List of conjectures
This is a list of notable mathematical conjectures.
Open problems
The following conjectures remain open. The column "cites" lists the number of results for a Google Scholar search for the term, in double quotes as of 2022.Conjectures now proved (theorems)
The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names.| Priority date | Proved by | Former name | Field | Comments |
| 1962 | Walter Feit and John G. Thompson | Burnside conjecture that, apart from cyclic groups, finite simple groups have even order | finite simple groups | Feit–Thompson theorem⇔trivially the "odd order theorem" that finite groups of odd order are solvable groups |
| 1968 | Gerhard Ringel and John William Theodore Youngs | Heawood conjecture | graph theory | Ringel-Youngs theorem |
| 1971 | Daniel Quillen | Adams conjecture | algebraic topology | On the J-homomorphism, proposed 1963 by Frank Adams |
| 1973 | Pierre Deligne | Weil conjectures | algebraic geometry | ⇒Ramanujan–Petersson conjecture Proposed by André Weil. Deligne's theorems completed around 15 years of work on the general case. |
| 1975 | Henryk Hecht and Wilfried Schmid | Blattner's conjecture | representation theory for semisimple groups | |
| 1975 | William Haboush | Mumford conjecture | geometric invariant theory | Haboush's theorem |
| 1976 | Kenneth Appel and Wolfgang Haken | Four color theorem | graph colouring | Traditionally called a "theorem", long before the proof. |
| 1976 | Daniel Quillen; and independently by Andrei Suslin | Serre's conjecture on projective modules | polynomial rings | Quillen–Suslin theorem |
| 1977 | Alberto Calderón | Denjoy's conjecture | rectifiable curves | A result claimed in 1909 by Arnaud Denjoy, proved by Calderón as a by-product of work on Cauchy singular operators |
| 1978 | Roger Heath-Brown and Samuel James Patterson | Kummer's conjecture on cubic Gauss sums | equidistribution | |
| 1983 | Gerd Faltings | Mordell conjecture | number theory | ⇐Faltings's theorem, the Shafarevich conjecture on finiteness of isomorphism classes of abelian varieties. The reduction step was by Alexey Parshin. |
| 1983 onwards | Neil Robertson and Paul D. Seymour | Wagner's conjecture | graph theory | Now generally known as the graph minor theorem. |
| 1983 | Michel Raynaud | Manin–Mumford conjecture | diophantine geometry | The Tate–Voloch conjecture is a quantitative derived conjecture for p-adic varieties. |
| c.1984 | Collective work | Smith conjecture | knot theory | Based on work of William Thurston on hyperbolic structures on 3-manifolds, with results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, also with Hyman Bass, Cameron Gordon, Peter Shalen, and Rick Litherland, written up by Bass and John Morgan. |
| 1984 | Louis de Branges de Bourcia | Bieberbach conjecture, 1916 | complex analysis | ⇐Robertson conjecture⇐Milin conjecture⇐de Branges's theorem |
| 1984 | Gunnar Carlsson | Segal's conjecture | homotopy theory | |
| 1984 | Haynes Miller | Sullivan conjecture | classifying spaces | Miller proved the version on mapping BG to a finite complex. |
| 1987 | Grigory Margulis | Oppenheim conjecture | diophantine approximation | Margulis proved the conjecture with ergodic theory methods. |
| 1989 | Vladimir I. Chernousov | Weil's conjecture on Tamagawa numbers | algebraic groups | The problem, based on Siegel's theory for quadratic forms, submitted to a long series of case analysis steps. |
| 1990 | Ken Ribet | epsilon conjecture | modular forms | |
| 1992 | Richard Borcherds | Conway–Norton conjecture | sporadic groups | Usually called monstrous moonshine |
| 1994 | David Harbater and Michel Raynaud | Abhyankar's conjecture | algebraic geometry | |
| 1994 | Andrew Wiles | Fermat's Last Theorem | number theory | ⇔The modularity theorem for semistable elliptic curves. Proof completed with Richard Taylor. |
| 1994 | Fred Galvin | Dinitz conjecture | combinatorics | |
| 1995 | Doron Zeilberger | Alternating sign matrix conjecture, | enumerative combinatorics | |
| 1996 | Vladimir Voevodsky | Milnor conjecture | algebraic K-theory | Voevodsky's theorem, ⇐norm residue isomorphism theorem⇔Beilinson–Lichtenbaum conjecture, Quillen–Lichtenbaum conjecture. The ambiguous term "Bloch-Kato conjecture" may refer to what is now the norm residue isomorphism theorem. |
| 1998 | Thomas Callister Hales | Kepler conjecture | sphere packing | |
| 1998 | Thomas Callister Hales and Sean McLaughlin | dodecahedral conjecture | Voronoi decompositions | |
| 2000 | Krzysztof Kurdyka, Tadeusz Mostowski, and Adam Parusiński | Gradient conjecture | gradient vector fields | Attributed to René Thom, c.1970. |
| 2001 | Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor | Taniyama–Shimura conjecture | elliptic curves | Now the modularity theorem for elliptic curves. Once known as the "Weil conjecture". |
| 2001 | Mark Haiman | n! conjecture | representation theory | |
| 2001 | Daniel Frohardt and Kay Magaard | Guralnick–Thompson conjecture | monodromy groups | |
| 2002 | Preda Mihăilescu | Catalan's conjecture, 1844 | exponential diophantine equations | ⇐Pillai's conjecture⇐abc conjecture Mihăilescu's theorem |
| 2002 | Maria Chudnovsky, Neil Robertson, Paul D. Seymour, and Robin Thomas | strong perfect graph conjecture | perfect graphs | Chudnovsky–Robertson–Seymour–Thomas theorem |
| 2002 | Grigori Perelman | Poincaré conjecture, 1904 | 3-manifolds | |
| 2003 | Grigori Perelman | geometrization conjecture of Thurston | 3-manifolds | ⇒spherical space form conjecture |
| 2003 | Ben Green; and independently by Alexander Sapozhenko | Cameron–Erdős conjecture | sum-free sets | |
| 2003 | Nils Dencker | Nirenberg–Treves conjecture | pseudo-differential operators | |
| 2004 | Nobuo Iiyori and Hiroshi Yamaki | Frobenius conjecture | group theory | A consequence of the classification of finite simple groups, completed in 2004 by the usual standards of pure mathematics. |
| 2004 | Adam Marcus and Gábor Tardos | Stanley–Wilf conjecture | permutation classes | Marcus–Tardos theorem |
| 2004 | Ualbai U. Umirbaev and Ivan P. Shestakov | Nagata's conjecture on automorphisms | polynomial rings | |
| 2004 | Ian Agol; and independently by Danny Calegari–David Gabai | tameness conjecture | geometric topology | ⇒Ahlfors measure conjecture |
| 2008 | Avraham Trahtman | Road coloring conjecture | graph theory | |
| 2008 | Chandrashekhar Khare and Jean-Pierre Wintenberger | Serre's modularity conjecture | modular forms | |
| 2009 | Jeremy Kahn and Vladimir Markovic | surface subgroup conjecture | 3-manifolds | ⇒Ehrenpreis conjecture on quasiconformality |
| 2009 | Jeremie Chalopin and Daniel Gonçalves | Scheinerman's conjecture | intersection graphs | |
| 2010 | Terence Tao and Van H. Vu | circular law | random matrix theory | |
| 2011 | Joel Friedman; and independently by Igor Mineyev | Hanna Neumann conjecture | group theory | |
| 2012 | Simon Brendle | Hsiang–Lawson's conjecture | differential geometry | |
| 2012 | Fernando Codá Marques and André Neves | Willmore conjecture | differential geometry | |
| 2013 | Yitang Zhang | bounded gap conjecture | number theory | The sequence of gaps between consecutive prime numbers has a finite lim inf. See Polymath Project#Polymath8 for quantitative results. |
| 2013 | Adam Marcus, Daniel Spielman and Nikhil Srivastava | Kadison–Singer problem | functional analysis | The original problem posed by Kadison and Singer was not a conjecture: its authors believed it false. As reformulated, it became the "paving conjecture" for Euclidean spaces, and then a question on random polynomials, in which latter form it was solved affirmatively. |
| 2015 | Jean Bourgain, Ciprian Demeter, and Larry Guth | Main conjecture in Vinogradov's mean-value theorem | analytic number theory | Bourgain–Demeter–Guth theorem, ⇐ decoupling theorem |
| 2018 | Karim Adiprasito | g-conjecture | combinatorics | |
| 2019 | Dimitris Koukoulopoulos and James Maynard | Duffin–Schaeffer conjecture | number theory | Rational approximation of irrational numbers |
Disproved (no longer conjectures)
The conjectures in following list were not necessarily generally accepted as true before being disproved.- Atiyah conjecture
- Borsuk's conjecture
- Chinese hypothesis
- Doomsday conjecture
- Euler's sum of powers conjecture
- Ganea conjecture
- Generalized Smith conjecture
- Hauptvermutung
- Hedetniemi's conjecture, counterexample announced 2019
- Hirsch conjecture
- Kaplansky unit conjecture
- Intersection graph conjecture
- Kelvin's conjecture
- Kouchnirenko's conjecture
- Mertens conjecture
- Pólya conjecture, 1919
- Ragsdale conjecture
- Schoenflies conjecture
- Tait's conjecture
- Von Neumann conjecture
- Weyl–Berry conjecture
- Williamson conjecture
Refutations of conventional wisdom
In contemporary mathematics, ideas are supposedly not accepted absent rigorous proof. Historically, some statements that were generally accepted were then shown to be false.- The idea of the Pythagoreans that all numbers can be expressed as a ratio of two whole numbers. This was disproved by one of Pythagoras' own disciples, Hippasus, who showed that the square root of two is what we today call an irrational number. One story claims that he was thrown off the ship in which he and some other Pythagoreans were sailing because his discovery was too heretical.
- Euclid's parallel postulate stated that if two lines cross a third in a plane in such a way that the sum of the "interior angles" is not 180° then the two lines meet. Furthermore, he implicitly assumed that two separate intersecting lines meet at only one point. These assumptions were believed to be true for more than 2000 years, but in light of General Relativity at least the second can no longer be considered true. In fact the very notion of a straight line in four-dimensional curved space-time has to be redefined, which one can do as a geodesic. It is now recognized that Euclidean geometry can be studied as a mathematical abstraction, but that the universe is non-Euclidean.
- Fermat conjectured that all numbers of the form were prime. However, this conjecture was disproved by Euler, who found that
- The idea that transcendental numbers were unusual. Disproved by Georg Cantor who showed that there are so many transcendental numbers that it is impossible to make a one-to-one mapping between them and the algebraic numbers. In other words, the cardinality of the set of transcendentals is greater than that of the set of algebraic numbers.
- Bernhard Riemann, at the end of his famous 1859 paper "On the Number of Primes Less Than a Given Magnitude", stated that the logarithmic integral gives a somewhat too high estimate of the prime-counting function. The evidence also seemed to indicate this. However, in 1914 J. E. Littlewood proved that this was not always the case, and in fact it is now known that the first x for which occurs somewhere before 10317. See Skewes' number for more detail.
- Naïvely it might be expected that a continuous function must have a derivative or else that the set of points where it is not differentiable should be "small" in some sense. This was disproved in 1872 by Karl Weierstrass, and in fact examples had been found earlier of functions that were nowhere differentiable. According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that such functions did not exist.
- It was conjectured in 1919 by George Pólya, based on the evidence, that most numbers less than any particular limit have an odd number of prime factors. However, this Pólya conjecture was disproved in 1958. It turns out that for some values of the limit, most numbers less than the limit have an even number of prime factors.