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 .

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			10
Andrews–Curtis conjecture	combinatorial group theory			358
Andrica's conjecture	number theory		Dorin Andrica	45
Artin conjecture (L-functions)	number theory			650
Artin's conjecture on primitive roots	number theory	⇐generalized Riemann hypothesis^[2] ⇐Selberg conjecture B^[3]		325
Bateman–Horn conjecture	number theory			245
Baum–Connes conjecture	operator K-theory	⇒Gromov-Lawson-Rosenberg conjecture^[4] ⇒Kaplansky-Kadison conjecture ⇒Novikov conjecture		2670
Beal's conjecture	number theory			142
Beilinson conjecture	number theory			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			2830
Birch–Tate conjecture	number theory			149
Birkhoff conjecture	integrable systems			345
Bloch–Beilinson conjectures	number theory			152
Bloch–Kato conjecture	algebraic K-theory			1620
Bochner–Riesz conjecture	harmonic analysis	⇒restriction conjecture⇒Kakeya maximal function conjecture⇒Kakeya dimension conjecture^[5]		236
Bombieri–Lang conjecture	diophantine geometry			181
Borel conjecture	geometric topology			981
Bost conjecture	geometric topology			65
Brennan conjecture	complex analysis		James E. Brennan	110
Brocard's conjecture	number theory			16
Brumer–Stark conjecture	number theory			208
Bunyakovsky conjecture	number theory			43
Carathéodory conjecture	differential geometry			173
Carmichael totient conjecture	number theory
Casas-Alvero conjecture	polynomials		Eduardo Casas-Alvero	56
Catalan–Dickson conjecture on aliquot sequences	number theory			46
Catalan's Mersenne conjecture	number theory		Eugène Charles Catalan
Cherlin–Zilber conjecture	group theory			86
Chowla conjecture	Möbius function	⇒Sarnak conjecture^[6] ^[7]
Collatz conjecture	number theory			1440
Cramér's conjecture	number theory			32
Conway's thrackle conjecture	graph theory			150
Deligne conjecture	monodromy			788
Dittert conjecture	combinatorics		Eric Dittert	11
Eilenberg−Ganea conjecture	algebraic topology			96
Elliott–Halberstam conjecture	number theory			300
Erdős–Faber–Lovász conjecture	graph theory			172
Erdős–Gyárfás conjecture	graph theory			37
Erdős–Straus conjecture	number theory			103
Farrell–Jones conjecture	geometric topology			545
Filling area conjecture	differential geometry		n/a	60
Firoozbakht's conjecture	number theory			33
Fortune's conjecture	number theory			16
Four exponentials conjecture	number theory		n/a	110
Frankl conjecture	combinatorics			83
Gauss circle problem	number theory			553
Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane	metric geometry
Gilbreath conjecture	number theory			34
Goldbach's conjecture	number theory	⇒The ternary Goldbach conjecture, which was the original formulation.^[8]		5880
Gold partition conjecture^[9]	order theory		n/a	25
Goldberg–Seymour conjecture	graph theory			57
Goormaghtigh conjecture	number theory			14
Green's conjecture	algebraic curves			150
Grimm's conjecture	number theory		Carl Albert Grimm	46
Grothendieck–Katz p-curvature conjecture	differential equations			98
Hadamard conjecture	combinatorics			858
Herzog–Schönheim conjecture	group theory		Marcel Herzog and Jochanan Schönheim	44
Hilbert–Smith conjecture	geometric topology			219
Hodge conjecture	algebraic geometry			2490
Homological conjectures in commutative algebra	commutative algebra		n/a
Hopf conjectures	geometry			476
Ibragimov–Iosifescu conjecture for φ-mixing sequences	probability theory		Ildar Ibragimov,
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			127
Kaplansky conjectures	ring theory			466
Keating–Snaith conjecture	number theory			48
Köthe conjecture	ring theory			167
Kung–Traub conjecture	iterative methods			332
Legendre's conjecture	number theory			110
Lemoine's conjecture	number theory			13
Lenstra–Pomerance–Wagstaff conjecture	number theory			32
Leopoldt's conjecture	number theory			773
List coloring conjecture	graph theory		n/a	300
Littlewood conjecture	diophantine approximation	⇐Margulis conjecture^[10]		1230
Lovász conjecture	graph theory			560
MNOP conjecture	algebraic geometry		n/a	63
Manin conjecture	diophantine geometry			338
Marshall Hall's conjecture	number theory			44
Mazur's conjectures	diophantine geometry			97
Montgomery's pair correlation conjecture	number theory			77
n conjecture	number theory		n/a	126
New Mersenne conjecture	number theory			47
Novikov conjecture	algebraic topology			3090
Oppermann's conjecture	number theory			12
Petersen coloring conjecture	graph theory			52
Pierce–Birkhoff conjecture	real algebraic geometry			96
Pillai's conjecture	number theory			33
De Polignac's conjecture	number theory			46
Quantum PCP conjecture	quantum information theory
quantum unique ergodicity conjecture	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	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			187
Rudin's conjecture	additive combinatorics			16
Sarnak conjecture	topological entropy			295
Sato–Tate conjecture	number theory			1080
Schanuel's conjecture	number theory			329
Schinzel's hypothesis H	number theory			49
Scholz conjecture	addition chains			41
Second Hardy–Littlewood conjecture	number theory			30
Selfridge's conjecture	number theory			6
Sendov's conjecture	complex polynomials			77
Serre's multiplicity conjectures	commutative algebra			221
Singmaster's conjecture	binomial coefficients			8
Standard conjectures on algebraic cycles	algebraic geometry		n/a	234
Tate conjecture	algebraic geometry
Toeplitz' conjecture	Jordan curves
Tuza's conjecture	graph theory		Zsolt Tuza
Twin prime conjecture	number theory		n/a	1700
Ulam's packing conjecture	packing
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
Virasoro conjecture	algebraic geometry
Vizing's conjecture	graph theory
Vojta's conjecture	number theory
Waring's conjecture	number theory
Weight monodromy conjecture	algebraic geometry		n/a
Weinstein conjecture	periodic orbits
Whitehead conjecture	algebraic topology
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^[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 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^[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]	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#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

• Deligne's conjecture on 1-motives^[19]
• Goldbach's weak conjecture (proved in 2013)
• Sensitivity conjecture (proved in 2019)

Disproved (no longer conjectures)

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

• Atiyah conjecture (not a conjecture to start with)
• Borsuk's conjecture
• Chinese hypothesis (not a conjecture to start with)
• Doomsday conjecture
• Euler's sum of powers conjecture
• Ganea conjecture
• Generalized Smith conjecture
• Hauptvermutung
• Hedetniemi's conjecture, counterexample announced 2019^[20]
• Hirsch conjecture (disproved in 2010)
• Intersection graph conjecture
• Kelvin's conjecture
• Kouchnirenko's conjecture
• Mertens conjecture
• Pólya conjecture, 1919 (1958)
• Ragsdale conjecture
• Schoenflies conjecture (disproved 1910)
• Tait's conjecture
• Von Neumann conjecture
• Weyl–Berry conjecture
• Williamson conjecture

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

2 ^{2 ^{m}} +1 (known as Fermat numbers) were prime. However, this conjecture was disproved by Euler, who found that 2 ^{2 ^{5}} +1=4 , 294 , 967 , 297 = 641 x 6 , 700 , 417 . ^[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

\beth _{1} ) is greater than that of the set of algebraic numbers (\aleph _{0} ).^[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

\pi(x) > li (x) 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

• Erdős conjectures
• Fuglede's conjecture
• Millennium Prize Problems
• Painlevé conjecture
• Mathematical fallacy
• Superseded theories in science
• List of incomplete proofs
• List of unsolved problems in mathematics
• List of disproved mathematical ideas
• List of unsolved problems
• List of lemmas
• List of theorems
• List of statements undecidable in ZFC

Further reading

• Book: Nash, J.F. . Rassias . M.T. . Open Problems in Mathematics . Springer International Publishing . 2016 . 978-3-319-32162-2 . 2024-07-20.
• Book: Guy, R. . Unsolved Problems in Number Theory . Springer New York . Problem Books in Mathematics . 2013 . 978-0-387-26677-0 . 2024-07-20.

External links

• Open Problem Garden

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.