This is a list of notable mathematical conjectures.
The following conjectures remain open. The (incomplete) column "cites" lists the number of results for a Google Scholar search for the term, in double quotes .
Conjecture | Field | Comments | Eponym(s) | Cites | |
---|---|---|---|---|---|
1/3–2/3 conjecture | order theory | n/a | 70 | ||
abc conjecture | number theory | ⇔Granville–Langevin conjecture, Vojta's conjecture in dimension 1 ⇒Erdős–Woods conjecture, Fermat–Catalan conjecture Formulated by David Masser and Joseph Oesterlé.[1] Proof claimed in 2012 by Shinichi Mochizuki | n/a | 2440 | |
Agoh–Giuga conjecture | number theory | Takashi Agoh and Giuseppe Giuga | 8 | ||
Agrawal's conjecture | number theory | 10 | |||
Andrews–Curtis conjecture | combinatorial group theory | 358 | |||
Andrica's conjecture | number theory | Dorin Andrica | 45 | ||
Artin conjecture (L-functions) | number theory | 650 | |||
Artin's conjecture on primitive roots | number theory | ⇐generalized Riemann hypothesis[2] ⇐Selberg conjecture B[3] | 325 | ||
Bateman–Horn conjecture | number theory | 245 | |||
Baum–Connes conjecture | operator K-theory | ⇒Gromov-Lawson-Rosenberg conjecture[4] ⇒Kaplansky-Kadison conjecture ⇒Novikov conjecture | 2670 | ||
Beal's conjecture | number theory | 142 | |||
Beilinson conjecture | number theory | 461 | |||
Berry–Tabor conjecture | geodesic flow | Michael Berry and Michael Tabor | 239 | ||
Big-line-big-clique conjecture | discrete geometry | ||||
Birch and Swinnerton-Dyer conjecture | number theory | 2830 | |||
Birch–Tate conjecture | number theory | 149 | |||
Birkhoff conjecture | integrable systems | 345 | |||
Bloch–Beilinson conjectures | number theory | 152 | |||
Bloch–Kato conjecture | algebraic K-theory | 1620 | |||
Bochner–Riesz conjecture | harmonic analysis | ⇒restriction conjecture⇒Kakeya maximal function conjecture⇒Kakeya dimension conjecture[5] | 236 | ||
Bombieri–Lang conjecture | diophantine geometry | 181 | |||
Borel conjecture | geometric topology | 981 | |||
Bost conjecture | geometric topology | 65 | |||
Brennan conjecture | complex analysis | James E. Brennan | 110 | ||
Brocard's conjecture | number theory | 16 | |||
Brumer–Stark conjecture | number theory | 208 | |||
Bunyakovsky conjecture | number theory | 43 | |||
Carathéodory conjecture | differential geometry | 173 | |||
Carmichael totient conjecture | number theory | ||||
Casas-Alvero conjecture | polynomials | Eduardo Casas-Alvero | 56 | ||
Catalan–Dickson conjecture on aliquot sequences | number theory | 46 | |||
Catalan's Mersenne conjecture | number theory | Eugène Charles Catalan | |||
Cherlin–Zilber conjecture | group theory | 86 | |||
Chowla conjecture | Möbius function | ⇒Sarnak conjecture[6] [7] | |||
Collatz conjecture | number theory | 1440 | |||
Cramér's conjecture | number theory | 32 | |||
Conway's thrackle conjecture | graph theory | 150 | |||
Deligne conjecture | monodromy | 788 | |||
Dittert conjecture | combinatorics | Eric Dittert | 11 | ||
Eilenberg−Ganea conjecture | algebraic topology | 96 | |||
Elliott–Halberstam conjecture | number theory | 300 | |||
Erdős–Faber–Lovász conjecture | graph theory | 172 | |||
Erdős–Gyárfás conjecture | graph theory | 37 | |||
Erdős–Straus conjecture | number theory | 103 | |||
Farrell–Jones conjecture | geometric topology | 545 | |||
Filling area conjecture | differential geometry | n/a | 60 | ||
Firoozbakht's conjecture | number theory | 33 | |||
Fortune's conjecture | number theory | 16 | |||
Four exponentials conjecture | number theory | n/a | 110 | ||
Frankl conjecture | combinatorics | 83 | |||
Gauss circle problem | number theory | 553 | |||
Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane | metric geometry | ||||
Gilbreath conjecture | number theory | 34 | |||
Goldbach's conjecture | number theory | ⇒The ternary Goldbach conjecture, which was the original formulation.[8] | 5880 | ||
Gold partition conjecture[9] | order theory | n/a | 25 | ||
Goldberg–Seymour conjecture | graph theory | 57 | |||
Goormaghtigh conjecture | number theory | 14 | |||
Green's conjecture | algebraic curves | 150 | |||
Grimm's conjecture | number theory | Carl Albert Grimm | 46 | ||
Grothendieck–Katz p-curvature conjecture | differential equations | 98 | |||
Hadamard conjecture | combinatorics | 858 | |||
Herzog–Schönheim conjecture | group theory | Marcel Herzog and Jochanan Schönheim | 44 | ||
Hilbert–Smith conjecture | geometric topology | 219 | |||
Hodge conjecture | algebraic geometry | 2490 | |||
Homological conjectures in commutative algebra | commutative algebra | n/a | |||
Hopf conjectures | geometry | 476 | |||
Ibragimov–Iosifescu conjecture for φ-mixing sequences | probability theory | Ildar Ibragimov, | |||
Invariant subspace problem | functional analysis | n/a | 2120 | ||
Jacobian conjecture | polynomials | Carl Gustav Jacob Jacobi (by way of the Jacobian determinant) | 2860 | ||
Jacobson's conjecture | ring theory | 127 | |||
Kaplansky conjectures | ring theory | 466 | |||
Keating–Snaith conjecture | number theory | 48 | |||
Köthe conjecture | ring theory | 167 | |||
Kung–Traub conjecture | iterative methods | 332 | |||
Legendre's conjecture | number theory | 110 | |||
Lemoine's conjecture | number theory | 13 | |||
Lenstra–Pomerance–Wagstaff conjecture | number theory | 32 | |||
Leopoldt's conjecture | number theory | 773 | |||
List coloring conjecture | graph theory | n/a | 300 | ||
Littlewood conjecture | diophantine approximation | ⇐Margulis conjecture[10] | 1230 | ||
Lovász conjecture | graph theory | 560 | |||
MNOP conjecture | algebraic geometry | n/a | 63 | ||
Manin conjecture | diophantine geometry | 338 | |||
Marshall Hall's conjecture | number theory | 44 | |||
Mazur's conjectures | diophantine geometry | 97 | |||
Montgomery's pair correlation conjecture | number theory | 77 | |||
n conjecture | number theory | n/a | 126 | ||
New Mersenne conjecture | number theory | 47 | |||
Novikov conjecture | algebraic topology | 3090 | |||
Oppermann's conjecture | number theory | 12 | |||
Petersen coloring conjecture | graph theory | 52 | |||
Pierce–Birkhoff conjecture | real algebraic geometry | 96 | |||
Pillai's conjecture | number theory | 33 | |||
De Polignac's conjecture | number theory | 46 | |||
Quantum PCP conjecture | quantum information theory | ||||
quantum unique ergodicity conjecture | dynamical systems | 2004, Elon Lindenstrauss, for arithmetic hyperbolic surfaces,[11] 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfaces[12] | n/a | 281 | |
Reconstruction conjecture | graph theory | n/a | 1040 | ||
Riemann hypothesis | number theory | ⇐Generalized Riemann hypothesis⇐Grand Riemann hypothesis ⇔De Bruijn–Newman constant=0 ⇒density hypothesis, Lindelöf hypothesis See Hilbert–Pólya conjecture. For other Riemann hypotheses, see the Weil conjectures (now theorems). | Bernhard Riemann | 24900 | |
Ringel–Kotzig conjecture | graph theory | 187 | |||
Rudin's conjecture | additive combinatorics | 16 | |||
Sarnak conjecture | topological entropy | 295 | |||
Sato–Tate conjecture | number theory | 1080 | |||
Schanuel's conjecture | number theory | 329 | |||
Schinzel's hypothesis H | number theory | 49 | |||
Scholz conjecture | addition chains | 41 | |||
Second Hardy–Littlewood conjecture | number theory | 30 | |||
Selfridge's conjecture | number theory | 6 | |||
Sendov's conjecture | complex polynomials | 77 | |||
Serre's multiplicity conjectures | commutative algebra | 221 | |||
Singmaster's conjecture | binomial coefficients | 8 | |||
Standard conjectures on algebraic cycles | algebraic geometry | n/a | 234 | ||
Tate conjecture | algebraic geometry | ||||
Toeplitz' conjecture | Jordan curves | ||||
Tuza's conjecture | graph theory | Zsolt Tuza | |||
Twin prime conjecture | number theory | n/a | 1700 | ||
Ulam's packing conjecture | packing | ||||
Unicity conjecture for Markov numbers | number theory | Andrey Markov (by way of Markov numbers) | |||
Uniformity conjecture | diophantine geometry | n/a | |||
Unique games conjecture | number theory | n/a | |||
Vandiver's conjecture | number theory | ||||
Virasoro conjecture | algebraic geometry | ||||
Vizing's conjecture | graph theory | ||||
Vojta's conjecture | number theory | ||||
Waring's conjecture | number theory | ||||
Weight monodromy conjecture | algebraic geometry | n/a | |||
Weinstein conjecture | periodic orbits | ||||
Whitehead conjecture | algebraic topology | ||||
Zauner's conjecture | operator theory | Gerhard Zauner |
The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names.
Priority date[13] | 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
| 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[14] | |
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 (diophantine approximation) 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[15] | |
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[16] | 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[17] | 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
| Cameron–Erdős conjecture | sum-free sets | ||
2003 | Nils Dencker | Nirenberg–Treves conjecture | pseudo-differential operators | ||
2004 (see comment) | 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
| 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[18] | |
2018 | Karim Adiprasito | g-conjecture | combinatorics | ||
2019 | Dimitris Koukoulopoulos and James Maynard | Duffin–Schaeffer conjecture | number theory | Rational approximation of irrational numbers |
The conjectures in following list were not necessarily generally accepted as true before being disproved.
In mathematics, ideas are supposedly not accepted as fact until they have been rigorously proved. However, there have been some ideas that were fairly accepted in the past but which were subsequently shown to be false. The following list is meant to serve as a repository for compiling a list of such ideas.
2m | |
2 |
+1
25 | |
2 |
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