List of conjectures

This is a list of mathematical conjectures.

Open problems

• 1/3–2/3 conjecture
• abc conjecture (proof claimed by Shinichi Mochizuki)
• Agoh–Giuga conjecture
• Agrawal's conjecture
• Andrews–Curtis conjecture
• Andrica's conjecture
• Artin conjecture (L-functions)
• Artin's conjecture on primitive roots
• Bateman–Horn conjecture
• Baum–Connes conjecture
• Beal's conjecture
• Beilinson conjecture
• Berry–Tabor conjecture
• Birch and Swinnerton-Dyer conjecture
• Birch–Tate conjecture
• Birkhoff conjecture
• Bloch–Beilinson conjectures
• Bloch–Kato conjecture
• Bombieri–Lang conjecture
• Borel conjecture
• Bost conjecture
• Brennan conjecture
• Brocard's conjecture
• Brumer–Stark conjecture
• Bunyakovsky conjecture
• Carathéodory conjecture
• Carmichael totient conjecture
• Casas-Alvero conjecture
• Catalan–Dickson conjecture on aliquot sequences
• Catalan's Mersenne conjecture
• Cherlin–Zilber conjecture
• Collatz conjecture
• Cramér's conjecture
• Conway's thrackle conjecture
• Deligne conjecture in monodromy
• Dittert conjecture
• Eilenberg−Ganea conjecture
• Elliott–Halberstam conjecture
• Erdős–Burr conjecture
• Erdős–Faber–Lovász conjecture
• Erdős–Gyárfás conjecture
• Erdős–Straus conjecture
• Farrell–Jones conjecture
• Filling area conjecture
• Firoozbakht's conjecture
• Fortune's conjecture
• Frankl conjecture
• Gauss circle problem
• Gilbreath conjecture
• Giuga's conjecture
• Goldbach's conjecture
• Gold partition conjecture^[1]
• Goormaghtigh conjecture
• Green's conjecture
• Grimm's conjecture
• Grothendieck–Katz p-curvature conjecture
• Guralnick–Thompson conjecture (Riemann surfaces)
• Hadamard conjecture
• Hedetniemi's conjecture
• Herzog–Schönheim conjecture
• Hilbert–Smith conjecture
• Hodge conjecture
• Homological conjectures in commutative algebra
• Hopf conjecture
• Invariant subspace problem
• Jacobian conjecture
• Jacobson's conjecture
• Kakeya conjectures
• Kakeya needle problem
• Kaplansky conjecture
• Keating–Snaith conjecture
• Köthe conjecture
• Kung–Traub conjecture
• Lawson's conjecture
• Lemoine's conjecture
• Lenstra–Pomerance–Wagstaff conjecture
• Leopoldt's conjecture
• Lichtenbaum conjecture
• List coloring conjecture
• Littlewood conjecture
• Lovász conjecture
• MNOP conjecture
• Manin conjecture
• Marshall Hall's conjecture
• Mazur's conjectures
• Montgomery's pair correlation conjecture
• n conjecture
• Nagata's conjecture on automorphisms
• New Mersenne conjecture
• Nirenberg–Treves conjecture
• Novikov conjecture
• Oppermann's conjecture
• Petersen coloring conjecture
• Pierce–Birkhoff conjecture
• Pillai's conjecture
• De Polignac's conjecture
• quantum unique ergodicity conjecture, (2004, Elon Lindenstrauss, for arithmetic hyperbolic surfaces,^[2] 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfaces^[3])
• Quillen–Lichtenbaum conjecture
• Reconstruction conjecture
• Riemann Hypotheses: see also Weil conjectures under Theorems
  • Riemann hypothesis
    • Generalized Riemann hypothesis
    • Grand Riemann hypothesis
  • Density hypothesis
  • Lindelöf hypothesis
  • Hilbert–Pólya conjecture on the Riemann hypothesis
• Ringel–Kotzig conjecture
• Rudin's conjecture
• Sato–Tate conjecture
• Schanuel's conjecture
• Schinzel's hypothesis H
• Scholz conjecture
• Second Hardy–Littlewood conjecture
• Selfridge's conjecture
• Sendov's conjecture
• Serre's multiplicity conjectures
• Singmaster's conjecture
• Standard conjectures on algebraic cycles
• Tate conjecture
• Toeplitz' conjecture
• Twin prime conjecture
• Ulam's packing conjecture
• Unicity conjecture for Markov numbers
• Uniformity conjecture
• Unique games conjecture
• Vandiver's conjecture
• Vizing's conjecture
• Waring's conjecture
• Weight monodromy conjecture
• Weinstein conjecture
• Whitehead conjecture
• Zauner's conjecture
• Zhou conjecture

Proved (now theorems)

The theorems may not be their official names.

• Abhyankar's conjecture
• Adams conjecture (J-homomorphism)
• Andrews–Curtis conjecture
• Angel problem (several proofs, 2007)
• Bieberbach conjecture, 1916 (de Branges's theorem, 1984)
• Blattner's conjecture (Blattner formula)
• Bounded gap conjecture (Zhang Yitang, 2013)
• Burnside conjecture (Feit–Thompson theorem)
• Cameron–Erdős conjecture
• Catalan's conjecture, 1844 (Mihăilescu's theorem, 2002)
• Circular law (proved in 2010)
• Conway–Norton conjecture (monstrous moonshine)
• Dinitz conjecture (Galvin's theorem)
• Deligne's conjecture on 1-motives
• Denjoy's conjecture
• Dodecahedral conjecture (Hales–McLaughlin theorem)
• Ehrenpreis conjecture (proved in 2011)
• Epsilon conjecture (Ribet's theorem)
• Fermat's Last Theorem (Andrew Wiles's proof in 1994)
• Four color theorem (Appel and Haken, 1976)
• Frobenius conjecture (Iiyori and Yamaki)
• Goldbach's weak conjecture (proved in 2013)
• Gradient conjecture (Kurdyka–Mostowski–Parusinski theorem)
• Hanna Neumann conjecture (proved in 2011)
• Heawood conjecture (Ringel–Youngs theorem)
• Kadison–Singer problem (Marcus–Spielman–Srivastava theorem)
• Kepler conjecture (proved in 1998)
• Kummer's conjecture on cubic Gauss sums (Kummer sum)
• Mahler–Manin conjecture
• Main conjecture in Vinogradov's mean-value theorem (Bourgain–Demeter–Guth theorem)
• Manin–Mumford conjecture (Raynaud's theorem)
• Milnor conjecture (Voevodsky's theorem)
• Mordell conjecture (Faltings's theorem)
• Mumford conjecture (Haboush's theorem)
• n! conjecture (proved in 2001)
• Poincaré conjecture, 1904 (theorem about the characterization of the three-dimensional sphere)
• Oppenheim conjecture (Margulis's theorem)
• Ramanujan–Petersson conjecture (consequence of the proof of the Weil conjectures)
• Road coloring conjecture, 1970 (2008)
• Scheinerman's conjecture (proved 2009)
• Segal's Burnside ring conjecture (Carlsson theorem)
• Serre's conjecture (Quillen–Suslin theorem)
• Serre's modularity conjecture (proved 2008)
• Seymour's conjecture
• Smith conjecture (Gordon's theorem)
• Stanley–Wilf conjecture (Marcus–Tardos theorem)
• Star height problem (Hashiguchi theorem)
• Strong perfect graph conjecture (Chudnovsky–Robertson–Seymour–Thomas theorem)
• Sullivan conjecture
• Surface subgroup conjecture (proved in 2009)
• Tameness conjecture (Agol or Calegari–Gabai theorem)
• Taniyama–Shimura conjecture (Modularity theorem)
• Thurston's geometrization conjecture (3-manifolds of the uniformization theorem for surfaces)
• Wagner's conjecture (Robertson–Seymour theorem)
• Weil conjectures (Deligne's theorems)
• Willmore conjecture (proved in 2012)

Disproved (no longer conjectures)

• Atiyah conjecture (not a conjecture to start with)
• Borsuk's conjecture
• Chinese hypothesis (not a conjecture to start with)
• Euler's sum of powers conjecture
• Ganea conjecture
• Generalized Smith conjecture
• Hauptvermutung
• Hirsch conjecture (disproved in 2010)
• Intersection graph conjecture
• Kelvin's conjecture
• Kouchnirenko's conjecture
• Mertens conjecture
• Pólya conjecture, 1919 (1958)
• Ragsdale conjecture
• Tait's conjecture
• Von Neumann conjecture
• Weyl–Berry conjecture

See also

• Erdős conjectures
• Millennium Prize Problems
• List of unsolved problems in mathematics
• List of disproved mathematical ideas
• List of unsolved problems
• List of lemmas
• List of theorems
• List of statements undecidable in ZFC

References

External links

• Open Problem Garden