List of conjectures

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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 conjecture order theory n/a 70
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 2440
Agoh–Giuga conjecture number theory Takashi Agoh and Giuseppe Giuga 8
Agrawal's conjecture number theory Manindra Agrawal 10
Andrews–Curtis conjecture combinatorial group theory James J. Andrews and Morton L. Curtis 358
Andrica's conjecture number theory Dorin Andrica 45
Artin conjecture (L-functions) number theory Emil Artin 650
Artin's conjecture on primitive roots number theory ⇐generalized Riemann hypothesis[2]
⇐Selberg conjecture B[3]		 Emil Artin 325
Bateman–Horn conjecture number theory Paul T. Bateman and Roger Horn 245
Baum–Connes conjecture operator K-theory ⇒Gromov-Lawson-Rosenberg conjecture[4]
⇒Kaplansky-Kadison conjecture[4]
⇒Novikov conjecture[4]		 Paul Baum and Alain Connes 2670
Beal's conjecture number theory Andrew Beal 142
Beilinson conjecture number theory Alexander Beilinson 461
Berry–Tabor conjecture geodesic flow Michael Berry and Michael Tabor 239
Big-line-big-clique conjecture discrete geometry
Birch and Swinnerton-Dyer conjecture number theory Bryan John Birch and Peter Swinnerton-Dyer 2830
Birch–Tate conjecture number theory Bryan John Birch and John Tate 149
Birkhoff conjecture integrable systems George David Birkhoff 345
Bloch–Beilinson conjectures number theory Spencer Bloch and Alexander Beilinson 152
Bloch–Kato conjecture algebraic K-theory Spencer Bloch and Kazuya Kato 1620
Bochner–Riesz conjecture harmonic analysis ⇒restriction conjecture⇒Kakeya maximal function conjecture⇒Kakeya dimension conjecture[5] Salomon Bochner and Marcel Riesz 236
Bombieri–Lang conjecture diophantine geometry Enrico Bombieri and Serge Lang 181
Borel conjecture geometric topology Armand Borel 981
Bost conjecture geometric topology Jean-Benoît Bost 65
Brennan conjecture complex analysis James E. Brennan 110
Brocard's conjecture number theory Henri Brocard 16
Brumer–Stark conjecture number theory Armand Brumer and Harold Stark 208
Bunyakovsky conjecture number theory Viktor Bunyakovsky 43
Carathéodory conjecture differential geometry Constantin Carathéodory 173
Carmichael totient conjecture number theory Robert Daniel Carmichael
Casas-Alvero conjecture polynomials Eduardo Casas-Alvero 56
Catalan–Dickson conjecture on aliquot sequences number theory Eugène Charles Catalan and Leonard Eugene Dickson 46
Catalan's Mersenne conjecture number theory Eugène Charles Catalan
Cherlin–Zilber conjecture group theory Gregory Cherlin and Boris Zilber 86
Chowla conjecture Möbius function ⇒Sarnak conjecture[6][7] Sarvadaman Chowla
Collatz conjecture number theory Lothar Collatz 1440
Cramér's conjecture number theory Harald Cramér 32
Conway's thrackle conjecture graph theory John Horton Conway 150
Deligne conjecture monodromy Pierre Deligne 788
Dittert conjecture combinatorics Eric Dittert 11
Eilenberg−Ganea conjecture algebraic topology Samuel Eilenberg and Tudor Ganea 96
Elliott–Halberstam conjecture number theory Peter D. T. A. Elliott and Heini Halberstam 300
Erdős–Faber–Lovász conjecture graph theory Paul Erdős, Vance Faber, and László Lovász 172
Erdős–Gyárfás conjecture graph theory Paul Erdős and András Gyárfás 37
Erdős–Straus conjecture number theory Paul Erdős and Ernst G. Straus 103
Farrell–Jones conjecture geometric topology F. Thomas Farrell and Lowell E. Jones 545
Filling area conjecture differential geometry n/a 60
Firoozbakht's conjecture number theory Farideh Firoozbakht 33
Fortune's conjecture number theory Reo Fortune 16
Four exponentials conjecture number theory n/a 110
Frankl conjecture combinatorics Péter Frankl 83
Gauss circle problem number theory Carl Friedrich Gauss 553
Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane metric geometry Edgar Gilbert and Henry O. Pollak
Gilbreath conjecture number theory Norman Laurence Gilbreath 34
Goldbach's conjecture number theory ⇒The ternary Goldbach conjecture, which was the original formulation.[8] Christian Goldbach 5880
Gold partition conjecture[9] order theory n/a 25
Goldberg–Seymour conjecture graph theory Mark K. Goldberg and Paul Seymour 57
Goormaghtigh conjecture number theory René Goormaghtigh 14
Green's conjecture algebraic curves Mark Lee Green 150
Grimm's conjecture number theory Carl Albert Grimm 46
Grothendieck–Katz p-curvature conjecture differential equations Alexander Grothendieck and Nick Katz 98
Hadamard conjecture combinatorics Jacques Hadamard 858
Herzog–Schönheim conjecture group theory Marcel Herzog and Jochanan Schönheim 44
Hilbert–Smith conjecture geometric topology David Hilbert and Paul Althaus Smith 219
Hodge conjecture algebraic geometry W. V. D. Hodge 2490
Homological conjectures in commutative algebra commutative algebra n/a
Hopf conjectures geometry Heinz Hopf 476
Ibragimov–Iosifescu conjecture for φ-mixing sequences probability theory Ildar Ibragimov, :ro:Marius Iosifescu
Invariant subspace problem functional analysis n/a 2120
Jacobian conjecture polynomials Carl Gustav Jacob Jacobi (by way of the Jacobian determinant) 2860
Jacobson's conjecture ring theory Nathan Jacobson 127
Kaplansky conjectures ring theory Irving Kaplansky 466
Keating–Snaith conjecture number theory Jonathan Keating and Nina Snaith 48
Köthe conjecture ring theory Gottfried Köthe 167
Kung–Traub conjecture iterative methods H. T. Kung and Joseph F. Traub 332
Legendre's conjecture number theory Adrien-Marie Legendre 110
Lemoine's conjecture number theory Émile Lemoine 13
Lenstra–Pomerance–Wagstaff conjecture number theory Hendrik Lenstra, Carl Pomerance, and Samuel S. Wagstaff Jr. 32
Leopoldt's conjecture number theory Heinrich-Wolfgang Leopoldt 773
List coloring conjecture graph theory n/a 300
Lovász conjecture graph theory László Lovász 560
MNOP conjecture algebraic geometry n/a 63
Manin conjecture diophantine geometry Yuri Manin 338
Marshall Hall's conjecture number theory Marshall Hall, Jr. 44
Mazur's conjectures diophantine geometry Barry Mazur 97
Montgomery's pair correlation conjecture number theory Hugh Lowell Montgomery 77
n conjecture number theory n/a 126
New Mersenne conjecture number theory Marin Mersenne 47
Novikov conjecture algebraic topology Sergei Novikov 3090
Oppermann's conjecture number theory Ludvig Oppermann 12
Petersen coloring conjecture graph theory Julius Petersen 52
Pierce–Birkhoff conjecture real algebraic geometry Richard S. Pierce and Garrett Birkhoff 96
Pillai's conjecture number theory Subbayya Sivasankaranarayana Pillai 33
De Polignac's conjecture number theory Alphonse de Polignac 46
Quantum PCP conjecture quantum information theory
quantum unique ergodicity conjecture dynamical systems 2004, Elon Lindenstrauss, for arithmetic hyperbolic surfaces,[10] 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfaces[11] n/a 281
Reconstruction conjecture graph theory n/a 1040
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 24900
Ringel–Kotzig conjecture graph theory Gerhard Ringel and Anton Kotzig 187
Rudin's conjecture additive combinatorics Walter Rudin 16
Sarnak conjecture topological entropy Peter Sarnak 295
Sato–Tate conjecture number theory Mikio Sato and John Tate 1080
Schanuel's conjecture number theory Stephen Schanuel 329
Schinzel's hypothesis H number theory Andrzej Schinzel 49
Scholz conjecture addition chains Arnold Scholz 41
Second Hardy–Littlewood conjecture number theory G. H. Hardy and John Edensor Littlewood 30
Selfridge's conjecture number theory John Selfridge 6
Sendov's conjecture complex polynomials Blagovest Sendov 77
Serre's multiplicity conjectures commutative algebra Jean-Pierre Serre 221
Singmaster's conjecture binomial coefficients David Singmaster 8
Standard conjectures on algebraic cycles algebraic geometry n/a 234
Tate conjecture algebraic geometry John Tate
Toeplitz' conjecture Jordan curves Otto Toeplitz
Tuza's conjecture graph theory Zsolt Tuza
Twin prime conjecture number theory n/a 1700
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)

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

Priority date[12] 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[13]
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 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 de Bourcia Bieberbach conjecture, 1916 complex analysis ⇐Robertson conjecture⇐Milin conjecture⇐de Branges's theorem[14]
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[15] 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[16] 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 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 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 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 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[17]
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)

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.[21]
  • 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 [math]\displaystyle{ 2^{2^m}+1 }[/math] (known as Fermat numbers) were prime. However, this conjecture was disproved by Euler, who found that [math]\displaystyle{ 2^{2^5}+1=4,294,967,297 = 641 \times 6,700,417. }[/math][22]
  • 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 [math]\displaystyle{ \beth_1 }[/math]) is greater than that of the set of algebraic numbers ([math]\displaystyle{ \aleph_0 }[/math]).[23]
  • 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 [math]\displaystyle{ \pi(x) \gt \mathrm{li}(x) }[/math] 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),[24][25] 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.[26]
  • 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.[27]

See also

References

  1. ↑ Weisstein, Eric W. (2002) (in en). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 13. ISBN 9781420035223. https://books.google.com/books?id=D_XKBQAAQBAJ&pg=PA13. 
  2. ↑ Frei, Günther; Lemmermeyer, Franz; Roquette, Peter J. (2014) (in en). Emil Artin and Helmut Hasse: The Correspondence 1923-1958. Springer Science & Business Media. p. 215. ISBN 9783034807159. https://books.google.com/books?id=6rDBBAAAQBAJ&pg=PA215. 
  3. ↑ Steuding, Jörn; Morel, J.-M.; Steuding, Jr̲n (2007) (in en). Value-Distribution of L-Functions. Springer Science & Business Media. p. 118. ISBN 9783540265269. https://books.google.com/books?id=gzUxpU-PXVoC&pg=PA118. 
  4. ↑ 4.0 4.1 4.2 Valette, Alain (2002) (in en). Introduction to the Baum-Connes Conjecture. Springer Science & Business Media. p. viii. ISBN 9783764367060. https://books.google.com/books?id=fRWjukxro3oC&pg=PR8. 
  5. ↑ Simon, Barry (2015) (in en). Harmonic Analysis. American Mathematical Soc.. p. 685. ISBN 9781470411022. https://books.google.com/books?id=YkUACwAAQBAJ&pg=PA685. 
  6. ↑ Tao, Terence (15 October 2012). "The Chowla conjecture and the Sarnak conjecture" (in en). https://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnak-conjecture/. 
  7. ↑ Ferenczi, Sébastien; Kułaga-Przymus, Joanna; Lemańczyk, Mariusz (2018) (in en). Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics: CIRM Jean-Morlet Chair, Fall 2016. Springer. p. 185. ISBN 9783319749082. https://books.google.com/books?id=wSpgDwAAQBAJ&pg=PA185. 
  8. ↑ Weisstein, Eric W. (2002) (in en). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 1203. ISBN 9781420035223. https://books.google.com/books?id=D_XKBQAAQBAJ&pg=PA1203. 
  9. ↑ M. Peczarski, The gold partition conjecture, it Order 23(2006): 89–95.
  10. ↑ "EMS Prizes". http://www.math.kth.se/4ecm/prizes.ecm.html. 
  11. ↑ "Archived copy". Archived from the original on 2011-07-24. https://web.archive.org/web/20110724181506/http://matematikkforeningen.no/INFOMAT/08/0810.pdf. Retrieved 2008-12-12. 
  12. ↑ 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.
  13. ↑ Dudziak, James (2011) (in en). Vitushkin's Conjecture for Removable Sets. Springer Science & Business Media. p. 39. ISBN 9781441967091. https://books.google.com/books?id=89A3gdCfPFUC&pg=PA39. 
  14. ↑ Weisstein, Eric W. (2002) (in en). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 218. ISBN 9781420035223. https://books.google.com/books?id=D_XKBQAAQBAJ&pg=PA218. 
  15. ↑ Weisstein, Eric W. (2002) (in en). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 65. ISBN 9781420035223. https://books.google.com/books?id=D_XKBQAAQBAJ&pg=PA65. 
  16. ↑ 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
  17. ↑ "Decoupling and the Bourgain-Demeter-Guth proof of the Vinogradov main conjecture" (in en). 10 December 2015. https://terrytao.wordpress.com/2015/12/10/decoupling-and-the-bourgain-demeter-guth-proof-of-the-vinogradov-main-conjecture/. 
  18. ↑ Holden, Helge; Piene, Ragni (2018) (in en). The Abel Prize 2013-2017. Springer. p. 51. ISBN 9783319990286. https://books.google.com/books?id=1NKJDwAAQBAJ&pg=PA51. 
  19. ↑ Kalai, Gil (10 May 2019). "A sensation in the morning news – Yaroslav Shitov: Counterexamples to Hedetniemi's conjecture." (in en). https://gilkalai.wordpress.com/2019/05/10/sansation-in-the-morning-news-yaroslav-shitov-counterexamples-to-hedetniemis-conjecture/. 
  20. ↑ Hazewinkel, Michiel, ed. (2001), "Schoenflies conjecture", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Schoenflies_conjecture 
  21. ↑ Farlow, Stanley J. (2014). Paradoxes in Mathematics. Courier Corporation. p. 57. ISBN 978-0-486-49716-7. https://books.google.com/books?id=d4YUAwAAQBAJ&pg=PA57. 
  22. ↑ 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. https://books.google.com/books?id=6JCBqZ0CMqgC&pg=PA1. 
  23. ↑ McQuarrie, Donald Allan (2003). Mathematical Methods for Scientists and Engineers. University Science Books. p. 711. https://books.google.com/books?id=FmAAwE8MSwoC&pg=PA711. 
  24. ↑ Lehman, R. S. (1960). "On Liouville's function". Mathematics of Computation 14 (72): 311–320. doi:10.1090/S0025-5718-1960-0120198-5. 
  25. ↑ 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. 
  26. ↑ Why mathematics is beautiful in New Scientist, 21 July 2007, p. 48
  27. ↑ 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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