List of conjectures

This is a list of notable mathematical conjectures.

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 . ConjectureFieldCommentsEponym(s)Cites1/3–2/3 conjectureorder theory n/a 70abc conjecturenumber theory⇔Granville–Langevin conjecture, Vojta's conjecture in dimension 1⇒Erdős–Woods conjecture, Fermat–Catalan conjectureFormulated by David Masser and Joseph Oesterlé.Proof claimed in 2012 by Shinichi MochizukiJoseph Oesterlé and David Masser under the name "Oesterlé–Masser conjecture" 2440Agoh–Giuga conjecturenumber theory Takashi Agoh and Giuseppe Giuga 8Agrawal's conjecturenumber theory Manindra Agrawal 10Andrews–Curtis conjecturecombinatorial group theory James J. Andrews and Morton L. Curtis 358Andrica's conjecturenumber theory Dorin Andrica 45Artin conjecture (L-functions)number theory Emil Artin 650Artin's conjecture on primitive rootsnumber theory⇐generalized Riemann hypothesis⇐Selberg conjecture B Emil Artin 325Bateman–Horn conjecturenumber theory Paul T. Bateman and Roger Horn 245Baum–Connes conjectureoperator K-theory⇒Gromov-Lawson-Rosenberg conjecture⇒Kaplansky-Kadison conjecture⇒Novikov conjecturePaul Baum and Alain Connes 2670Beal's conjecturenumber theory Andrew Beal 142Beilinson conjecturenumber theory Alexander Beilinson 461Berry–Tabor conjecturegeodesic flow Michael Berry and Michael Tabor 239Big-line-big-clique conjecturediscrete geometry Birch and Swinnerton-Dyer conjecturenumber theory Bryan John Birch and Peter Swinnerton-Dyer 2830Birch–Tate conjecturenumber theory Bryan John Birch and John Tate 149Birkhoff conjectureintegrable systems George David Birkhoff 345Bloch–Beilinson conjecturesnumber theory Spencer Bloch and Alexander Beilinson 152Bloch–Kato conjecturealgebraic K-theory Spencer Bloch and Kazuya Kato 1620Bochner–Riesz conjectureharmonic analysis ⇒restriction conjecture⇒Kakeya maximal function conjecture⇒Kakeya dimension conjectureSalomon Bochner and Marcel Riesz 236Bombieri–Lang conjecturediophantine geometry Enrico Bombieri and Serge Lang 181Borel conjecturegeometric topology Armand Borel 981Bost conjecturegeometric topology Jean-Benoît Bost 65Brennan conjecturecomplex analysis James E. Brennan 110Brocard's conjecturenumber theory Henri Brocard 16Brumer–Stark conjecturenumber theory Armand Brumer and Harold Stark 208Bunyakovsky conjecturenumber theory Viktor Bunyakovsky 43Carmichael totient conjecturenumber theory Robert Daniel Carmichael Casas-Alvero conjecturepolynomials Eduardo Casas-Alvero 56Catalan–Dickson conjecture on aliquot sequencesnumber theory Eugène Charles Catalan and Leonard Eugene Dickson 46Catalan's Mersenne conjecturenumber theory Eugène Charles Catalan Cherlin–Zilber conjecturegroup theory Gregory Cherlin and Boris Zilber 86Chowla conjectureMöbius function⇒Sarnak conjecture Sarvadaman Chowla Collatz conjecturenumber theory Lothar Collatz 1440Cramér's conjecturenumber theory Harald Cramér 32Conway's thrackle conjecturegraph theory John Horton Conway 150Deligne conjecturemonodromy Pierre Deligne 788Dittert conjecturecombinatorics Eric Dittert 11Eilenberg−Ganea conjecturealgebraic topology Samuel Eilenberg and Tudor Ganea 96Elliott–Halberstam conjecturenumber theory Peter D. T. A. Elliott and Heini Halberstam 300Erdős–Faber–Lovász conjecturegraph theory Paul Erdős, Vance Faber, and László Lovász 172Erdős–Gyárfás conjecturegraph theory Paul Erdős and András Gyárfás 37Erdős–Straus conjecturenumber theory Paul Erdős and Ernst G. Straus 103Farrell–Jones conjecturegeometric topology F. Thomas Farrell and Lowell E. Jones 545Filling area conjecturedifferential geometry n/a 60Firoozbakht's conjecturenumber theory Farideh Firoozbakht 33Fortune's conjecturenumber theory Reo Fortune 16Four exponentials conjecturenumber theory n/a 110Frankl conjecturecombinatorics Péter Frankl 83Gauss circle problemnumber theory Carl Friedrich Gauss 553Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean planemetric geometry Edgar Gilbert and Henry O. Pollak Gilbreath conjecturenumber theory Norman Laurence Gilbreath 34Goldbach's conjecturenumber theory⇒The ternary Goldbach conjecture, which was the original formulation. Christian Goldbach 5880Gold partition conjectureM. Peczarski, The gold partition conjecture, it Order 23(2006): 89–95.order theory n/a 25Goldberg–Seymour conjecturegraph theory Mark K. Goldberg and Paul Seymour 57Goormaghtigh conjecturenumber theory René Goormaghtigh 14Green's conjecturealgebraic curves Mark Lee Green 150Grimm's conjecturenumber theory Carl Albert Grimm 46Grothendieck–Katz p-curvature conjecturedifferential equations Alexander Grothendieck and Nick Katz 98Hadamard conjecturecombinatorics Jacques Hadamard 858Herzog–Schönheim conjecturegroup theory Marcel Herzog and Jochanan Schönheim 44Hilbert–Smith conjecturegeometric topology David Hilbert and Paul Althaus Smith 219Homological conjectures in commutative algebracommutative algebra n/a Ibragimov–Iosifescu conjecture for φ-mixing sequencesprobability theory Ildar Ibragimov, :ro:Marius Iosifescu Invariant subspace problemfunctional analysis n/a 2120Jacobson's conjecturering theory Nathan Jacobson 127Kaplansky conjecturesring theory Irving Kaplansky 466Keating–Snaith conjecturenumber theory Jonathan Keating and Nina Snaith 48Köthe conjecturering theory Gottfried Köthe 167Kung–Traub conjectureiterative methods H. T. Kung and Joseph F. Traub 332Legendre's conjecturenumber theory Adrien-Marie Legendre 110Lemoine's conjecturenumber theory Émile Lemoine 13Lenstra–Pomerance–Wagstaff conjecturenumber theory Hendrik Lenstra, Carl Pomerance, and Samuel S. Wagstaff Jr. 32Leopoldt's conjecturenumber theory Heinrich-Wolfgang Leopoldt 773 List coloring conjecturegraph theory n/a 300 Littlewood conjecturediophantine approximation⇐Margulis conjecture John Edensor Littlewood 1230Lovász conjecturegraph theory László Lovász 560MNOP conjecturealgebraic geometry n/a 63Manin conjecturediophantine geometry Yuri Manin 338Marshall Hall's conjecturenumber theory Marshall Hall, Jr. 44 Mazur's conjecturesdiophantine geometry Barry Mazur 97Montgomery's pair correlation conjecturenumber theory Hugh Lowell Montgomery 77n conjecturenumber theory n/a 126 New Mersenne conjecturenumber theory Marin Mersenne 47Novikov conjecturealgebraic topology Sergei Novikov 3090Oppermann's conjecturenumber theory Ludvig Oppermann 12Petersen coloring conjecturegraph theory Julius Petersen 52Pierce–Birkhoff conjecturereal algebraic geometry Richard S. Pierce and Garrett Birkhoff 96Pillai's conjecturenumber theory Subbayya Sivasankaranarayana Pillai 33De Polignac's conjecturenumber theory Alphonse de Polignac 46Quantum PCP conjecturecomputational complexity quantum unique ergodicity conjecturedynamical systems 2004, Elon Lindenstrauss, for arithmetic hyperbolic surfaces, 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfacesn/a 281Reconstruction conjecturegraph theory n/a 1040Riemann hypothesisnumber theory⇐Generalized Riemann hypothesis⇐Grand Riemann hypothesis⇔De Bruijn–Newman constant=0⇒density hypothesis, Lindelöf hypothesisSee Hilbert–Pólya conjecture. For other Riemann hypotheses, see the Weil conjectures (now theorems).Bernhard Riemann24900Ringel–Kotzig conjecturegraph theory Gerhard Ringel and Anton Kotzig 187Rudin's conjectureadditive combinatorics Walter Rudin 16Sarnak conjecturetopological entropy Peter Sarnak 295Sato–Tate conjecturenumber theory Mikio Sato and John Tate 1080Schanuel's conjecturenumber theory Stephen Schanuel 329Schinzel's hypothesis Hnumber theory Andrzej Schinzel 49Scholz conjectureaddition chains Arnold Scholz 41Second Hardy–Littlewood conjecturenumber theory G. H. Hardy and John Edensor Littlewood 30Selfridge's conjecturenumber theory John Selfridge 6Singmaster's conjecturebinomial coefficients David Singmaster 8Spectral gap conjectureergodic theory Standard conjectures on algebraic cyclesalgebraic geometry n/a 234 Tate conjecturealgebraic geometry John Tate Toeplitz' conjectureJordan curves Otto Toeplitz Tuza's conjecturegraph theory Zsolt TuzaTwin prime conjecturenumber theory n/a 1700Ulam's packing conjecturepacking Stanislaw Ulam Unicity conjecture for Markov numbersnumber theory Andrey Markov (by way of Markov numbers) Uniformity conjecturediophantine geometry n/a Unique games conjecturecomputational complexity n/a Kummer–Vandiver conjecturenumber theory Ernst Kummer and Harry Vandiver Virasoro conjecturealgebraic geometry Miguel Ángel Virasoro Vizing's conjecturegraph theory Vadim G. Vizing Vojta's conjecturenumber theory⇒abc conjecture Paul Vojta Waring's conjecturenumber theory Edward Waring Weight monodromy conjecturealgebraic geometry n/a Whitehead conjecturealgebraic topology J. H. C. Whitehead Zauner's conjectureoperator theory Gerhard Zauner

Conjectures with resolved and unresolved cases The following conjectures are treated in reliable sources as having distinct recognized cases, settings, or parameter regimes with different resolution statuses. At least one such case must have been completely resolved, either affirmatively or negatively, while at least one other remains open. Results concerning only isolated special cases, individual instances, or weaker or conditional forms do not by themselves qualify. The column "cites" lists the number of results for a Google Scholar search for the term, in double quotes . ConjectureFieldCommentsEponym(s)Cites Carathéodory conjecture Differential geometry Disproved for convex surfaces; whether the real-analytic case has been proved remains disputed Constantin Carathéodory 185 Cartan–Hadamard conjecture Riemannian geometry Proved in dimensions 2, 3, and 4; open in higher dimensions Élie Cartan and Jacques Hadamard 103 Dixmier conjecture Abstract algebra The generalized conjecture is false for with ; the cases and remain open Jacques Dixmier 267 Fuglede's conjecture Harmonic analysis False in dimensions 3 and higher; open in dimensions 1 and 2 Bent Fuglede 152 Generalized Poincaré conjecture Topology True in the topological category in all dimensions; in the PL and smooth categories some cases are resolved while others remain open Henri Poincaré 787 Hodge conjecture Algebraic geometry Proved for projective complex manifolds of dimension at most 3 and for various other classes; open in general W. V. D. Hodge 4020 Hopf conjecture Differential geometry The curvature-sign conjecture is proved in dimensions 2 and 4; open in even dimensions 6 and higher Heinz Hopf 581 Jacobian conjecture Algebraic geometry True for ; open for ; false for Carl Gustav Jacob Jacobi (by way of the Jacobian determinant) 3450 Kakeya conjecture Geometric measure theory Proved in dimensions at most 3; open in dimensions 4 and higher Sōichi Kakeya 776 Lonely runner conjecture Diophantine approximation Proved for 13 runners or fewer; open in general — 253 Mahler conjecture Convex geometry Proved in dimensions 2 and 3; open in higher dimensions Kurt Mahler 580 Serre's conjecture II Number theory Proved over perfect fields; over imperfect fields, cases involving anisotropic groups of types , , , and trialitarian remain open Jean-Pierre Serre 52 Serre's multiplicity conjectures Commutative algebra The dimension inequality, non-negativity, and vanishing conjectures are proved; the positivity conjecture remains open in general Jean-Pierre Serre 7 Torsion conjecture Arithmetic geometry Proved for elliptic curves over number fields; the higher-dimensional abelian-variety case remains largely open — 219 Weinstein conjecture Symplectic geometry Proved for all closed 3-manifolds and several broader classes; open in general Alan Weinstein 1320

Conjectures now proved (theorems)

The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names.