List of conjectures

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This is a list of mathematical conjectures.

Open problems[edit]

Conjecture Field Comments Eponym(s)
1/3–2/3 conjecture order theory n/a
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
Agoh–Giuga conjecture number theory Takashi Agoh and Giuseppe Giuga
Agrawal's conjecture number theory Manindra Agrawal
Andrews–Curtis conjecture combinatorial group theory James J. Andrews and Morton L. Curtis
Andrica's conjecture number theory Dorin Andrica
Artin conjecture (L-functions) number theory Emil Artin
Artin's conjecture on primitive roots number theory generalized Riemann hypothesis[2]
Selberg conjecture B[3]
Emil Artin
Bateman–Horn conjecture number theory Paul T. Bateman and Roger Horn
Baum–Connes conjecture operator K-theory Gromov-Lawson-Rosenberg conjecture[4]
Kaplansky-Kadison conjecture[4]
Novikov conjecture[4]
Paul Baum and Alain Connes
Beal's conjecture number theory
Beilinson conjecture number theory
Berry–Tabor conjecture geodesic flow
Birch and Swinnerton-Dyer conjecture number theory
Birch–Tate conjecture number theory
Birkhoff conjecture integrable systems
Bloch–Beilinson conjectures number theory
Bloch–Kato conjecture algebraic K-theory
Bochner–Riesz conjecture harmonic analysis ⇒restriction conjecture⇒Kakeya maximal function conjectureKakeya dimension conjecture[5]
Bombieri–Lang conjecture diophantine geometry Enrico Bombieri and Serge Lang
Borel conjecture geometric topology Armand Borel
Bost conjecture geometric topology
Brennan conjecture complex analysis
Brocard's conjecture number theory
Brumer–Stark conjecture number theory
Bunyakovsky conjecture number theory
Carathéodory conjecture differential geometry
Carmichael totient conjecture number theory
Casas-Alvero conjecture polynomials
Catalan–Dickson conjecture on aliquot sequences number theory
Catalan's Mersenne conjecture number theory
Cherlin–Zilber conjecture group theory
Chowla conjecture Möbius function Sarnak conjecture[6] Sarvadaman Chowla
Collatz conjecture number theory
Cramér's conjecture number theory
Conway's thrackle conjecture graph theory John Horton Conway
Deligne conjecture monodromy Pierre Deligne
Dittert conjecture combinatorics
Eilenberg−Ganea conjecture algebraic topology
Elliott–Halberstam conjecture number theory Peter D. T. A. Elliott and Heini Halberstam
Erdős–Faber–Lovász conjecture graph theory
Erdős–Gyárfás conjecture graph theory
Erdős–Straus conjecture number theory
Farrell–Jones conjecture geometric topology
Filling area conjecture differential geometry
Firoozbakht's conjecture number theory
Fortune's conjecture number theory
Four exponentials conjecture number theory
Frankl conjecture combinatorics
Gauss circle problem number theory
Gilbreath conjecture number theory
Goldbach's conjecture number theory ⇒The ternary Goldbach conjecture, which was the original formulation.[7] Christian Goldbach
Gold partition conjecture[8] order theory
Goldberg–Seymour conjecture graph theory
Goormaghtigh conjecture number theory
Green's conjecture algebraic curves
Grimm's conjecture number theory
Grothendieck–Katz p-curvature conjecture differential equations Alexander Grothendieck and Nicholas Katz
Hadamard conjecture combinatorics
Herzog–Schönheim conjecture group theory
Hilbert–Smith conjecture geometric topology
Hodge conjecture algebraic geometry
Homological conjectures in commutative algebra commutative algebra
Hopf conjectures geometry Heinz Hopf
Invariant subspace problem functional analysis n/a
Jacobian conjecture polynomials
Jacobson's conjecture ring theory Nathan Jacobson
Kaplansky conjectures ring theory Irving Kaplansky
Keating–Snaith conjecture number theory
Köthe conjecture ring theory
Kung–Traub conjecture iterative methods
Legendre's conjecture number theory
Lemoine's conjecture number theory
Lenstra–Pomerance–Wagstaff conjecture number theory
Leopoldt's conjecture number theory
List coloring conjecture graph theory n/a
Littlewood conjecture diophantine approximation Margulis conjecture[9] John Edensor Littlewood
Lovász conjecture graph theory
MNOP conjecture algebraic geometry n/a
Manin conjecture diophantine geometry Yuri Manin
Marshall Hall's conjecture number theory Marshall Hall, Jr.
Mazur's conjectures diophantine geometry
Montgomery's pair correlation conjecture number theory Hugh Montgomery
n conjecture number theory n/a
New Mersenne conjecture number theory
Novikov conjecture algebraic topology Sergei Novikov
Oppermann's conjecture number theory
Petersen coloring conjecture graph theory
Pierce–Birkhoff conjecture real algebraic geometry
Pillai's conjecture number theory
De Polignac's conjecture number theory
quantum unique ergodicity conjecture dynamical systems 2004, Elon Lindenstrauss, for arithmetic hyperbolic surfaces,[10] 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfaces[11] n/a
Reconstruction conjecture graph theory n/a
Riemann hypothesis number theory Generalized 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
Ringel–Kotzig conjecture graph theory
Rudin's conjecture additive combinatorics Walter Rudin
Sarnak conjecture topological entropy Peter Sarnak
Sato–Tate conjecture number theory
Schanuel's conjecture number theory
Schinzel's hypothesis H number theory Andrzej Schinzel
Scholz conjecture addition chains
Second Hardy–Littlewood conjecture number theory G. H. Hardy and J. E. Littlewood
Selfridge's conjecture number theory
Sendov's conjecture complex polynomials
Serre's multiplicity conjectures commutative algebra Jean-Pierre Serre
Singmaster's conjecture binomial coefficients David Singmaster
Standard conjectures on algebraic cycles algebraic geometry n/a
Tate conjecture algebraic geometry John Tate
Toeplitz' conjecture Jordan curves Otto Toeplitz
Twin prime conjecture number theory n/a
Ulam's packing conjecture packing Stanislas Ulam
Unicity conjecture for Markov numbers number theory n/a
Uniformity conjecture diophantine geometry n/a
Unique games conjecture number theory n/a
Vandiver's conjecture number theory
Vizing's conjecture graph theory
Waring's conjecture number theory Edward Waring
Weight monodromy conjecture algebraic geometry n/a
Weinstein conjecture periodic orbits
Whitehead conjecture algebraic topology J. H. C. Whitehead
Zauner's conjecture operator theory

Conjectures now proved (theorems)[edit]

For a more complete list of problems solved, not restricted to so-called conjectures, see List of unsolved problems in mathematics#Problems solved since 1995

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

Priority date[12] Proved by Former name Field Comments
1962 Walter Feit, John 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
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
1976 Kenneth Appel and Wolfgang Haken Four color theorem graph colouring Traditionally called a "theorem", long before the proof.
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[13]
1978 Roger Heath-Brown and S. J. 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 Bieberbach conjecture, 1916 complex analysis Robertson conjectureMilin conjecturede Branges's theorem[14]
1984 Gunnar Carlsson Segal's conjecture homotopy theory
1987 Grigory Margulis Oppenheim conjecture diophantine approximation Margulis proved the conjecture with ergodic theory methods.
1989 V. 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[15] Alternating sign matrix conjecture, enumerative combinatorics
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".
2002 Preda Mihăilescu Catalan's conjecture, 1844 exponential diophantine equations Pillai's conjectureabc conjecture
Mihăilescu's 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
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.
2008 Avraham Trahtman Road coloring conjecture graph theory
2009 Jeremy Kahn, 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 Igor Mineyev, independently Hanna Neumann conjecture group theory
2012 Simon Brendle Hsiang–Lawson's conjecture differential geometry
2013 Zhang Yitang 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[16]

Disproved (no longer conjectures)[edit]

See also[edit]


  1. ^ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 13. ISBN 9781420035223.
  2. ^ Frei, Günther; Lemmermeyer, Franz; Roquette, Peter J. (2014). Emil Artin and Helmut Hasse: The Correspondence 1923-1958. Springer Science & Business Media. p. 215. ISBN 9783034807159.
  3. ^ Steuding, Jörn; Morel, J.-M.; Steuding, Jr̲n (2007). Value-Distribution of L-Functions. Springer Science & Business Media. p. 118. ISBN 9783540265269.
  4. ^ a b c Valette, Alain (2002). Introduction to the Baum-Connes Conjecture. Springer Science & Business Media. p. viii. ISBN 9783764367060.
  5. ^ Simon, Barry (2015). Harmonic Analysis. American Mathematical Soc. p. 685. ISBN 9781470411022.
  6. ^ Tao, Terence (15 October 2012). "The Chowla conjecture and the Sarnak conjecture". What's new.
  7. ^ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 1203. ISBN 9781420035223.
  8. ^ M. Peczarski, The gold partition conjecture, it Order 23(2006): 89–95.
  9. ^ Burger, Marc; Iozzi, Alessandra (2013). Rigidity in Dynamics and Geometry: Contributions from the Programme Ergodic Theory, Geometric Rigidity and Number Theory, Isaac Newton Institute for the Mathematical Sciences Cambridge, United Kingdom, 5 January – 7 July 2000. Springer Science & Business Media. p. 408. ISBN 9783662047439.
  10. ^ "EMS Prizes".
  11. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2011-07-24. Retrieved 2008-12-12.CS1 maint: Archived copy as title (link)
  12. ^ In the terms normally used for scientific priority, priority claims are typically understood to be settled by publication date. That approach is certainly flawed in contemporary mathematics, because lead times for publication in mathematical journals can run to several years. The understanding in intellectual property is that the priority claim is established by a filing date. Practice in mathematics adheres more closely to that idea, with an early manuscript submission to a journal, or circulation of a preprint, establishing a "filing date" that would be generally accepted.
  13. ^ Dudziak, James (2011). Vitushkin’s Conjecture for Removable Sets. Springer Science & Business Media. p. 39. ISBN 9781441967091.
  14. ^ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 218. ISBN 9781420035223.
  15. ^ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 65. ISBN 9781420035223.
  16. ^ "Decoupling and the Bourgain-Demeter-Guth proof of the Vinogradov main conjecture". What's new. 10 December 2015.
  17. ^ Holden, Helge; Piene, Ragni (2018). The Abel Prize 2013-2017. Springer. p. 51. ISBN 9783319990286.
  18. ^ Daniel Frohardt and Kay Magaard, Composition Factors of Monodromy Groups, Annals of Mathematics Second Series, Vol. 154, No. 2 (Sep., 2001), pp. 327–345. Published by: Mathematics Department, Princeton University DOI: 10.2307/3062099 JSTOR 3062099
  19. ^ Kalai, Gil (10 May 2019). "A sensation in the morning news – Yaroslav Shitov: Counterexamples to Hedetniemi's conjecture". Combinatorics and more.
  20. ^ Hazewinkel, Michiel, ed. (2001) [1994], "Schoenflies conjecture", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

External links[edit]