List of mathematical conjectures explained

This is a list of notable mathematical conjectures.

Open problems

The following conjectures remain open. The (incomplete) column "cites" lists the number of results for a Google Scholar search for the term, in double quotes .

ConjectureFieldCommentsEponym(s)Cites
1/3–2/3 conjectureorder theoryn/a 70
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 2440
Agoh–Giuga conjecturenumber theoryTakashi Agoh and Giuseppe Giuga 8
Agrawal's conjecturenumber theory10
Andrews–Curtis conjecturecombinatorial group theory358
Andrica's conjecturenumber theoryDorin Andrica 45
Artin conjecture (L-functions)number theory650
Artin's conjecture on primitive rootsnumber theorygeneralized Riemann hypothesis[2]
⇐Selberg conjecture B[3]
325
Bateman–Horn conjecturenumber theory245
Baum–Connes conjectureoperator K-theory⇒Gromov-Lawson-Rosenberg conjecture[4]
Kaplansky-Kadison conjecture
Novikov conjecture
2670
Beal's conjecturenumber theory142
Beilinson conjecturenumber theory461
Berry–Tabor conjecturegeodesic flowMichael Berry and Michael Tabor 239
Big-line-big-clique conjecturediscrete geometry
Birch and Swinnerton-Dyer conjecturenumber theory2830
Birch–Tate conjecturenumber theory149
Birkhoff conjectureintegrable systems345
Bloch–Beilinson conjecturesnumber theory152
Bloch–Kato conjecturealgebraic K-theory1620
Bochner–Riesz conjectureharmonic analysis⇒restriction conjecture⇒Kakeya maximal function conjecture⇒Kakeya dimension conjecture[5] 236
Bombieri–Lang conjecturediophantine geometry181
Borel conjecturegeometric topology981
Bost conjecturegeometric topology65
Brennan conjecturecomplex analysisJames E. Brennan 110
Brocard's conjecturenumber theory16
Brumer–Stark conjecturenumber theory208
Bunyakovsky conjecturenumber theory43
Carathéodory conjecturedifferential geometry173
Carmichael totient conjecturenumber theory
Casas-Alvero conjecturepolynomialsEduardo Casas-Alvero 56
Catalan–Dickson conjecture on aliquot sequencesnumber theory46
Catalan's Mersenne conjecturenumber theoryEugène Charles Catalan
Cherlin–Zilber conjecturegroup theory86
Chowla conjectureMöbius function⇒Sarnak conjecture[6] [7]
Collatz conjecturenumber theory1440
Cramér's conjecturenumber theory32
Conway's thrackle conjecturegraph theory150
Deligne conjecturemonodromy788
Dittert conjecturecombinatoricsEric Dittert 11
Eilenberg−Ganea conjecturealgebraic topology96
Elliott–Halberstam conjecturenumber theory300
Erdős–Faber–Lovász conjecturegraph theory172
Erdős–Gyárfás conjecturegraph theory37
Erdős–Straus conjecturenumber theory103
Farrell–Jones conjecturegeometric topology545
Filling area conjecturedifferential geometryn/a 60
Firoozbakht's conjecturenumber theory33
Fortune's conjecturenumber theory16
Four exponentials conjecturenumber theoryn/a 110
Frankl conjecturecombinatorics83
Gauss circle problemnumber theory553
Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean planemetric geometry
Gilbreath conjecturenumber theory34
Goldbach's conjecturenumber theory⇒The ternary Goldbach conjecture, which was the original formulation.[8] 5880
Gold partition conjecture[9] order theoryn/a 25
Goldberg–Seymour conjecturegraph theory57
Goormaghtigh conjecturenumber theory14
Green's conjecturealgebraic curves150
Grimm's conjecturenumber theoryCarl Albert Grimm 46
Grothendieck–Katz p-curvature conjecturedifferential equations98
Hadamard conjecturecombinatorics858
Herzog–Schönheim conjecturegroup theoryMarcel Herzog and Jochanan Schönheim 44
Hilbert–Smith conjecturegeometric topology219
Hodge conjecturealgebraic geometry2490
Homological conjectures in commutative algebracommutative algebran/a
Hopf conjecturesgeometry476
Ibragimov–Iosifescu conjecture for φ-mixing sequencesprobability theoryIldar Ibragimov,
Invariant subspace problemfunctional analysisn/a 2120
Jacobian conjecturepolynomialsCarl Gustav Jacob Jacobi (by way of the Jacobian determinant) 2860
Jacobson's conjecturering theory127
Kaplansky conjecturesring theory466
Keating–Snaith conjecturenumber theory48
Köthe conjecturering theory167
Kung–Traub conjectureiterative methods332
Legendre's conjecturenumber theory110
Lemoine's conjecturenumber theory13
Lenstra–Pomerance–Wagstaff conjecturenumber theory32
Leopoldt's conjecturenumber theory773
List coloring conjecturegraph theoryn/a 300
Littlewood conjecturediophantine approximation⇐Margulis conjecture[10] 1230
Lovász conjecturegraph theory560
MNOP conjecturealgebraic geometryn/a 63
Manin conjecturediophantine geometry338
Marshall Hall's conjecturenumber theory44
Mazur's conjecturesdiophantine geometry97
Montgomery's pair correlation conjecturenumber theory77
n conjecturenumber theoryn/a 126
New Mersenne conjecturenumber theory47
Novikov conjecturealgebraic topology3090
Oppermann's conjecturenumber theory12
Petersen coloring conjecturegraph theory52
Pierce–Birkhoff conjecturereal algebraic geometry96
Pillai's conjecturenumber theory33
De Polignac's conjecturenumber theory46
Quantum PCP conjecturequantum information 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 281
Reconstruction conjecturegraph theoryn/a 1040
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 Riemann24900
Ringel–Kotzig conjecturegraph theory187
Rudin's conjectureadditive combinatorics16
Sarnak conjecturetopological entropy295
Sato–Tate conjecturenumber theory1080
Schanuel's conjecturenumber theory329
Schinzel's hypothesis Hnumber theory49
Scholz conjectureaddition chains41
Second Hardy–Littlewood conjecturenumber theory30
Selfridge's conjecturenumber theory6
Sendov's conjecturecomplex polynomials77
Serre's multiplicity conjecturescommutative algebra221
Singmaster's conjecturebinomial coefficients8
Standard conjectures on algebraic cyclesalgebraic geometryn/a 234
Tate conjecturealgebraic geometry
Toeplitz' conjectureJordan curves
Tuza's conjecturegraph theoryZsolt Tuza
Twin prime conjecturenumber theoryn/a 1700
Ulam's packing conjecturepacking
Unicity conjecture for Markov numbersnumber theoryAndrey Markov (by way of Markov numbers)
Uniformity conjecturediophantine geometryn/a
Unique games conjecturenumber theoryn/a
Vandiver's conjecturenumber theory
Virasoro conjecturealgebraic geometry
Vizing's conjecturegraph theory
Vojta's conjecturenumber theory
Waring's conjecturenumber theory
Weight monodromy conjecturealgebraic geometryn/a
Weinstein conjectureperiodic orbits
Whitehead conjecturealgebraic topology
Zauner's conjectureoperator theoryGerhard Zauner

Conjectures now proved (theorems)

The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names.

Priority date[13] Proved byFormer nameFieldComments
1962Walter Feit and John G. 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 John William Theodore 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 independently by Andrei Suslin
Serre'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 Samuel James 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 Branges de BourciaBieberbach 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.
1989Vladimir 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 theorem⇔Beilinson–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 Parusiński Gradient 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 equations⇐Pillai's conjecture⇐abc conjecture
Mihăilescu's theorem
2002Maria Chudnovsky, Neil Robertson, Paul D. 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 Sapozhenko
Cameron–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 Gabai
tameness conjecturegeometric topologyAhlfors measure conjecture
2008Avraham TrahtmanRoad coloring conjecturegraph theory
2008Chandrashekhar Khare and Jean-Pierre WintenbergerSerre's modularity conjecturemodular forms
2009Jeremy Kahn and 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 independently by Igor MineyevHanna Neumann conjecturegroup theory
2012Simon BrendleHsiang–Lawson's conjecturedifferential geometry
2012Fernando Codá Marques and André NevesWillmore conjecturedifferential geometry
2013Yitang Zhangbounded 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]
2018Karim Adiprasitog-conjecturecombinatorics
2019Dimitris Koukoulopoulos and James MaynardDuffin–Schaeffer conjecturenumber theoryRational approximation of irrational numbers

Disproved (no longer conjectures)

The conjectures in following list were not necessarily generally accepted as true before being disproved.

In mathematics, ideas are supposedly not accepted as fact until they have been rigorously proved. However, there have been some ideas that were fairly accepted in the past but which were subsequently shown to be false. The following list is meant to serve as a repository for compiling a list of such ideas.

2m
2

+1

(known as Fermat numbers) were prime. However, this conjecture was disproved by Euler, who found that
25
2

+1=4,294,967,297=641 x 6,700,417.

[22]

\beth1

) is greater than that of the set of algebraic numbers (

\aleph0

).[23]

\pi(x)>li(x)

occurs somewhere before 10317. See Skewes' number for more detail.

See also

Further reading

External links

Notes and References

  1. Book: Weisstein . Eric W. . CRC Concise Encyclopedia of Mathematics . 2002 . CRC Press . 9781420035223 . 13 . en.
  2. Book: Frei . Günther . Lemmermeyer . Franz . Roquette . Peter J. . Emil Artin and Helmut Hasse: The Correspondence 1923-1958 . 2014 . Springer Science & Business Media . 9783034807159 . 215 . en.
  3. Book: Steuding . Jörn . Morel . J.-M. . Steuding . Jr̲n . Value-Distribution of L-Functions . 2007 . Springer Science & Business Media . 9783540265269 . 118 . en.
  4. Book: Valette . Alain . Introduction to the Baum-Connes Conjecture . 2002 . Springer Science & Business Media . 9783764367060 . viii . en.
  5. Book: Simon . Barry . Harmonic Analysis . 2015 . American Mathematical Soc. . 9781470411022 . 685 . en.
  6. Web site: Tao . Terence . The Chowla conjecture and the Sarnak conjecture . What's new . en . 15 October 2012.
  7. Book: Ferenczi . Sébastien . Kułaga-Przymus . Joanna . Lemańczyk . Mariusz . Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics: CIRM Jean-Morlet Chair, Fall 2016 . 2018 . Springer . 9783319749082 . 185 . en.
  8. Book: Weisstein . Eric W. . CRC Concise Encyclopedia of Mathematics . 2002 . CRC Press . 9781420035223 . 1203 . en.
  9. M. Peczarski, The gold partition conjecture, it Order 23(2006): 89–95.
  10. Book: Burger . Marc . Iozzi . Alessandra. Alessandra Iozzi . 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 . 2013 . Springer Science & Business Media . 9783662047439 . 408 . en.
  11. Web site: EMS Prizes. www.math.kth.se.
  12. Web site: Archived copy . 2008-12-12 . dead . https://web.archive.org/web/20110724181506/http://matematikkforeningen.no/INFOMAT/08/0810.pdf . 2011-07-24 .
  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. Book: Dudziak . James . Vitushkin's Conjecture for Removable Sets . 2011 . Springer Science & Business Media . 9781441967091 . 39 . en.
  15. Book: Weisstein . Eric W. . CRC Concise Encyclopedia of Mathematics . 2002 . CRC Press . 9781420035223 . 218 . en.
  16. Book: Weisstein . Eric W. . CRC Concise Encyclopedia of Mathematics . 2002 . CRC Press . 9781420035223 . 65 . en.
  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
  18. Web site: Decoupling and the Bourgain-Demeter-Guth proof of the Vinogradov main conjecture . What's new . en . 10 December 2015.
  19. Book: Holden . Helge . Piene . Ragni . The Abel Prize 2013-2017 . 2018 . Springer . 9783319990286 . 51 . en.
  20. Web site: Kalai . Gil . A sensation in the morning news – Yaroslav Shitov: Counterexamples to Hedetniemi's conjecture. . Combinatorics and more . en . 10 May 2019.
  21. Book: Farlow, Stanley J. . Paradoxes in Mathematics . 2014 . . 57 . 978-0-486-49716-7.
  22. Book: Krizek . Michal . Luca . Florian . Somer . Lawrence . 17 Lectures on Fermat Numbers: From Number Theory to Geometry . 2001 . . 1 . 0-387-95332-9 . 10.1007/978-0-387-21850-2.
  23. Book: McQuarrie, Donald Allan . Mathematical Methods for Scientists and Engineers . 2003 . 711 . University Science Books.
  24. Lehman . R. S. . 1960 . On Liouville's function . . 10.1090/S0025-5718-1960-0120198-5 . free . 0120198 . 2003890 . 14 . 72 . 311–320.
  25. Tanaka . M. . 1980 . A Numerical Investigation on Cumulative Sum of the Liouville Function . . 10.3836/tjm/1270216093 . 0584557 . 3 . 1 . 187–189. free.
  26. https://www.newscientist.com/channel/fundamentals/mg19526131.900-interview-why-mathematics-is-beautiful.html Why mathematics is beautiful
  27. Neeman . Amnon . 2002 . A counterexample to a 1961 “theorem” in homological algebra . Inventiones mathematicae . 10.1007/s002220100197 . 148 . 397–420.