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 as of September 2022.

Conjecture Field Comments Eponym(s) Cites
1/3–2/3 conjectureorder theoryn/a70
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/a2440
Agoh–Giuga conjecturenumber theoryTakashi Agoh and Giuseppe Giuga8
Agrawal's conjecturenumber theoryManindra Agrawal10
Andrews–Curtis conjecturecombinatorial group theoryJames J. Andrews and Morton L. Curtis358
Andrica's conjecturenumber theoryDorin Andrica45
Artin conjecture (L-functions)number theoryEmil Artin650
Artin's conjecture on primitive rootsnumber theorygeneralized Riemann hypothesis[2]
⇐Selberg conjecture B[3]
Emil Artin325
Bateman–Horn conjecturenumber theoryPaul T. Bateman and Roger Horn245
Baum–Connes conjectureoperator K-theory⇒Gromov-Lawson-Rosenberg conjecture[4]
Kaplansky-Kadison conjecture[4]
Novikov conjecture[4]
Paul Baum and Alain Connes2670
Beal's conjecturenumber theoryAndrew Beal142
Beilinson conjecturenumber theoryAlexander Beilinson461
Berry–Tabor conjecturegeodesic flowMichael Berry and Michael Tabor239
Big-line-big-clique conjecturediscrete geometry
Birch and Swinnerton-Dyer conjecturenumber theoryBryan John Birch and Peter Swinnerton-Dyer2830
Birch–Tate conjecturenumber theoryBryan John Birch and John Tate149
Birkhoff conjectureintegrable systemsGeorge David Birkhoff345
Bloch–Beilinson conjecturesnumber theorySpencer Bloch and Alexander Beilinson152
Bloch–Kato conjecturealgebraic K-theorySpencer Bloch and Kazuya Kato1620
Bochner–Riesz conjectureharmonic analysis⇒restriction conjecture⇒Kakeya maximal function conjectureKakeya dimension conjecture[5]Salomon Bochner and Marcel Riesz236
Bombieri–Lang conjecturediophantine geometryEnrico Bombieri and Serge Lang181
Borel conjecturegeometric topologyArmand Borel981
Bost conjecturegeometric topologyJean-Benoît Bost65
Brennan conjecturecomplex analysisJames E. Brennan110
Brocard's conjecturenumber theoryHenri Brocard16
Brumer–Stark conjecturenumber theoryArmand Brumer and Harold Stark208
Bunyakovsky conjecturenumber theoryViktor Bunyakovsky43
Carathéodory conjecturedifferential geometryConstantin Carathéodory173
Carmichael totient conjecturenumber theoryRobert Daniel Carmichael
Casas-Alvero conjecturepolynomialsEduardo Casas-Alvero56
Catalan–Dickson conjecture on aliquot sequencesnumber theoryEugène Charles Catalan and Leonard Eugene Dickson46
Catalan's Mersenne conjecturenumber theoryEugène Charles Catalan
Cherlin–Zilber conjecturegroup theoryGregory Cherlin and Boris Zilber86
Chowla conjectureMöbius function⇒Sarnak conjecture[6][7]Sarvadaman Chowla
Collatz conjecturenumber theoryLothar Collatz1440
Cramér's conjecturenumber theoryHarald Cramér32
Conway's thrackle conjecturegraph theoryJohn Horton Conway150
Deligne conjecturemonodromyPierre Deligne788
Dittert conjecturecombinatoricsEric Dittert11
Eilenberg−Ganea conjecturealgebraic topologySamuel Eilenberg and Tudor Ganea96
Elliott–Halberstam conjecturenumber theoryPeter D. T. A. Elliott and Heini Halberstam300
Erdős–Faber–Lovász conjecturegraph theoryPaul Erdős, Vance Faber, and László Lovász172
Erdős–Gyárfás conjecturegraph theoryPaul Erdős and András Gyárfás37
Erdős–Straus conjecturenumber theoryPaul Erdős and Ernst G. Straus103
Farrell–Jones conjecturegeometric topologyF. Thomas Farrell and Lowell E. Jones545
Filling area conjecturedifferential geometryn/a60
Firoozbakht's conjecturenumber theoryFarideh Firoozbakht33
Fortune's conjecturenumber theoryReo Fortune16
Four exponentials conjecturenumber theoryn/a110
Frankl conjecturecombinatoricsPéter Frankl83
Gauss circle problemnumber theoryCarl Friedrich Gauss553
Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean planemetric geometryEdgar Gilbert and Henry O. Pollak
Gilbreath conjecturenumber theoryNorman Laurence Gilbreath34
Goldbach's conjecturenumber theory⇒The ternary Goldbach conjecture, which was the original formulation.[8]Christian Goldbach5880
Gold partition conjecture[9]order theoryn/a25
Goldberg–Seymour conjecturegraph theoryMark K. Goldberg and Paul Seymour57
Goormaghtigh conjecturenumber theoryRené Goormaghtigh14
Green's conjecturealgebraic curvesMark Lee Green150
Grimm's conjecturenumber theoryCarl Albert Grimm46
Grothendieck–Katz p-curvature conjecturedifferential equationsAlexander Grothendieck and Nick Katz98
Hadamard conjecturecombinatoricsJacques Hadamard858
Herzog–Schönheim conjecturegroup theoryMarcel Herzog and Jochanan Schönheim44
Hilbert–Smith conjecturegeometric topologyDavid Hilbert and Paul Althaus Smith219
Hodge conjecturealgebraic geometryW. V. D. Hodge2490
Homological conjectures in commutative algebracommutative algebran/a
Hopf conjecturesgeometryHeinz Hopf476
Ibragimov–Iosifescu conjecture for φ-mixing sequencesprobability theoryIldar Ibragimov, ro:Marius Iosifescu
Invariant subspace problemfunctional analysisn/a2120
Jacobian conjecturepolynomialsCarl Gustav Jacob Jacobi (by way of the Jacobian determinant)2860
Jacobson's conjecturering theoryNathan Jacobson127
Kaplansky conjecturesring theoryIrving Kaplansky466
Keating–Snaith conjecturenumber theoryJonathan Keating and Nina Snaith48
Köthe conjecturering theoryGottfried Köthe167
Kung–Traub conjectureiterative methodsH. T. Kung and Joseph F. Traub332
Legendre's conjecturenumber theoryAdrien-Marie Legendre110
Lemoine's conjecturenumber theoryÉmile Lemoine13
Lenstra–Pomerance–Wagstaff conjecturenumber theoryHendrik Lenstra, Carl Pomerance, and Samuel S. Wagstaff Jr.32
Leopoldt's conjecturenumber theoryHeinrich-Wolfgang Leopoldt773
List coloring conjecturegraph theoryn/a300
Littlewood conjecturediophantine approximation⇐Margulis conjecture[10]John Edensor Littlewood1230
Lovász conjecturegraph theoryLászló Lovász560
MNOP conjecturealgebraic geometryn/a63
Manin conjecturediophantine geometryYuri Manin338
Marshall Hall's conjecturenumber theoryMarshall Hall, Jr.44
Mazur's conjecturesdiophantine geometryBarry Mazur97
Montgomery's pair correlation conjecturenumber theoryHugh Lowell Montgomery77
n conjecturenumber theoryn/a126
New Mersenne conjecturenumber theoryMarin Mersenne47
Novikov conjecturealgebraic topologySergei Novikov3090
Oppermann's conjecturenumber theoryLudvig Oppermann12
Petersen coloring conjecturegraph theoryJulius Petersen52
Pierce–Birkhoff conjecturereal algebraic geometryRichard S. Pierce and Garrett Birkhoff96
Pillai's conjecturenumber theorySubbayya Sivasankaranarayana Pillai33
De Polignac's conjecturenumber theoryAlphonse de Polignac46
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/a281
Reconstruction conjecturegraph theoryn/a1040
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 theoryGerhard Ringel and Anton Kotzig187
Rudin's conjectureadditive combinatoricsWalter Rudin16
Sarnak conjecturetopological entropyPeter Sarnak295
Sato–Tate conjecturenumber theoryMikio Sato and John Tate1080
Schanuel's conjecturenumber theoryStephen Schanuel329
Schinzel's hypothesis Hnumber theoryAndrzej Schinzel49
Scholz conjectureaddition chainsArnold Scholz41
Second Hardy–Littlewood conjecturenumber theoryG. H. Hardy and John Edensor Littlewood30
Selfridge's conjecturenumber theoryJohn Selfridge6
Sendov's conjecturecomplex polynomialsBlagovest Sendov77
Serre's multiplicity conjecturescommutative algebraJean-Pierre Serre221
Singmaster's conjecturebinomial coefficientsDavid Singmaster8
Standard conjectures on algebraic cyclesalgebraic geometryn/a234
Tate conjecturealgebraic geometryJohn Tate
Toeplitz' conjectureJordan curvesOtto Toeplitz
Tuza's conjecturegraph theoryZsolt Tuza
Twin prime conjecturenumber theoryn/a1700
Ulam's packing conjecturepackingStanislaw Ulam
Unicity conjecture for Markov numbersnumber theoryAndrey Markov (by way of Markov numbers)
Uniformity conjecturediophantine geometryn/a
Unique games conjecturenumber theoryn/a
Vandiver's conjecturenumber theoryErnst Kummer and Harry Vandiver
Virasoro conjecturealgebraic geometryMiguel Ángel Virasoro
Vizing's conjecturegraph theoryVadim G. Vizing
Vojta's conjecturenumber theoryabc conjecturePaul Vojta
Waring's conjecturenumber theoryEdward Waring
Weight monodromy conjecturealgebraic geometryn/a
Weinstein conjectureperiodic orbitsAlan Weinstein
Whitehead conjecturealgebraic topologyJ. H. C. Whitehead
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 by Former name Field Comments
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 SuslinSerre'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 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 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 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 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.

  • The idea of the Pythagoreans that all numbers can be expressed as a ratio of two whole numbers. This was disproved by one of Pythagoras' own disciples, Hippasus, who showed that the square root of two is what we today call an irrational number. One story claims that he was thrown off the ship in which he and some other Pythagoreans were sailing because his discovery was too heretical.[22]
  • Euclid's parallel postulate stated that if two lines cross a third in a plane in such a way that the sum of the "interior angles" is not 180° then the two lines meet. Furthermore, he implicitly assumed that two separate intersecting lines meet at only one point. These assumptions were believed to be true for more than 2000 years, but in light of General Relativity at least the second can no longer be considered true. In fact the very notion of a straight line in four-dimensional curved space-time has to be redefined, which one can do as a geodesic. (But the notion of a plane does not carry over.) It is now recognized that Euclidean geometry can be studied as a mathematical abstraction, but that the universe is non-Euclidean.
  • Fermat conjectured that all numbers of the form (known as Fermat numbers) were prime. However, this conjecture was disproved by Euler, who found that [23]
  • The idea that transcendental numbers were unusual. Disproved by Georg Cantor who showed that there are so many transcendental numbers that it is impossible to make a one-to-one mapping between them and the algebraic numbers. In other words, the cardinality of the set of transcendentals (denoted ) is greater than that of the set of algebraic numbers ().[24]
  • Bernhard Riemann, at the end of his famous 1859 paper "On the Number of Primes Less Than a Given Magnitude", stated (based on his results) that the logarithmic integral gives a somewhat too high estimate of the prime-counting function. The evidence also seemed to indicate this. However, in 1914 J. E. Littlewood proved that this was not always the case, and in fact it is now known that the first x for which occurs somewhere before 10317. See Skewes' number for more detail.
  • Naïvely it might be expected that a continuous function must have a derivative or else that the set of points where it is not differentiable should be "small" in some sense. This was disproved in 1872 by Karl Weierstrass, and in fact examples had been found earlier of functions that were nowhere differentiable (see Weierstrass function). According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that such functions did not exist.
  • It was conjectured in 1919 by George Pólya, based on the evidence, that most numbers less than any particular limit have an odd number of prime factors. However, this Pólya conjecture was disproved in 1958. It turns out that for some values of the limit (such as values a bit more than 906 million),[25][26] most numbers less than the limit have an even number of prime factors.
  • Erik Christopher Zeeman tried for 7 years to prove that one cannot untie a knot on a 4-sphere. Then one day he decided to try to prove the opposite, and he succeeded in a few hours.[27]
  • A "theorem" of Jan-Erik Roos in 1961 stated that in an [AB4*] abelian category, lim1 vanishes on Mittag-Leffler sequences. This "theorem" was used by many people since then, but it was disproved by counterexample in 2002 by Amnon Neeman.[28]

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. 1 2 3 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]
  22. Farlow, Stanley J. (2014). Paradoxes in Mathematics. Courier Corporation. p. 57. ISBN 978-0-486-49716-7.
  23. Krizek, Michal; Luca, Florian; Somer, Lawrence (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. Springer. p. 1. doi:10.1007/978-0-387-21850-2. ISBN 0-387-95332-9.
  24. McQuarrie, Donald Allan (2003). Mathematical Methods for Scientists and Engineers. University Science Books. p. 711.
  25. Lehman, R. S. (1960). "On Liouville's function". Mathematics of Computation. 14 (72): 311–320. doi:10.1090/S0025-5718-1960-0120198-5. JSTOR 2003890. MR 0120198.
  26. Tanaka, M. (1980). "A Numerical Investigation on Cumulative Sum of the Liouville Function". Tokyo Journal of Mathematics. 3 (1): 187–189. doi:10.3836/tjm/1270216093. MR 0584557.
  27. Why mathematics is beautiful in New Scientist, 21 July 2007, p. 48
  28. Neeman, Amnon (2002). "A counterexample to a 1961 "theorem" in homological algebra". Inventiones mathematicae. 148: 397–420. doi:10.1007/s002220100197.
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