List of conjectures

This is a list of mathematical conjectures.

Open problems

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	n/a
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 conjecture⇒Kakeya 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][7]	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.[8]	Christian Goldbach
Gold partition conjecture[9]	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[10]	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,[11] 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfaces[12]	n/a
Reconstruction conjecture	graph theory		n/a
Riemann hypothesis	number theory	⇐Generalized Riemann hypothesis⇐Grand 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)

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[13]	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
1968	Gerhard Ringel and Ted Youngs	Heawood conjecture	graph theory	Ringel-Youngs theorem
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
1975	William Haboush	Mumford conjecture	geometric invariant theory	Haboush's theorem
1976	Kenneth Appel and Wolfgang Haken	Four color theorem	graph colouring	Traditionally called a "theorem", long before the proof.
1976	Daniel Quillen and Andrei Suslin independently	Serre's conjecture on projective modules	polynomial rings	Quillen–Suslin theorem
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[14]
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 conjecture⇐Milin conjecture⇐de Branges's theorem[15]
1984	Gunnar Carlsson	Segal's conjecture	homotopy theory
1984	Haynes Miller	Sullivan conjecture	classifying spaces	Miller proved the version on mapping BG to a finite complex.
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[16]	Alternating sign matrix conjecture,	enumerative combinatorics
1996	Vladimir Voevodsky	Milnor conjecture	algebraic K-theory	Voevodsky's theorem, ⇐norm residue isomorphism theorem⇔Beilinson–Lichtenbaum conjecture, Quillen–Lichtenbaum conjecture. The ambiguous term "Bloch-Kato conjecture" may refer to what is now the norm residue isomorphism theorem.
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".
2001	Mark Haiman	n! conjecture	representation theory
2001	Daniel Frohardt and Kay Magaard[17]	Guralnick–Thompson conjecture	monodromy groups
2002	Preda Mihăilescu	Catalan's conjecture, 1844	exponential diophantine equations	⇐Pillai's conjecture⇐abc conjecture Mihăilescu's theorem
2002	Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas	strong perfect graph conjecture	perfect graphs	Chudnovsky–Robertson–Seymour–Thomas 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
2003	Nils Dencker	Nirenberg–Treves conjecture	pseudo-differential operators
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.
2004	Adam Marcus and Gábor Tardos	Stanley–Wilf conjecture	permutation classes	Marcus–Tardos theorem
2004	Ualbai U. Umirbaev and Ivan P. Shestakov	Nagata's conjecture on automorphisms	polynomial rings
2004	Ian Agol and independently by Danny Calegari–David Gabai	tameness conjecture	geometric topology	⇒Ahlfors measure conjecture
2008	Avraham Trahtman	Road coloring conjecture	graph theory
2008	Chandrashekhar Khare, Jean-Pierre Wintenberger	Serre's modularity conjecture	modular forms
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
2012	Fernando Codá Marques and André Neves	Willmore 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[18]
2019	Dimitris Koukoulopoulos and James Maynard	Duffin–Schaeffer conjecture	number theory	Rational approximation of irrational numbers

• Deligne's conjecture on 1-motives[19]
• Goldbach's weak conjecture (proved in 2013)
• Sensitivity conjecture (proved in 2019)

Disproved (no longer conjectures)

• Atiyah conjecture (not a conjecture to start with)
• Borsuk's conjecture
• Chinese hypothesis (not a conjecture to start with)
• Doomsday conjecture
• Euler's sum of powers conjecture
• Ganea conjecture
• Generalized Smith conjecture
• Hauptvermutung
• Hedetniemi's conjecture, counterexample announced 2019[20]
• Hirsch conjecture (disproved in 2010)
• Intersection graph conjecture
• Kelvin's conjecture
• Kouchnirenko's conjecture
• Mertens conjecture
• Pólya conjecture, 1919 (1958)
• Ragsdale conjecture
• Schoenflies conjecture (disproved 1910)[21]
• Tait's conjecture
• Von Neumann conjecture
• Weyl–Berry conjecture
• Williamson conjecture

See also

• Erdős conjectures
• Fuglede's conjecture
• Millennium Prize Problems
• Painlevé conjecture
• 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

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. 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. Ferenczi, Sébastien; Kułaga-Przymus, Joanna; Lemańczyk, Mariusz (2018). Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics: CIRM Jean-Morlet Chair, Fall 2016. Springer. p. 185. ISBN 9783319749082.
8. Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 1203. ISBN 9781420035223.
9. M. Peczarski, The gold partition conjecture, it Order 23(2006): 89–95.
10. 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.
11. "EMS Prizes". www.math.kth.se.
12. "Archived copy" (PDF). Archived from the original (PDF) on 2011-07-24. Retrieved 2008-12-12.CS1 maint: archived copy as title (link)
13. 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.
14. Dudziak, James (2011). Vitushkin's Conjecture for Removable Sets. Springer Science & Business Media. p. 39. ISBN 9781441967091.
15. Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 218. ISBN 9781420035223.
16. Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 65. ISBN 9781420035223.
17. 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
18. "Decoupling and the Bourgain-Demeter-Guth proof of the Vinogradov main conjecture". What's new. 10 December 2015.
19. Holden, Helge; Piene, Ragni (2018). The Abel Prize 2013-2017. Springer. p. 51. ISBN 9783319990286.
20. Kalai, Gil (10 May 2019). "A sensation in the morning news – Yaroslav Shitov: Counterexamples to Hedetniemi's conjecture". Combinatorics and more.
21. "Schoenflies conjecture", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

External links

• Open Problem Garden