List of 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

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

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

Goldbach's weak conjecture (proof claimed by Harald Helfgott)

Gold partition conjecture

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, 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfaces )

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

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

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.

Adams conjecture (J-homomorphism)

Andrews–Curtis conjecture

Angel problem (several proofs, 2007)

Bieberbach conjecture, 1916 (De Branges' 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' proof in 1994)

Four color theorem (Appel and Haken, 1976)

Frobenius conjecture (Iiyori and Yamaki)

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' 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' 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

Kouchnirenko's conjecture

Mertens conjecture

Pólya conjecture, 1919 (1958)

Ragsdale conjecture

Tait's conjecture

Von Neumann conjecture

Weyl–Berry conjecture