List of conjectures

Last updated

This is a list of notable mathematical conjectures .

Contents

Open problems

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 as of September 2022.

ConjectureFieldCommentsEponym(s)Cites
1/3–2/3 conjecture order theoryn/a70
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/a2440
Agoh–Giuga conjecture number theoryTakashi Agoh and Giuseppe Giuga8
Agrawal's conjecture number theory Manindra Agrawal 10
Andrews–Curtis conjecture combinatorial group theory James J. Andrews and Morton L. Curtis 358
Andrica's conjecture number theoryDorin Andrica45
Artin conjecture (L-functions) number theory Emil Artin 650
Artin's conjecture on primitive roots number theorygeneralized Riemann hypothesis [2]
Selberg conjecture B [3]
Emil Artin 325
Bateman–Horn conjecture number theory Paul T. Bateman and Roger Horn 245
Baum–Connes conjecture operator K-theoryGromov-Lawson-Rosenberg conjecture [4]
Kaplansky-Kadison conjecture [4]
Novikov conjecture [4]
Paul Baum and Alain Connes 2670
Beal's conjecture number theory Andrew Beal 142
Beilinson conjecture number theory Alexander Beilinson 461
Berry–Tabor conjecture geodesic flow Michael Berry and Michael Tabor239
Big-line-big-clique conjecture discrete geometry
Birch and Swinnerton-Dyer conjecture number theory Bryan John Birch and Peter Swinnerton-Dyer 2830
Birch–Tate conjecture number theory Bryan John Birch and John Tate 149
Birkhoff conjecture integrable systems George David Birkhoff 345
Bloch–Beilinson conjectures number theory Spencer Bloch and Alexander Beilinson 152
Bloch–Kato conjecture algebraic K-theory Spencer Bloch and Kazuya Kato 1620
Bochner–Riesz conjecture harmonic analysis⇒restriction conjecture⇒Kakeya maximal function conjectureKakeya dimension conjecture [5] Salomon Bochner and Marcel Riesz 236
Bombieri–Lang conjecture diophantine geometry Enrico Bombieri and Serge Lang 181
Borel conjecture geometric topology Armand Borel 981
Bost conjecture geometric topology Jean-Benoît Bost 65
Brennan conjecture complex analysisJames E. Brennan110
Brocard's conjecture number theory Henri Brocard 16
Brumer–Stark conjecture number theoryArmand Brumer and Harold Stark 208
Bunyakovsky conjecture number theory Viktor Bunyakovsky 43
Carathéodory conjecture differential geometry Constantin Carathéodory 173
Carmichael totient conjecture number theory Robert Daniel Carmichael
Casas-Alvero conjecture polynomialsEduardo Casas-Alvero56
Catalan–Dickson conjecture on aliquot sequences number theory Eugène Charles Catalan and Leonard Eugene Dickson 46
Catalan's Mersenne conjecture number theory Eugène Charles Catalan
Cherlin–Zilber conjecture group theoryGregory Cherlin and Boris Zilber 86
Chowla conjecture Möbius functionSarnak conjecture [6] [7] Sarvadaman Chowla
Collatz conjecture number theory Lothar Collatz 1440
Cramér's conjecture number theory Harald Cramér 32
Conway's thrackle conjecture graph theory John Horton Conway 150
Deligne conjecture monodromy Pierre Deligne 788
Dittert conjecture combinatoricsEric Dittert11
Eilenberg−Ganea conjecture algebraic topology Samuel Eilenberg and Tudor Ganea 96
Elliott–Halberstam conjecture number theory Peter D. T. A. Elliott and Heini Halberstam 300
Erdős–Faber–Lovász conjecture graph theory Paul Erdős, Vance Faber, and László Lovász 172
Erdős–Gyárfás conjecture graph theory Paul Erdős and András Gyárfás 37
Erdős–Straus conjecture number theory Paul Erdős and Ernst G. Straus 103
Farrell–Jones conjecture geometric topology F. Thomas Farrell and Lowell E. Jones 545
Filling area conjecture differential geometryn/a60
Firoozbakht's conjecture number theory Farideh Firoozbakht 33
Fortune's conjecture number theory Reo Fortune 16
Four exponentials conjecture number theoryn/a110
Frankl conjecture combinatorics Péter Frankl 83
Gauss circle problem number theory Carl Friedrich Gauss 553
Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane metric geometry Edgar Gilbert and Henry O. Pollak
Gilbreath conjecture number theory Norman Laurence Gilbreath 34
Goldbach's conjecture number theory⇒The ternary Goldbach conjecture, which was the original formulation. [8] Christian Goldbach 5880
Gold partition conjecture [9] order theoryn/a25
Goldberg–Seymour conjecture graph theoryMark K. Goldberg and Paul Seymour 57
Goormaghtigh conjecture number theory René Goormaghtigh 14
Green's conjecture algebraic curves Mark Lee Green 150
Grimm's conjecture number theoryCarl Albert Grimm46
Grothendieck–Katz p-curvature conjecture differential equations Alexander Grothendieck and Nick Katz 98
Hadamard conjecture combinatorics Jacques Hadamard 858
Herzog–Schönheim conjecture group theoryMarcel Herzog and Jochanan Schönheim44
Hilbert–Smith conjecture geometric topology David Hilbert and Paul Althaus Smith 219
Hodge conjecture algebraic geometry W. V. D. Hodge 2490
Homological conjectures in commutative algebra commutative algebran/a
Hopf conjectures geometry Heinz Hopf 476
Ibragimov–Iosifescu conjecture for φ-mixing sequences probability theory Ildar Ibragimov, ro:Marius Iosifescu
Invariant subspace problem functional analysisn/a2120
Jacobian conjecture polynomials Carl Gustav Jacob Jacobi (by way of the Jacobian determinant)2860
Jacobson's conjecture ring theory Nathan Jacobson 127
Kaplansky conjectures ring theory Irving Kaplansky 466
Keating–Snaith conjecture number theory Jonathan Keating and Nina Snaith 48
Köthe conjecture ring theory Gottfried Köthe 167
Kung–Traub conjecture iterative methods H. T. Kung and Joseph F. Traub 332
Legendre's conjecture number theory Adrien-Marie Legendre 110
Lemoine's conjecture number theory Émile Lemoine 13
Lenstra–Pomerance–Wagstaff conjecture number theory Hendrik Lenstra, Carl Pomerance, and Samuel S. Wagstaff Jr. 32
Leopoldt's conjecture number theory Heinrich-Wolfgang Leopoldt 773
List coloring conjecture graph theoryn/a300
Littlewood conjecture diophantine approximationMargulis conjecture [10] John Edensor Littlewood 1230
Lovász conjecture graph theory László Lovász 560
MNOP conjecture algebraic geometryn/a63
Manin conjecture diophantine geometry Yuri Manin 338
Marshall Hall's conjecture number theory Marshall Hall, Jr. 44
Mazur's conjectures diophantine geometry Barry Mazur 97
Montgomery's pair correlation conjecture number theory Hugh Lowell Montgomery 77
n conjecture number theoryn/a126
New Mersenne conjecture number theory Marin Mersenne 47
Novikov conjecture algebraic topology Sergei Novikov 3090
Oppermann's conjecture number theory Ludvig Oppermann 12
Petersen coloring conjecture graph theory Julius Petersen 52
Pierce–Birkhoff conjecture real algebraic geometryRichard S. Pierce and Garrett Birkhoff 96
Pillai's conjecture number theory Subbayya Sivasankaranarayana Pillai 33
De Polignac's conjecture number theory Alphonse de Polignac 46
Quantum PCP conjecture quantum information theory
quantum unique ergodicity conjecture dynamical systems2004, 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/a281
Reconstruction conjecture graph theoryn/a1040
Riemann hypothesis number theoryGeneralized Riemann hypothesisGrand 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 Gerhard Ringel and Anton Kotzig 187
Rudin's conjecture additive combinatorics Walter Rudin 16
Sarnak conjecture topological entropy Peter Sarnak 295
Sato–Tate conjecture number theory Mikio Sato and John Tate 1080
Schanuel's conjecture number theory Stephen Schanuel 329
Schinzel's hypothesis H number theory Andrzej Schinzel 49
Scholz conjecture addition chains Arnold Scholz 41
Second Hardy–Littlewood conjecture number theory G. H. Hardy and John Edensor Littlewood 30
Selfridge's conjecture number theory John Selfridge 6
Sendov's conjecture complex polynomials Blagovest Sendov 77
Serre's multiplicity conjectures commutative algebra Jean-Pierre Serre 221
Singmaster's conjecture binomial coefficients David Singmaster 8
Standard conjectures on algebraic cycles algebraic geometryn/a234
Tate conjecture algebraic geometry John Tate
Toeplitz' conjecture Jordan curves Otto Toeplitz
Tuza's conjecture graph theoryZsolt Tuza
Twin prime conjecturenumber theoryn/a1700
Ulam's packing conjecture packing Stanislaw Ulam
Unicity conjecture for Markov numbers number theory Andrey Markov (by way of Markov numbers)
Uniformity conjecture diophantine geometryn/a
Unique games conjecture number theoryn/a
Vandiver's conjecture number theory Ernst Kummer and Harry Vandiver
Virasoro conjecture algebraic geometry Miguel Ángel Virasoro
Vizing's conjecture graph theory Vadim G. Vizing
Vojta's conjecture number theoryabc conjecture Paul Vojta
Waring's conjecture number theory Edward Waring
Weight monodromy conjecture algebraic geometryn/a
Weinstein conjecture periodic orbits Alan Weinstein
Whitehead conjecture algebraic topology J. H. C. Whitehead
Zauner's conjecture operator theoryGerhard Zauner

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 [13] Proved byFormer nameFieldComments
1962 Walter Feit and John G. Thompson Burnside conjecture that, apart from cyclic groups, finite simple groups have even orderfinite 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 theoryRingel-Youngs theorem
1971 Daniel Quillen Adams conjecture algebraic topologyOn the J-homomorphism, proposed 1963 by Frank Adams
1973 Pierre Deligne Weil conjectures algebraic geometryRamanujan–Petersson conjecture
Proposed by André Weil. Deligne's theorems completed around 15 years of work on the general case.
1975Henryk Hecht and Wilfried Schmid Blattner's conjecture representation theory for semisimple groups
1975 William Haboush Mumford conjecturegeometric invariant theory Haboush's theorem
1976 Kenneth Appel and Wolfgang Haken Four color theorem graph colouringTraditionally called a "theorem", long before the proof.
1976 Daniel Quillen; and independently by Andrei Suslin Serre's conjecture on projective modulespolynomial rings Quillen–Suslin theorem
1977 Alberto Calderón Denjoy's conjecturerectifiable curvesA 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 theoryFaltings'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 theoryNow generally known as the graph minor theorem.
1983 Michel Raynaud Manin–Mumford conjecture diophantine geometryThe Tate–Voloch conjecture is a quantitative (diophantine approximation) derived conjecture for p-adic varieties.
c.1984Collective work Smith conjecture knot theoryBased 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, 1916complex analysisRobertson conjectureMilin conjecturede Branges's theorem [15]
1984 Gunnar Carlsson Segal's conjecture homotopy theory
1984 Haynes Miller Sullivan conjecture classifying spacesMiller proved the version on mapping BG to a finite complex.
1987 Grigory Margulis Oppenheim conjecture diophantine approximationMargulis proved the conjecture with ergodic theory methods.
1989Vladimir I. Chernousov Weil's conjecture on Tamagawa numbers algebraic groupsThe 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 groupsUsually 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-theoryVoevodsky's theorem, ⇐norm residue isomorphism theoremBeilinson–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
2000Krzysztof Kurdyka, Tadeusz Mostowski, and Adam Parusiński Gradient conjecture gradient vector fieldsAttributed to René Thom, c.1970.
2001 Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor Taniyama–Shimura conjecture elliptic curvesNow the modularity theorem for elliptic curves. Once known as the "Weil conjecture".
2001 Mark Haiman n! conjecture representation theory
2001Daniel Frohardt and Kay Magaard [17] Guralnick–Thompson conjecture monodromy groups
2002 Preda Mihăilescu Catalan's conjecture, 1844exponential diophantine equationsPillai's conjectureabc 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, 19043-manifolds
2003 Grigori Perelman geometrization conjecture of Thurston3-manifoldsspherical 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 (see comment)Nobuo Iiyori and Hiroshi Yamaki Frobenius conjecture group theoryA 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
2004Ualbai U. Umirbaev and Ivan P. Shestakov Nagata's conjecture on automorphisms polynomial rings
2004 Ian Agol; and independently by Danny CalegariDavid Gabai tameness conjecturegeometric topologyAhlfors 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-manifoldsEhrenpreis conjecture on quasiconformality
2009Jeremie Chalopin and Daniel Gonçalves Scheinerman's conjecture intersection graphs
2010 Terence Tao and Van H. Vu circular law random matrix theory
2011Joel 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 conjecturenumber theoryThe 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 analysisThe 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 theoryBourgain–Demeter–Guth theorem, ⇐ decoupling theorem [18]
2018 Karim Adiprasito g-conjecture combinatorics
2019 Dimitris Koukoulopoulos and James Maynard Duffin–Schaeffer conjecture number theoryRational approximation of irrational numbers

Disproved (no longer conjectures)

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.

See also

Related Research Articles

<span class="mw-page-title-main">Conjecture</span> Proposition in mathematics that is unproven

In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's Last Theorem, have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.

In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n many kth powers of positive integers is itself a kth power, then n is greater than or equal to k:

In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with integer coefficients. The best-known transcendental numbers are π and e. The quality of a number being transcendental is called transcendence.

<span class="mw-page-title-main">Theorem</span> In mathematics, a statement that has been proved

In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.

A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy."

The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.

In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof, which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.

In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.

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<span class="mw-page-title-main">Schanuel's conjecture</span> Conjecture on the transcendence degree of field extensions to the rational numbers

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<span class="mw-page-title-main">Pólya conjecture</span> Disproved conjecture in number theory

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<span class="mw-page-title-main">Fermat's Last Theorem</span> 17th-century conjecture proved by Andrew Wiles in 1994

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.

In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states that, for every n there is at least one other integer m ≠ n such that φ(m) = φ(n). Robert Carmichael first stated this conjecture in 1907, but as a theorem rather than as a conjecture. However, his proof was faulty, and in 1922, he retracted his claim and stated the conjecture as an open problem.

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<span class="mw-page-title-main">Wiles's proof of Fermat's Last Theorem</span> 1995 publication in mathematics

Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using current knowledge by almost all current mathematicians at the time.

The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. The conjecture is that if the sum of some k-th powers equals the sum of some other k-th powers, then the total number of terms in both sums combined must be at least k.

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