List of conjectures

This is a list of mathematical conjectures.

Open problems

Conjecture Field Comments Eponym(s)
1/3–2/3 conjectureorder theoryn/a
abc conjecturenumber 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 conjecturenumber theoryTakashi Agoh and Giuseppe Giuga
Agrawal's conjecturenumber theoryManindra Agrawal
Andrews–Curtis conjecturecombinatorial group theoryJames J. Andrews and Morton L. Curtis
Andrica's conjecturenumber theoryDorin Andrica
Artin conjecture (L-functions)number theoryEmil Artin
Artin's conjecture on primitive rootsnumber theorygeneralized Riemann hypothesis[2]
Selberg conjecture B[3]
Emil Artin
Bateman–Horn conjecturenumber theoryPaul T. Bateman and Roger Horn
Baum–Connes conjectureoperator K-theoryGromov-Lawson-Rosenberg conjecture[4]
Kaplansky-Kadison conjecture[4]
Novikov conjecture[4]
Paul Baum and Alain Connes
Beal's conjecturenumber theory
Beilinson conjecturenumber theory
Berry–Tabor conjecturegeodesic flow
Birch and Swinnerton-Dyer conjecturenumber theory
Birch–Tate conjecturenumber theory
Birkhoff conjectureintegrable systems
Bloch–Beilinson conjecturesnumber theory
Bloch–Kato conjecturealgebraic K-theory
Bochner–Riesz conjectureharmonic analysis⇒restriction conjecture⇒Kakeya maximal function conjectureKakeya dimension conjecture[5]
Bombieri–Lang conjecturediophantine geometryEnrico Bombieri and Serge Lang
Borel conjecturegeometric topologyArmand Borel
Bost conjecturegeometric topology
Brennan conjecturecomplex analysis
Brocard's conjecturenumber theory
Brumer–Stark conjecturenumber theory
Bunyakovsky conjecturenumber theory
Carathéodory conjecturedifferential geometry
Carmichael totient conjecturenumber theory
Casas-Alvero conjecturepolynomials
Catalan–Dickson conjecture on aliquot sequencesnumber theory
Catalan's Mersenne conjecturenumber theory
Cherlin–Zilber conjecturegroup theory
Chowla conjectureMöbius functionSarnak conjecture[6][7]Sarvadaman Chowla
Collatz conjecturenumber theory
Cramér's conjecturenumber theory
Conway's thrackle conjecturegraph theoryJohn Horton Conway
Deligne conjecturemonodromyPierre Deligne
Dittert conjecturecombinatorics
Eilenberg−Ganea conjecturealgebraic topology
Elliott–Halberstam conjecturenumber theoryPeter D. T. A. Elliott and Heini Halberstam
Erdős–Faber–Lovász conjecturegraph theory
Erdős–Gyárfás conjecturegraph theory
Erdős–Straus conjecturenumber theory
Farrell–Jones conjecturegeometric topology
Filling area conjecturedifferential geometry
Firoozbakht's conjecturenumber theory
Fortune's conjecturenumber theory
Four exponentials conjecturenumber theory
Frankl conjecturecombinatorics
Gauss circle problemnumber theory
Gilbreath conjecturenumber theory
Goldbach's conjecturenumber theory⇒The ternary Goldbach conjecture, which was the original formulation.[8]Christian Goldbach
Gold partition conjecture[9]order theory
Goldberg–Seymour conjecturegraph theory
Goormaghtigh conjecturenumber theory
Green's conjecturealgebraic curves
Grimm's conjecturenumber theory
Grothendieck–Katz p-curvature conjecturedifferential equationsAlexander Grothendieck and Nicholas Katz
Hadamard conjecturecombinatorics
Herzog–Schönheim conjecturegroup theory
Hilbert–Smith conjecturegeometric topology
Hodge conjecturealgebraic geometry
Homological conjectures in commutative algebracommutative algebra
Hopf conjecturesgeometryHeinz Hopf
Invariant subspace problemfunctional analysisn/a
Jacobian conjecturepolynomials
Jacobson's conjecturering theoryNathan Jacobson
Kaplansky conjecturesring theoryIrving Kaplansky
Keating–Snaith conjecturenumber theory
Köthe conjecturering theory
Kung–Traub conjectureiterative methods
Legendre's conjecturenumber theory
Lemoine's conjecturenumber theory
Lenstra–Pomerance–Wagstaff conjecturenumber theory
Leopoldt's conjecturenumber theory
List coloring conjecturegraph theoryn/a
Littlewood conjecturediophantine approximationMargulis conjecture[10]John Edensor Littlewood
Lovász conjecturegraph theory
MNOP conjecturealgebraic geometryn/a
Manin conjecturediophantine geometryYuri Manin
Marshall Hall's conjecturenumber theoryMarshall Hall, Jr.
Mazur's conjecturesdiophantine geometry
Montgomery's pair correlation conjecturenumber theoryHugh Montgomery
n conjecturenumber theoryn/a
New Mersenne conjecturenumber theory
Novikov conjecturealgebraic topologySergei Novikov
Oppermann's conjecturenumber theory
Petersen coloring conjecturegraph theory
Pierce–Birkhoff conjecturereal algebraic geometry
Pillai's conjecturenumber theory
De Polignac's conjecturenumber theory
quantum unique ergodicity conjecturedynamical systems2004, 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 conjecturegraph theoryn/a
Riemann hypothesisnumber theoryGeneralized 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 conjecturegraph theory
Rudin's conjectureadditive combinatoricsWalter Rudin
Sarnak conjecturetopological entropyPeter Sarnak
Sato–Tate conjecturenumber theory
Schanuel's conjecturenumber theory
Schinzel's hypothesis Hnumber theoryAndrzej Schinzel
Scholz conjectureaddition chains
Second Hardy–Littlewood conjecturenumber theoryG. H. Hardy and J. E. Littlewood
Selfridge's conjecturenumber theory
Sendov's conjecturecomplex polynomials
Serre's multiplicity conjecturescommutative algebraJean-Pierre Serre
Singmaster's conjecturebinomial coefficientsDavid Singmaster
Standard conjectures on algebraic cyclesalgebraic geometryn/a
Tate conjecturealgebraic geometryJohn Tate
Toeplitz' conjectureJordan curvesOtto Toeplitz
Twin prime conjecturenumber theoryn/a
Ulam's packing conjecturepackingStanislas Ulam
Unicity conjecture for Markov numbersnumber theoryn/a
Uniformity conjecturediophantine geometryn/a
Unique games conjecturenumber theoryn/a
Vandiver's conjecturenumber theory
Virasoro conjecturealgebraic geometryn/a
Vizing's conjecturegraph theory
Waring's conjecturenumber theoryEdward Waring
Weight monodromy conjecturealgebraic geometryn/a
Weinstein conjectureperiodic orbits
Whitehead conjecturealgebraic topologyJ. H. C. Whitehead
Zauner's conjectureoperator 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
1962Walter Feit, John ThompsonBurnside conjecture that, apart from cyclic groups, finite simple groups have even orderfinite simple groupsFeit–Thompson theorem⇔trivially the "odd order theorem" that finite groups of odd order are solvable groups
1968Gerhard Ringel and Ted YoungsHeawood conjecturegraph theoryRingel-Youngs theorem
1971Daniel QuillenAdams conjecturealgebraic topologyOn the J-homomorphism, proposed 1963 by Frank Adams
1973Pierre DeligneWeil conjecturesalgebraic geometryRamanujan–Petersson conjecture
Proposed by André Weil. Deligne's theorems completed around 15 years of work on the general case.
1975Henryk Hecht and Wilfried SchmidBlattner's conjecturerepresentation theory for semisimple groups
1975William HaboushMumford conjecturegeometric invariant theoryHaboush's theorem
1976Kenneth Appel and Wolfgang HakenFour color theoremgraph colouringTraditionally called a "theorem", long before the proof.
1976Daniel Quillen and Andrei Suslin independentlySerre's conjecture on projective modulespolynomial ringsQuillen–Suslin theorem
1977Alberto CalderónDenjoy's conjecturerectifiable curvesA result claimed in 1909 by Arnaud Denjoy, proved by Calderón as a by-product of work on Cauchy singular operators[14]
1978Roger Heath-Brown and S. J. PattersonKummer's conjecture on cubic Gauss sumsequidistribution
1983Gerd FaltingsMordell conjecturenumber theoryFaltings's theorem, the Shafarevich conjecture on finiteness of isomorphism classes of abelian varieties. The reduction step was by Alexey Parshin.
1983 onwardsNeil Robertson and Paul D. SeymourWagner's conjecturegraph theoryNow generally known as the graph minor theorem.
1983Michel RaynaudManin–Mumford conjecturediophantine geometryThe Tate–Voloch conjecture is a quantitative (diophantine approximation) derived conjecture for p-adic varieties.
c.1984Collective workSmith conjectureknot theoryBased 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.
1984Louis de BrangesBieberbach conjecture, 1916complex analysisRobertson conjectureMilin conjecturede Branges's theorem[15]
1984Gunnar CarlssonSegal's conjecturehomotopy theory
1984Haynes MillerSullivan conjectureclassifying spacesMiller proved the version on mapping BG to a finite complex.
1987Grigory MargulisOppenheim conjecturediophantine approximationMargulis proved the conjecture with ergodic theory methods.
1989V. I. ChernousovWeil's conjecture on Tamagawa numbersalgebraic groupsThe problem, based on Siegel's theory for quadratic forms, submitted to a long series of case analysis steps.
1990Ken Ribetepsilon conjecturemodular forms
1992Richard BorcherdsConway–Norton conjecturesporadic groupsUsually called monstrous moonshine
1994David Harbater and Michel RaynaudAbhyankar's conjecturealgebraic geometry
1994Andrew WilesFermat's Last Theoremnumber theory⇔The modularity theorem for semistable elliptic curves.
Proof completed with Richard Taylor.
1994Fred GalvinDinitz conjecturecombinatorics
1995Doron Zeilberger[16]Alternating sign matrix conjecture,enumerative combinatorics
1996Vladimir VoevodskyMilnor conjecturealgebraic K-theoryVoevodsky's theorem, ⇐norm residue isomorphism theoremBeilinson–Lichtenbaum conjecture, Quillen–Lichtenbaum conjecture.
The ambiguous term "Bloch-Kato conjecture" may refer to what is now the norm residue isomorphism theorem.
1998Thomas Callister HalesKepler conjecturesphere packing
1998Thomas Callister Hales and Sean McLaughlindodecahedral conjectureVoronoi decompositions
2000Krzysztof Kurdyka, Tadeusz Mostowski and Adam ParusińskiGradient conjecturegradient vector fieldsAttributed to René Thom, c.1970.
2001Christophe Breuil, Brian Conrad, Fred Diamond and Richard TaylorTaniyama–Shimura conjectureelliptic curvesNow the modularity theorem for elliptic curves. Once known as the "Weil conjecture".
2001Mark Haimann! conjecturerepresentation theory
2001Daniel Frohardt and Kay Magaard[17]Guralnick–Thompson conjecturemonodromy groups
2002Preda MihăilescuCatalan's conjecture, 1844exponential diophantine equationsPillai's conjectureabc conjecture
Mihăilescu's theorem
2002Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomasstrong perfect graph conjectureperfect graphsChudnovsky–Robertson–Seymour–Thomas theorem
2002Grigori PerelmanPoincaré conjecture, 19043-manifolds
2003Grigori Perelmangeometrization conjecture of Thurston3-manifoldsspherical space form conjecture
2003Ben Green; and independently by Alexander SapozhenkoCameron–Erdős conjecturesum-free sets
2003Nils DenckerNirenberg–Treves conjecturepseudo-differential operators
2004 (see comment)Nobuo Iiyori and Hiroshi YamakiFrobenius conjecturegroup theoryA consequence of the classification of finite simple groups, completed in 2004 by the usual standards of pure mathematics.
2004Adam Marcus and Gábor TardosStanley–Wilf conjecturepermutation classesMarcus–Tardos theorem
2004Ualbai U. Umirbaev and Ivan P. ShestakovNagata's conjecture on automorphismspolynomial rings
2004Ian Agol and independently by Danny CalegariDavid Gabaitameness conjecturegeometric topologyAhlfors measure conjecture
2008Avraham TrahtmanRoad coloring conjecturegraph theory
2008Chandrashekhar Khare, Jean-Pierre WintenbergerSerre's modularity conjecturemodular forms
2009Jeremy Kahn, Vladimir Markovicsurface subgroup conjecture3-manifoldsEhrenpreis conjecture on quasiconformality
2009Jeremie Chalopin and Daniel GonçalvesScheinerman's conjectureintersection graphs
2010Terence Tao and Van H. Vucircular lawrandom matrix theory
2011Joel Friedman and Igor Mineyev, independentlyHanna Neumann conjecturegroup theory
2012Simon BrendleHsiang–Lawson's conjecturedifferential geometry
2012Fernando Codá Marques and André NevesWillmore conjecturedifferential geometry
2013Zhang Yitangbounded gap conjecturenumber theoryThe sequence of gaps between consecutive prime numbers has a finite lim inf. See Polymath Project#Polymath8 for quantitative results.
2013Adam Marcus, Daniel Spielman and Nikhil SrivastavaKadison–Singer problemfunctional analysisThe 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.
2015Jean Bourgain, Ciprian Demeter, and Larry GuthMain conjecture in Vinogradov's mean-value theoremanalytic number theoryBourgain–Demeter–Guth theorem, ⇐ decoupling theorem[18]
2019Dimitris Koukoulopoulos and James MaynardDuffin–Schaeffer conjecturenumber theoryRational approximation of irrational numbers

Disproved (no longer conjectures)

See also

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]
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