List of conjectures

From Wikipedia, the free encyclopedia

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]
⇐[3]		 Emil Artin
Bateman–Horn conjecture number theory Paul T. Bateman and Roger Horn
Baum–Connes conjecture operator K-theory ⇒[4]
⇒Kaplansky-Kadison conjecture[4]
⇒Novikov conjecture[4]		 Paul Baum and Alain Connes
Beal's conjecture number theory Andrew Beal
Beilinson conjecture number theory Alexander Beilinson
Berry–Tabor conjecture geodesic flow Michael Berry and Michael Tabor
Birch and Swinnerton-Dyer conjecture number theory Bryan John Birch and Peter Swinnerton-Dyer
Birch–Tate conjecture number theory Bryan John Birch and John Tate
integrable systems George David Birkhoff
Bloch–Beilinson conjectures number theory Spencer Bloch and Alexander Beilinson
Bloch–Kato conjecture algebraic K-theory Spencer Bloch and Kazuya Kato
Bochner–Riesz conjecture harmonic analysis ⇒restriction conjecture⇒Kakeya maximal function conjecture⇒Kakeya dimension conjecture[5] Salomon Bochner and Marcel Riesz
Bombieri–Lang conjecture diophantine geometry Enrico Bombieri and Serge Lang
Borel conjecture geometric topology Armand Borel
Bost conjecture geometric topology Jean-Benoît Bost
Brennan conjecture complex analysis James E. Brennan
Brocard's conjecture number theory Henri Brocard
Brumer–Stark conjecture number theory Armand Brumer and Harold Stark
Bunyakovsky conjecture number theory Viktor Bunyakovsky
Carathéodory conjecture differential geometry Constantin Carathéodory
Carmichael totient conjecture number theory Robert Daniel Carmichael
Casas-Alvero conjecture polynomials Eduardo Casas-Alvero
Catalan–Dickson conjecture on aliquot sequences number theory Eugène Charles Catalan and Leonard Eugene Dickson
Catalan's Mersenne conjecture number theory Eugène Charles Catalan
Cherlin–Zilber conjecture group theory Gregory Cherlin and Boris Zilber
Möbius function ⇒[6][7] Sarvadaman Chowla
Collatz conjecture number theory Lothar Collatz
Cramér's conjecture number theory Harald Cramér
Conway's thrackle conjecture graph theory John Horton Conway
Deligne conjecture monodromy Pierre Deligne
Dittert conjecture combinatorics Eric Dittert
Eilenberg−Ganea conjecture algebraic topology Samuel Eilenberg and Tudor Ganea
Elliott–Halberstam conjecture number theory Peter D. T. A. Elliott and Heini Halberstam
Erdős–Faber–Lovász conjecture graph theory Paul Erdős, Vance Faber, and László Lovász
Erdős–Gyárfás conjecture graph theory Paul Erdős and András Gyárfás
Erdős–Straus conjecture number theory Paul Erdős and Ernst G. Straus
Farrell–Jones conjecture geometric topology F. Thomas Farrell and Lowell E. Jones
Filling area conjecture differential geometry n/a
Firoozbakht's conjecture number theory Farideh Firoozbakht
Fortune's conjecture number theory Reo Fortune
Four exponentials conjecture number theory n/a
Frankl conjecture combinatorics Péter Frankl
Gauss circle problem number theory Carl Friedrich Gauss
Gilbreath conjecture number theory Norman Laurence Gilbreath
Goldbach's conjecture number theory ⇒The ternary Goldbach conjecture, which was the original formulation.[8] Christian Goldbach
Gold partition conjecture[9] order theory n/a
Goldberg–Seymour conjecture graph theory Mark K. Goldberg and Paul Seymour
Goormaghtigh conjecture number theory René Goormaghtigh
Green's conjecture algebraic curves Mark Lee Green
Grimm's conjecture number theory Carl Albert Grimm
Grothendieck–Katz p-curvature conjecture differential equations Alexander Grothendieck and Nick Katz
Hadamard conjecture combinatorics Jacques Hadamard
Herzog–Schönheim conjecture group theory Marcel Herzog and Jochanan Schönheim
Hilbert–Smith conjecture geometric topology David Hilbert and Paul Althaus Smith
Hodge conjecture algebraic geometry W. V. D. Hodge
Homological conjectures in commutative algebra commutative algebra n/a
Hopf conjectures geometry Heinz Hopf
Invariant subspace problem functional analysis n/a
Jacobian conjecture polynomials Carl Gustav Jacob Jacobi (by way of the Jacobian determinant)
Jacobson's conjecture ring theory Nathan Jacobson
Kaplansky conjectures ring theory Irving Kaplansky
number theory Jonathan Keating and Nina Snaith
Köthe conjecture ring theory Gottfried Köthe
iterative methods H. T. Kung and Joseph F. Traub
Legendre's conjecture number theory Adrien-Marie Legendre
Lemoine's conjecture number theory Émile Lemoine
Lenstra–Pomerance–Wagstaff conjecture number theory Hendrik Lenstra, Carl Pomerance, and Samuel S. Wagstaff Jr.
Leopoldt's conjecture number theory Heinrich-Wolfgang Leopoldt
List coloring conjecture graph theory n/a
Littlewood conjecture diophantine approximation ⇐[10] John Edensor Littlewood
Lovász conjecture graph theory László Lovász
algebraic geometry n/a
Manin conjecture diophantine geometry Yuri Manin
Marshall Hall's conjecture number theory Marshall Hall, Jr.
diophantine geometry Barry Mazur
Montgomery's pair correlation conjecture number theory Hugh Lowell Montgomery
n conjecture number theory n/a
New Mersenne conjecture number theory Marin Mersenne
Novikov conjecture algebraic topology Sergei Novikov
Oppermann's conjecture number theory Ludvig Oppermann
Petersen coloring conjecture graph theory Julius Petersen
Pierce–Birkhoff conjecture real algebraic geometry Richard S. Pierce and Garrett Birkhoff
Pillai's conjecture number theory Subbayya Sivasankaranarayana Pillai
De Polignac's conjecture number theory Alphonse de Polignac
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
⇒, Lindelöf hypothesis
See Hilbert–Pólya conjecture. For other Riemann hypotheses, see the Weil conjectures (now theorems).		 Bernhard Riemann
Ringel–Kotzig conjecture graph theory Gerhard Ringel and Anton Kotzig
Rudin's conjecture additive combinatorics Walter Rudin
topological entropy Peter Sarnak
Sato–Tate conjecture number theory Mikio Sato and John Tate
Schanuel's conjecture number theory Stephen Schanuel
Schinzel's hypothesis H number theory Andrzej Schinzel
Scholz conjecture addition chains Arnold Scholz
Second Hardy–Littlewood conjecture number theory G. H. Hardy and John Edensor Littlewood
Selfridge's conjecture number theory John Selfridge
Sendov's conjecture complex polynomials Blagovest Sendov
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 Stanislaw Ulam
Unicity conjecture for Markov numbers number theory Andrey Markov (by way of Markov numbers)
Uniformity conjecture diophantine geometry n/a
Unique games conjecture number theory n/a
Vandiver's conjecture number theory Ernst Kummer and Harry Vandiver
Virasoro conjecture algebraic geometry Miguel Ángel Virasoro
Vizing's conjecture graph theory Vadim G. Vizing
Vojta's conjecture number theory ⇒abc conjecture Paul Vojta
Waring's conjecture number theory Edward Waring
Weight monodromy conjecture algebraic geometry n/a
Weinstein conjecture periodic orbits Alan Weinstein
Whitehead conjecture algebraic topology J. H. C. Whitehead
Zauner's conjecture operator theory Gerhard Zauner

Conjectures now proved (theorems)[]

Further information: 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 and John G. 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 John William Theodore 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 independently by Andrei Suslin 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 Samuel James 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 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 de Bourcia 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 Vladimir 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] 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 D. 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 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 and Jean-Pierre Wintenberger Serre's modularity conjecture modular forms
2009 Jeremy Kahn and 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 independently by Igor Mineyev 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 Yitang Zhang 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]
2018 Karim Adiprasito g-conjecture combinatorics
2019 Dimitris Koukoulopoulos and James Maynard Duffin–Schaeffer conjecture number theory Rational 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. ^ a b c 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.{{cite web}}: 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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