems by pointing me to all sorts of resources about mathematics and the arts. Samuel R. Kaplan, for helping me with the history of the 'Dottie Number', which illustrates the Lindemann–Weierstrass Theorem. Donald Keedwell, for alerting me to the fact that the Lagrange Property for Moufang Loops had been proved. And for helpfully checking my presentation of this result. 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. Clark Kimberling for enlightening me on the history of Zeckendorf's Theorem. Tinne Hoff Kjeldsen, for providing me with a pdf of her wonderful paper on von Nuemann's Minimax Theorem to link to from that theorem. Anthony Knapp has been a generous source of advice on everything from the Cauchy Problem to cohomology (and seems to have written awardwinning textbooks or monographs on most of it!) The Cameos page here was his idea too. Ron Knott for helpful comments and suggestions regarding Fibonaccirelated theorems, in particular Zeckendorf's Theorem. Erik Koelink for suggesting the automated theorem display facility. Robert Lockhart has made detailed and invaluable comments on many of the theorems. William McCune for alerting me to the amazingly short singleaxiom 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. Gérard Michon for the nice idea to add a 'heraldry' section to the Cameos page; and for providing the image of Legendre for the illustration of the Law of Quadratic Reciprocity. 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. Peter M. Neumann for telling me about Praeger's Theorem on Bounded Movement and for helping me get it right. Emil Vaughan helped too. James Oxley for helping me to elucidate somewhat the more obscure bits of my presentation of the Piff–Welsh Theorem. Lynn Pearson for providing background on the mathematical tiling pattern used to illustrate the Handshaking Lemma. Andrew Rich for helping me get a correct formal definition of the Sharkovsky ordering in Sharkovsky's Theorem; pointing out a nonsequitur 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. Jon Selig who has commented on numerous entries and suggested quite a few, including the Contraction Mapping Theorem and the Greibach Normal Form Theorem. David Singerman for drawing my attention to the Theorem on Modular Fibonacci Periodicity and for showing me his 'book' proof of even periodicity. 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). Morwen Thistlethwaite for some informative comments on Reidemeister's Theorem, and for providing me with a jpeg solution to his unknot puzzle, which I used to illustrated Haken's Unknot Theorem. 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. Zach Teitler gave me a lot of help getting my presentation right for Sylvester's Catalecticant. Petr Vojtěchovský for drawing my attention to Gagola III and Hall's independent proof of the Lagrange Property for Moufang Loops. Bert Wehrfritz for helping me to choose and present, as a memorial tribute, Gruenberg's Theorem on Nilpotent Groups. Dominic Welsh for telling me about the counterexample showing that graphs are not wellquasiordered by topological minors (the Robertson–Seymour Graph Minor Theorem). Jon Woodring for permission to use his ripples graphic to illustrate the First Isomorphism Theorem. Taoyang Wu for suggesting the Rotation Distance Bound as a memorial tribute for William Thurston, and for helping get it right. 