List of conjectures

This is an incomplete list of mathematical conjectures. They are divided into four sections, according to their current status.

See also:

• Erdős conjecture, which lists conjectures of Paul Erdős and his collaborators
• Unsolved problems in mathematics
• List of unsolved problems
• Millennium Prize Problems

and, for proved results,

• List of theorems, list of lemmas;

also

• List of statements undecidable in ZFC

for problems not subject to conventional proof nor disproof.

Proved (now theorems)

The theorems may not be their official names.

• Adams conjecture (J-homomorphism)
• Andrews–Curtis conjecture
• Bieberbach conjecture, 1916 (De Branges' theorem, 1984)
• Blattner's conjecture (Blattner formula)
• Burnside conjecture (Feit–Thompson theorem)
• Cameron–Erdős conjecture
• Catalan's conjecture, 1844 (Mihăilescu's theorem, 2002)
• Conway–Norton conjecture (monstrous moonshine)
• Dinitz conjecture (Galvin's theorem)
• Deligne's conjecture on 1-motives
• Denjoy's conjecture
• Dodecahedral conjecture (Hales–McLaughlin theorem)
• Epsilon conjecture (Ribet's theorem)
• Fermat's Last Theorem (Andrew Wiles' proof in 1994)
• Frobenius conjecture (Iiyori and Yamaki)
• Gradient conjecture (Kurdyka–Mostowski–Parusinski theorem)
• Heawood conjecture (Ringel–Youngs' theorem)
• Kummer's conjecture on cubic Gauss sums (Kummer sum)
• Mahler–Manin conjecture
• Manin–Mumford conjecture (Raynaud's theorem)
• Milnor conjecture (Voevodsky's theorem)
• Mordell conjecture (Faltings' theorem)
• Mumford conjecture (Haboush's theorem)
• Poincaré conjecture, 1904 (theorem about the characterization of the three-dimensional sphere)
• Oppenheim conjecture (Margulis' 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)
• 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
• 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)

Disproved (no longer conjectures)

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

Recent work

• Bloch–Kato conjecture
• Casas-Alvero conjecture
• Erdős–Straus conjecture
• Guralnick–Thompson conjecture (Riemann surfaces)
• Herzog–Schönheim conjecture
• Kepler conjecture
• Nagata's conjecture on automorphisms
• Nirenberg–Treves conjecture
• quantum unique ergodicity conjecture, (2004, Elon Lindenstrauss, for arithmetic hyperbolic surfaces,^[1] 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfaces ^[2])
• Sato–Tate conjecture

Open problems

• The 1/3 − 2/3 conjecture
• abc conjecture
• Andrews–Curtis conjecture
• Angel problem
• Agoh–Giuga conjecture
• Andrica's conjecture
• Artin conjectures
• 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
• Bombieri-Lang conjecture
• Borel conjecture
• Bost conjecture
• Brocard's conjecture
• Brumer–Stark conjecture
• Bunyakovsky conjecture
• Catalan–Dickson conjecture on aliquot sequences
• Carathéodory conjecture
• Carmichael totient conjecture
• Catalan's Mersenne conjecture
• Cherlin–Zilber conjecture
• Circular Law
• Collatz conjecture
• Cramér's conjecture
• Conway's thrackle conjecture
• Deligne conjecture disambiguation
• Eilenberg−Ganea conjecture
• Elliott–Halberstam conjecture
• Erdős–Burr conjecture
• Erdős–Faber–Lovász conjecture
• Erdős–Gyárfás conjecture
• Farrell–Jones conjecture
• Filling area conjecture
• Frankl conjecture
• Gauss circle problem
• Gilbreath conjecture
• Giuga's conjecture
• Goldbach's conjecture
• Goldbach's weak conjecture
• Gold partition conjecture^[3]
• Goormaghtigh conjecture
• Green's conjecture
• Grimm's conjecture
• Grothendieck–Katz p-curvature conjecture
• Hadamard conjecture
• Hanna Neumann conjecture
• Hedetniemi's conjecture
• Hilbert–Smith conjecture
• Hirsch conjecture
• Hopf conjecture
• Hodge conjecture
• Homological conjectures in commutative algebra
• Jacobian conjecture
• Jacobson's conjecture
• Kakeya conjecture(s)(five problems)
• Kakeya needle problem
• Kaplansky conjecture
• Keating–Snaith conjecture
• Köthe conjecture
• Lawson's conjecture
• Lenstra–Pomerance–Wagstaff conjecture
• Lichtenbaum conjecture
• List coloring conjecture
• Littlewood conjecture
• Lovász conjecture
• Marshall Hall's conjecture
• Mazur's conjectures
• Deligne conjecture in monodromy
• Montgomery's pair correlation conjecture
• New Mersenne conjecture
• Novikov conjecture
• Oppermann's conjecture
• Petersen coloring conjecture
• Pierce–Birkhoff conjecture
• Pillai's conjecture
• De Polignac's conjecture
• Quillen–Lichtenbaum conjecture
• Reconstruction conjecture
• Riemann Hypotheses: see also Weil conjectures, above
  • Riemann hypothesis
    • Generalized Riemann hypothesis
    • Grand Riemann hypothesis
  • Density hypothesis
  • Lindelöf hypothesis
  • Hilbert–Pólya conjecture on the Riemann hypothesis
• Ringel-Kotzig conjecture
• Schanuel's conjecture
• Schinzel's hypothesis H
• Scholz conjecture
• Second Hardy–Littlewood conjecture
• Selfridge's conjecture
• Sendov's conjecture
• Serre conjecture (number theory)
• Serre's multiplicity conjectures
• Singmaster's conjecture
• Standard conjectures on algebraic cycles
• Tate conjecture
• Toeplitz' conjecture
• Twin prime conjecture
• Uniformity conjecture
• Vandiver's conjecture
• Vizing's conjecture
• Waring's conjecture
• Weight-monodromy conjecture
• Weinstein conjecture
• Whitehead conjecture
• Zhou conjecture

References

External links

• Open Problem Garden