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Since the ], every century has seen the solution of more mathematical problems than the century before, and yet many mathematical problems, both major and minor, still remain unsolved.<ref>Eves, ''An Introduction to the History of Mathematics'' 6th Edition, Thomson, 1990, ISBN 978-0-03-029558-4.</ref> Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems receive considerable attention such as the list of ] of and problems of ], ]) {{as of|2015|08|lc=y}}, ], ] and ] ], ], ], ], ] and ], ]s, ], and ], ], ], ], ] and ] theories, as well as miscellaneous unsolved problems.


{{Dynamic list}}
==Lists of unsolved problems in mathematics==
Over the course of time, several lists of unsolved mathematical problems have appeared.


Many ] have been stated but not yet solved. These problems come from many ], such as ], ], ], ], ], ], ], ] and ], ], ], ], ], ], ], ]s, and ]s. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the ], receive considerable attention.
{| class="wikitable"

This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance.

== Lists of unsolved problems in mathematics ==
Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.

{| class="wikitable sortable"
|- |-
! List !! Number of problems !! Proposed by !! Proposed in ! List !! Number of<br />problems !! Number unsolved <br /> or incompletely solved !! Proposed by !! Proposed<br />in
|- |-
| ]<ref>{{citation|last=Thiele|first=Rüdiger|chapter=On Hilbert and his twenty-four problems|title=Mathematics and the historian's craft. The Kenneth O. May Lectures|pages=243–295|isbn=978-0-387-25284-1|editor-last=Van Brummelen|editor-first=Glen|year=2005|series=] Books in Mathematics/Ouvrages de Mathématiques de la SMC |volume=21|title-link=Kenneth May}}</ref> || 23 || 15 || ] || 1900
| ] || 23 || ] || 1900
|- |-
| ]<ref>{{citation|title=Unsolved Problems in Number Theory|first=Richard|last=Guy|author-link=Richard K. Guy|edition=2nd|publisher=Springer|year=1994|page=vii|url=https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PR7|isbn=978-1-4899-3585-4|access-date=2016-09-22|archive-url=https://web.archive.org/web/20190323220345/https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PR7|archive-date=2019-03-23|url-status=live}}.</ref> || 4 || 4 || ] || 1912
| ] || 4 || ] || 1912
|- |-
| ]<ref>{{cite journal | last = Shimura | first = G. | authorlink = Goro Shimura | title = Yutaka Taniyama and his time | journal = Bulletin of the London Mathematical Society | volume = 21 | issue = 2 | pages = 186–196 | year = 1989 | url = http://blms.oxfordjournals.org/content/21/2/186 | doi = 10.1112/blms/21.2.186}}</ref> || 36 || ] || 1955 | ]<ref>{{cite journal | last = Shimura | first = G. | author-link = Goro Shimura | title = Yutaka Taniyama and his time | journal = Bulletin of the London Mathematical Society | volume = 21 | issue = 2 | pages = 186–196 | year = 1989 | doi = 10.1112/blms/21.2.186 }}</ref> || 36 || - || ] || 1955
|- |-
| ]<ref>{{cite journal
| Thurston's 24 questions<ref>http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/dmv_091514.pdf</ref><ref></ref> || 24 || ] || 1982
| last = Friedl | first = Stefan
| doi = 10.1365/s13291-014-0102-x
| issue = 4
| journal = Jahresbericht der Deutschen Mathematiker-Vereinigung
| mr = 3280572
| pages = 223–241
| title = Thurston's vision and the virtual fibering theorem for 3-manifolds
| volume = 116
| year = 2014| s2cid = 56322745
}}</ref><ref>{{cite journal
| last = Thurston | first = William P.
| doi = 10.1090/S0273-0979-1982-15003-0
| issue = 3
| journal = Bulletin of the American Mathematical Society
| mr = 648524
| pages = 357–381
| series = New Series
| title = Three-dimensional manifolds, Kleinian groups and hyperbolic geometry
| volume = 6
| year = 1982}}</ref> || 24 || - || ] || 1982
|- |-
| ] || 18 || ] || 1998 | ] || 18 || 14 || ] || 1998
|- |-
| ] || 7 || 6<ref name="auto1">{{cite web |title=Millennium Problems |url=http://claymath.org/millennium-problems |archive-url=https://web.archive.org/web/20170606121331/http://claymath.org/millennium-problems |archive-date=2017-06-06 |access-date=2015-01-20 |website=claymath.org}}</ref>|| ] || 2000
| ] || 7 || ] || 2000
|- |-
| ] || 15 || <12<ref>{{cite web |url=http://www2.cnrs.fr/en/2435.htm?debut=8&theme1=12 |title=Fields Medal awarded to Artur Avila |website=] |date=2014-08-13 |access-date=2018-07-07 |archive-url=https://web.archive.org/web/20180710010437/http://www2.cnrs.fr/en/2435.htm?debut=8&theme1=12 |archive-date=2018-07-10 }}</ref><ref name="guardian">{{cite web |url=https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani |title=Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained |website=] |last=Bellos |first=Alex |date=2014-08-13 |access-date=2018-07-07 |archive-url=https://web.archive.org/web/20161021115900/https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/13/fields-medals-2014-maths-avila-bhargava-hairer-mirzakhani |archive-date=2016-10-21 |url-status=live }}</ref> || ] || 2000
| ]<ref>{{cite book | last1 = Abe | first1 = Jair Minoro | last2 = Tanaka | first2 = Shotaro | title = Unsolved Problems in Mathematics for the 21st Century | publisher = IOS Press | year = 2001 | url = http://books.google.de/books?id=yHzfbqtVGLIC&printsec=frontcover&dq=unsolved+problems+in+mathematics&hl=de&sa=X&ei=tmX0ULyDHMjJsga1n4D4CQ&ved=0CDIQ6AEwAA#v=onepage&q&f=false | isbn = 9051994907}}</ref> || 22 || Jair Minoro Abe, Shotaro Tanaka || 2001
|- |-
| ]<ref>{{cite book | last1 = Abe | first1 = Jair Minoro | last2 = Tanaka | first2 = Shotaro | title = Unsolved Problems on Mathematics for the 21st Century | publisher = IOS Press | year = 2001 | url = https://books.google.com/books?id=yHzfbqtVGLIC&q=unsolved+problems+in+mathematics | isbn = 978-90-5199-490-2}}</ref> || 22 || - || Jair Minoro Abe, Shotaro Tanaka || 2001
| ]<ref>{{cite web | title = DARPA invests in math | publisher = ] | date = 2008-10-14 | url = http://edition.cnn.com/2008/TECH/science/10/09/darpa.challenges/index.html | accessdate = 2013-01-14 | archiveurl = http://web.archive.org/web/20090304121240/http://edition.cnn.com/2008/TECH/science/10/09/darpa.challenges/index.html | archivedate = 2009-03-04}}</ref><ref>{{cite web | title = Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO) | publisher = DARPA | date = 2007-09-10 | url = http://www.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html | accessdate = 2013-06-25 | archiveurl = http://web.archive.org/web/20121001111057/http://www.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html
|-
| archivedate = 2012-10-01}}</ref> || 23 || ] || 2007
| ]'s math challenges<ref>{{cite web | title = DARPA invests in math | publisher = ] | date = 2008-10-14 | url = http://edition.cnn.com/2008/TECH/science/10/09/darpa.challenges/index.html | access-date = 2013-01-14 | archive-url = https://web.archive.org/web/20090304121240/http://edition.cnn.com/2008/TECH/science/10/09/darpa.challenges/index.html | archive-date = 2009-03-04}}</ref><ref>{{cite web | title = Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO) | publisher = DARPA | date = 2007-09-10 | url = http://www.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html | access-date = 2013-06-25 | archive-url = https://web.archive.org/web/20121001111057/http://www.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html
| archive-date = 2012-10-01}}</ref> || 23 || - || ] || 2007
|-
| ]<ref>{{cite web|url=https://www.erdosproblems.com/|title=Erdős Problems|first=Thomas|last=Bloom|author-link=Thomas Bloom|access-date=2024-08-25}}</ref> || >893 || 603 || ] || Over six decades of Erdős' career, from the 1930s to 1990s
|} |}
], subject of the ]<ref>{{Cite web |title=Math Problems Guide: From Simple to Hardest Math Problems Tips & Examples. |url=https://blendedlearningmath.com/math-word-problems-to-challenge-university-students/ |access-date=2024-11-28 |website=blendedlearningmath |language=en-US}}</ref>]]


===Millennium Prize Problems=== === Millennium Prize Problems ===
Of the original seven ] set by the ], six have yet to be solved, as of 2016:<ref>{{cite web|url=http://claymath.org/millennium-problems|title=Millennium Problems|publisher=}}</ref> Of the original seven ] listed by the ] in 2000, six remain unsolved to date:<ref name="auto1"/>


* ]
* ]
* ] * ]
* ]
* ]
* ] * ]
* ] * ]
* ]
* ]


The seventh problem, the ], has been solved.<ref>{{cite web |title=Poincaré Conjecture |url=http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archiveurl=http://wayback.archive.org/web/20131215120130/http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archivedate=2013-12-15 |website=Clay Mathematics Institute}}</ref> The ]—that is, whether a four-dimensional topological sphere can have two or more inequivalent ]s—is still unsolved.<ref>{{cite web|url=http://www.openproblemgarden.org/?q=op/smooth_4_dimensional_poincare_conjecture|title=Smooth 4-dimensional Poincare conjecture|publisher=}}</ref> The seventh problem, the ], was solved by ] in 2003.<ref>{{cite web |title=Poincaré Conjecture |url=http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archive-url=https://web.archive.org/web/20131215120130/http://www.claymath.org/millenium-problems/poincar%C3%A9-conjecture |archive-date=2013-12-15 |website=Clay Mathematics Institute}}</ref> However, a generalization called the ]—that is, whether a ''four''-dimensional ] can have two or more inequivalent ]s—is unsolved.<ref>{{cite web |last=rybu |date=November 7, 2009 |title=Smooth 4-dimensional Poincare conjecture |url=http://www.openproblemgarden.org/?q=op/smooth_4_dimensional_poincare_conjecture |url-status=live |archive-url=https://web.archive.org/web/20180125203721/http://www.openproblemgarden.org/?q=op%2Fsmooth_4_dimensional_poincare_conjecture |archive-date=2018-01-25 |access-date=2019-08-06 |website=Open Problem Garden}}</ref>


==Unsolved problems== === Notebooks ===
* The ] Notebook ({{Langx|ru|Коуровская тетрадь}}) is a collection of unsolved problems in ], first published in 1965 and updated many times since.<ref>{{citation |last1=Khukhro |first1=Evgeny I. |title=Unsolved Problems in Group Theory. The Kourovka Notebook |year=2019 |arxiv=1401.0300v16 |last2=Mazurov |first2=Victor D. |author-link2=Victor Mazurov}}</ref>
* The ] Notebook ({{Langx|ru|Свердловская тетрадь}}) is a collection of unsolved problems in ], first published in 1965 and updated every 2 to 4 years since.<ref>{{Cite book |last1=RSFSR |first1=MV i SSO |url=https://books.google.com/books?id=nKwgzgEACAAJ |title=Свердловская тетрадь: нерешенные задачи теории подгрупп |last2=Russie) |first2=Uralʹskij gosudarstvennyj universitet im A. M. Gorʹkogo (Ekaterinbourg |date=1969 |publisher=S. l. |language=ru}}</ref><ref>{{cite book| title = Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп |location= ] |date = 1979 |publisher= ] }}</ref><ref>{{cite book| title = Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп |location= ] |date = 1989 |publisher= ] }}</ref>
* The ] Notebook ({{Langx|ru|Днестровская тетрадь}}) lists several hundred unsolved problems in algebra, particularly ] and ].<ref>{{citation |title=ДНЕСТРОВСКАЯ ТЕТРАДЬ |url=http://math.nsc.ru/LBRT/a1/files/dnestr93.pdf |year=1993 |trans-title=DNIESTER NOTEBOOK |publisher=The Russian Academy of Sciences |language=ru}}</ref><ref>{{citation |title=DNIESTER NOTEBOOK: Unsolved Problems in the Theory of Rings and Modules |url=https://math.usask.ca/~bremner/research/publications/dniester.pdf |website=] |access-date=2019-08-15}}</ref>
* The ] Notebook ({{Langx|ru|Эрлагольская тетрадь}}) lists unsolved problems in algebra and ].<ref>{{citation |title=Эрлагольская тетрадь |url=http://uamt.conf.nstu.ru/erl_note.pdf |year=2018 |trans-title=Erlagol notebook |publisher=The Novosibirsk State University |language=ru}}</ref>


== Unsolved problems ==
===]===
{{seealso|Problems involving arithmetic progressions}}
* ]
* ]
* ]
* The values of ''g''(''k'') and ''G''(''k'') in ]
* ] (3''n''&nbsp;+&nbsp;1 conjecture)
* ]
* ]s
* ]
* ]
* ]
* ]
* ]
* Determine growth rate of ''r''<sub>''k''</sub>(''N'') (see ])
* ]


===]=== === Algebra ===
{{Main|Algebra}}
] representation of a ], a ] forms a ]. Zauner conjectured that analogous structures exist in complex ]s of all finite dimensions.]]

* ] on the relation between the order of the ] of the ] of the ] of a ] to the field's ].
* ]s on densities of rational points of ]s and ] defined on ] and their ]s.
* ] in ] theory
* ]: the ] of a complex function <math>f</math> applied to a complex matrix <math>A</math> is at most twice the ] of <math>|f(z)|</math> over the ] of <math>A</math>.
* ] on the ] of the sum of two ].
* ]: a group with ] 2 also has a 2-dimensional ] <math>K(G, 1)</math>.
* ] on whether certain ]s are ].
** ]: a specific case of the Farrell–Jones conjecture
* ]: is every finite ] isomorphic to the ] of some finite ]?<ref>{{cite journal |last1=Dowling |first1=T. A. |title=A class of geometric lattices based on finite groups|journal=] |series=Series B |date=February 1973 |volume=14 |issue=1 |pages=61–86 |doi=10.1016/S0095-8956(73)80007-3 | doi-access=free }}</ref>
* ] on the ] of certain ].
* ]: the ] of a non-] is determined by the extent to which it, as a ], has ].
* ]: a conjectured ] for ].
* ]: for every positive integer <math>k</math>, a ] of order <math>4k</math> exists.
** ]: the problem of finding Williamson matrices, which can be used to construct Hadamard matrices.
* ]: what is the largest ] of a matrix with entries all equal to 1 or –1?
* ]: put ] on a rigorous foundation.
* ]: what are the possible configurations of the ] of ]?
* ] * ]
* ]: the intersection of all powers of the ] of a left-and-right ] is precisely 0.
* ]
* ] * ]
* ]: if a ring has no ] other than <math>\{0\}</math>, then it has no nil ] other than <math>\{0\}</math>.
* ]
* ] * ] on ] ]s
* Existence of ] and associated ] * Existence of ]s and associated ]
* ]: every piecewise-polynomial <math>f:\mathbb{R}^{n}\rightarrow\mathbb{R}</math> is the maximum of a finite set of minimums of finite collections of polynomials.
* ]: existence of ]s in all dimensions
* ]: for matroids of rank <math>n</math> with <math>n</math> disjoint bases <math>B_{i}</math>, it is possible to create an <math>n \times n</math> matrix whose rows are <math>B_{i}</math> and whose columns are also bases.
* ]: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems
* ]: if <math>G</math> is a ] ] over a perfect ] of ] at most <math>2</math>, then the ] set <math>H^{1}(F, G)</math> is zero.
* ]
* ] that if <math>R</math> is a commutative ], and <math>P, Q</math> are ]s of <math>R</math>, then <math>\dim (R/P) + \dim (R/Q) = \dim (R)</math> implies <math>\chi(R/P, R/Q) > 0</math>.
* ]
* ]: do ]s of ] <math>g \geq 2</math> over ] <math>K</math> have at most some bounded number <math>N(K, g)</math> of <math>K</math>-]s?
* ]
* ]s: problems involving classification of pairs of <math>n\times n</math> matrices under simultaneous conjugation.
* ]
* ]: for a ] <math>V</math> with ] <math>R</math>, if the ] of <math>R</math> are a ] over <math>R</math>, then <math>V</math> is ].
* ]
* Zauner's conjecture: do ]s exist in all dimensions?
* ]
*] that if <math>X</math> is a mixed ] or ] defined over <math>\mathbb{C}</math>, and <math>V \subseteq X</math> is a subvariety, then <math>V</math> contains only finitely many atypical subvarieties.
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]


===]=== ==== Group theory ====
{{Main|Group theory }}
* ]
] <math>B(2,3)</math> is finite; in its ], shown here, each of its 27 elements is represented by a vertex. The question of which other groups <math>B(m,n)</math> are finite remains open.]]
* ]
* ]: every balanced ] of the ] can be transformed into a trivial presentation by a sequence of ]s on ] and conjugations of relators
* ]
* ]: for which positive integers ''m'', ''n'' is the free Burnside group {{nowrap|B(''m'',''n'')}} finite? In particular, is {{nowrap|B(2, 5)}} finite?
* ]
* Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems<ref>{{citation |last=Aschbacher |first=Michael |author-link=Michael Aschbacher |title=On Conjectures of Guralnick and Thompson |journal=] |volume=135 |issue=2 |pages=277–343 |year=1990 |doi=10.1016/0021-8693(90)90292-V}}</ref>
* ]
* ]: if a finite system of left ]s of subgroups of a group <math>G</math> form a partition of <math>G</math>, then the finite indices of said subgroups cannot be distinct.
* ]
* The ]: is every finite group the Galois group of a Galois extension of the rationals?
* ]
* Are there an infinite number of ]s?
* ]
* Does ] exist?
* ]
* Is every ] ] finite?
* ]
* Is every group ]?
* ]
* Is every discrete, countable group ]?
* ]
* ] consider generalizations of groups
* ]
* ]
* ]


===]=== ==== Representation theory ====
* ]
* Are there infinitely many ] with ] (])?
* ] relating the numbers of ] of ] of a finite group to the numbers of characters of blocks of local ]s.
* Characterize all algebraic number fields that have some ].
* ] on ] of ]s over the integers.
* ] (including ])
* ] relating the values of the ]s at 1 with ] of complex ] and ].
* ]: in a group <math>G</math>, the number of ] of degree not divisible by a ] <math>p</math> is equal to the number of irreducible complex characters of the ] of any ] within <math>G</math>.


===]=== === Analysis ===
{{Main|Mathematical analysis}}
* The ]
* The ]: estimating the integral of powers of the moduli of the derivative of ]s into the open unit disk, on certain subsets of <math>\mathbb{C}</math>
* ] and ]
* ] on whether nonconvex sets in <math>\mathbb{R}</math> and <math>\mathbb{R}^{2}</math> are spectral if and only if they tile by ].
* ]
* ] on the coefficients of ]s
* ]
* ] – does every ] on a complex ] send some non-trivial ] subspace to itself?
* Are <math>\gamma</math> (the ]), ]&nbsp;+&nbsp;'']'', {{pi}}&nbsp;&minus;&nbsp;''e'', {{pi}}''e'', {{pi}}/''e'', {{pi}}<sup>''e''</sup>, {{pi}}<sup>]</sup>, {{pi}}<sup>{{pi}}</sup>, e<sup>{{pi}}<sup>2</sup></sup>, ]&nbsp;{{pi}}, 2<sup>''e''</sup>, ''e''<sup>''e''</sup>, ] or ] rational, ] irrational, or ]? What is the ] of each of these numbers?<ref>For background on the numbers that are the focus of this problem, see articles by Eric W. Weisstein, on pi (), e (), Khinchin's Constant (), irrational numbers (), transcendental numbers (), and irrationality measures () at Wolfram ''MathWorld'', all articles accessed 15 December 2014.</ref><ref>Michel Waldschmidt, 2008, "An introduction to irrationality and transcendence methods," at The University of Arizona The Southwest Center for Arithmetic Geometry 2008 Arizona Winter School, March 15–19, 2008 (Special Functions and Transcendence), see , accessed 15 December 2014.</ref><ref>John Albert, posting date unknown, "Some unsolved problems in number theory" , in University of Oklahoma Math 4513 course materials, see , accessed 15 December 2014.</ref>
* Kung–Traub conjecture on the optimal order of a multipoint iteration without memory<ref>{{citation |last1=Kung |first1=H. T. |last2=Traub |first2=Joseph Frederick |author-link1=H. T. Kung |author-link2=Joseph F. Traub |title=Optimal order of one-point and multipoint iteration |journal=] |year=1974 |volume=21 |number=4 |pages=643–651|doi=10.1145/321850.321860 |s2cid=74921 }}</ref>
* ]
* ] on the Mahler measure of non-cyclotomic polynomials<ref>{{citation | first=Chris | last=Smyth | chapter=The Mahler measure of algebraic numbers: a survey | pages=322–349 | editor1-first=James | editor1-last=McKee | editor2-last=Smyth | editor2-first=Chris | title=Number Theory and Polynomials | series=London Mathematical Society Lecture Note Series | volume=352 | publisher=] | year=2008 | isbn=978-0-521-71467-9 }}</ref>
* ]
* The ]: given a ] ] <math>f</math> of ] <math>d \ge 2</math> and a complex number <math>z</math>, is there a ] <math>c</math> of <math>f</math> such that <math>|f(z)-f(c)| \le |f'(z)||z-c|</math>?
* ]
* The ] on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy<ref>{{SpringerEOM|title=Pompeiu problem|id=Pompeiu_problem&oldid=14506|author-last1=Berenstein|author-first1=Carlos A.}}</ref>
* ]: if a complex polynomial with degree at least <math>2</math> has all roots in the closed ], then each root is within distance <math>1</math> from some ].
* ] on compact subsets of <math>\mathbb{C}</math> with ] <math>0</math>
* What is the exact value of ], including ]?


* Regularity of solutions of ]
===]===
* Convergence of ]
* Number of ]s {{OEIS|id=A006052}}
* Regularity of solutions of ]
* Number of magic tori {{OEIS|id=A270876}}
* Finding a formula for the probability that two elements chosen at random generate the ] <math>S_n</math>
* Frankl's ]: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
* The ]: if <math>k+1</math> runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance <math>1/(k+1)</math> from each other runner) at some time?
* ]: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
* The ]: does every finite ] that is not ] contain two elements ''x'' and ''y'' such that the probability that ''x'' appears before ''y'' in a random ] is between 1/3 and 2/3?
* ]


===]=== === Combinatorics ===
{{Main|Combinatorics}}
* Solving the ] for arbitrary <math>n</math>
* The ] – does every finite ] that is not ] contain two elements ''x'' and ''y'' such that the probability that ''x'' appears before ''y'' in a random ] is between 1/3 and 2/3?<ref>{{citation
* Finding matching upper and lower bounds for ] and halving lines
| last1 = Brightwell | first1 = Graham R.
* The ] on covering ''n''-dimensional convex bodies with at most 2<sup>''n''</sup> smaller copies
| last2 = Felsner | first2 = Stefan
* The ] on triangles in line arrangements
| last3 = Trotter | first3 = William T.
* The ] on projectively transforming sets of points into ]
| doi = 10.1007/BF01110378
* ] about the identity of the worst-packing convex solid
| mr = 1368815
* ]
| issue = 4
* ]
| journal = ]
* ]
| pages = 327–349
* ] for dimensions other than 1, 2, 3, 4, 8 and 24
| title = Balancing pairs and the cross product conjecture
| volume = 12
| year = 1995| citeseerx = 10.1.1.38.7841
| s2cid = 14793475
}}.</ref>
* The ] concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
* ] – open questions concerning ]
* The ] – if <math>k</math> runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance <math>1/k</math> from each other runner) at some time?<ref>{{cite journal
| last=Tao | first=Terence | author-link=Terence Tao
| title=Some remarks on the lonely runner conjecture
| journal=Contributions to Discrete Mathematics
| volume=13
| issue=2
| pages=1–31
| date=2018
| arxiv=1701.02048
| doi=10.11575/cdm.v13i2.62728 | doi-access=free}}</ref>
* ] – various problems in map folding and stamp folding.
* ] – how many points can be placed in the <math>n \times n</math> grid so that no three of them lie on a line?
* ] on the number of squares in finite ]s<ref>{{cite journal|journal=LMS Journal of Computation and Mathematics|volume=17|issue=1|year=2014|pages=58–76|title=On a conjecture of Rudin on squares in arithmetic progressions|author=González-Jiménez, Enrique|author2=Xarles, Xavier|doi=10.1112/S1461157013000259|arxiv=1301.5122|s2cid=11615385 }}</ref>
* The ] – can the number of <math>k</math> size sets required for the existence of a sunflower of <math>r</math> sets be bounded by an exponential function in <math>k</math> for every fixed <math>r>2</math>?
* Frankl's ] – for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets<ref>{{citation
| last1 = Bruhn
| first1 = Henning
| last2 = Schaudt
| first2 = Oliver
| doi = 10.1007/s00373-014-1515-0
| issue = 6
| journal = Graphs and Combinatorics
| mr = 3417215
| pages = 2043–2074
| title = The journey of the union-closed sets conjecture
| url = http://www.zaik.uni-koeln.de/~schaudt/UCSurvey.pdf
| volume = 31
| year = 2015
| arxiv = 1309.3297
| s2cid = 17531822
| access-date = 2017-07-18
| archive-url = https://web.archive.org/web/20170808104232/http://www.zaik.uni-koeln.de/~schaudt/UCSurvey.pdf
| archive-date = 2017-08-08
| url-status = live
}}</ref>


* Give a combinatorial interpretation of the ]s<ref>{{citation
===]===
| last = Murnaghan | first = F. D.
* The ] – does there exist a two-dimensional shape that forms the ] for an ], but not for any periodic tiling?<ref>{{citation
| doi = 10.2307/2371542
| last1 = Socolar | first1 = Joshua E. S.
| last2 = Taylor | first2 = Joan M.
| arxiv = 1009.1419
| doi = 10.1007/s00283-011-9255-y
| issue = 1 | issue = 1
| journal = The Mathematical Intelligencer | journal = ]
| mr = 2902144 | mr = 1507301
| pages = 18–28 | pages = 44–65
| title = The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups
| title = Forcing nonperiodicity with a single tile
| volume = 34 | volume = 60
| year = 2012}}</ref> | year = 1938| pmc = 1076971
| pmid=16577800
* ] – does every Jordan curve have an inscribed square?<ref name="matschke">{{citation |last=Matschke |first=Benjamin |date=2014 |title=A survey on the square peg problem |journal=] |volume=61 |issue=4 |pages=346–253 |doi=10.1090/noti1100}}</ref>
| jstor = 2371542
* ] – what is the smallest area of a shape that can cover every unit-length curve in the plane?<ref>{{citation
}}</ref>
| last1 = Norwood | first1 = Rick | author1-link = Rick Norwood
* The values of the ]s <math>M(n)</math> for <math>n \ge 10</math><ref>{{Cite web |url=http://www.sfu.ca/~tyusun/ThesisDedekind.pdf |title=Dedekind Numbers and Related Sequences |access-date=2020-04-30 |archive-date=2015-03-15 |archive-url=https://web.archive.org/web/20150315021125/http://www.sfu.ca/~tyusun/ThesisDedekind.pdf }}</ref>
| last2 = Poole | first2 = George
* The values of the ], particularly <math>R(5, 5)</math>
| last3 = Laidacker | first3 = Michael
* The values of the ]s
| doi = 10.1007/BF02187832
* Finding a function to model n-step ]s<ref>{{Cite journal|last1=Liśkiewicz|first1=Maciej|last2=Ogihara|first2=Mitsunori|last3=Toda|first3=Seinosuke|date=2003-07-28|title=The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes|journal=Theoretical Computer Science|volume=304|issue=1|pages=129–156|doi=10.1016/S0304-3975(03)00080-X|s2cid=33806100 }}</ref>

=== Dynamical systems ===
{{Main|Dynamical system}}
]. It is not known whether the Mandelbrot set is ] or not.]]
* ] and ] – relating symplectic geometry to Morse theory.
* ] in ]
* ] problem – is there an ] with simple Lebesgue spectrum?<ref>S. M. Ulam, Problems in Modern Mathematics. Science Editions John Wiley & Sons, Inc., New York, 1964, page 76.</ref>
* ] conjecture – if a ] is strictly convex and integrable, is its boundary necessarily an ellipse?<ref>{{cite journal |last1=Kaloshin |first1=Vadim |author-link1=Vadim Kaloshin |last2=Sorrentino |first2=Alfonso |title=On the local Birkhoff conjecture for convex billiards |doi=10.4007/annals.2018.188.1.6 |volume=188 |number=1 |year=2018 |pages=315–380 |journal=]|arxiv=1612.09194 |s2cid=119171182 }}</ref>
* ] (also known as the <math>3n + 1</math> conjecture)
* ] that the ] of the local ]s on the global ] is achieved on a stationary point or an unstable periodic orbit embedded into the attractor.
* ] conjecture: every component of the ] of an ] ] function is unbounded.
* ] that a quadratic family of maps from the ] to itself is hyperbolic for an open dense set of parameters.
* ] conjecture – is every invariant and ] measure for the <math>\times 2,\times 3</math> action on the circle either Lebesgue or atomic?
* ] on the dimension of an ] in terms of its ]s
* ] conjecture – measure classification for diagonalizable actions in higher-rank groups.
* ] – is the Mandelbrot set locally connected?
* Many problems concerning an ], for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.
* Quantum unique ergodicity conjecture on the distribution of large-frequency ]s of the ] on a ] ]<ref>{{citation |last=Sarnak |first=Peter |author-link=Peter Sarnak |title=Recent progress on the quantum unique ergodicity conjecture |journal=] |volume=48 |issue=2 |year=2011 |pages=211–228 |doi=10.1090/S0273-0979-2011-01323-4 |mr=2774090|doi-access=free }}</ref>
* ] multiple mixing problem – are all ] systems also strongly 3-mixing?<ref>Paul Halmos, Ergodic theory. Chelsea, New York, 1956.</ref>
* ] – does a regular compact ] ] of a ] on a ] carry at least one periodic orbit of the Hamiltonian flow?

* Does every positive integer generate a ] terminating at 1?
* ] – For what classes of ], describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion?
* Is every ] in three or more dimensions locally reversible?<ref>{{cite conference |last=Kari |first=Jarkko |author-link=Jarkko Kari |year=2009 |title=Structure of Reversible Cellular Automata |conference=International Conference on Unconventional Computation |series=] |publisher=Springer |volume=5715 |page=6 |bibcode=2009LNCS.5715....6K |doi=10.1007/978-3-642-03745-0_5 |isbn=978-3-642-03744-3 |doi-access=free |contribution=Structure of reversible cellular automata}}</ref>

=== Games and puzzles ===
{{Main|Game theory}}

====Combinatorial games====
{{Main|Combinatorial game theory}}
* ]:
** How many puzzles have exactly one solution?<ref name="openq"/>
** How many puzzles with exactly one solution are ]?<ref name="openq"/>
** What is the ] for a ] puzzle?<ref name="openq">{{Cite web |title=Open Q - Solving and rating of hard Sudoku |url=http://english.log-it-ex.com/2.html |archive-url=https://web.archive.org/web/20171110030932/http://english.log-it-ex.com/2.html |archive-date=10 November 2017 |website=english.log-it-ex.com}}</ref>
* ]:
** Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also ] and ])<ref>{{cite web |url=https://www.youtube.com/watch?v=FwJZa-helig |title=Higher-Dimensional Tic-Tac-Toe |website=] |publisher=] |date=2017-09-21 |access-date=2018-07-29 |archive-url=https://web.archive.org/web/20171011000653/https://www.youtube.com/watch?v=FwJZa-helig |archive-date=2017-10-11 |url-status=live }}</ref>
* ]:
** What is the outcome of a perfectly played game of chess? (See also ])
* ]:
** What is the perfect value of ]?
* Are the nim-sequences of all finite ]s eventually periodic?
* Is the nim-sequence of ] eventually periodic?

====Games with imperfect information====
* ]

=== Geometry ===
{{Main|Geometry}}

==== Algebraic geometry ====
{{Main|Algebraic geometry}}
* ]: if the ] of a ] with ] is ], then it is semiample.
* ] on the ] of certain ].
* ] relating ]s over a ] ] and over the ] <math>A</math>.
* ]: any one of numerous named for ].
** ] about the ]ic structure on ].
* ]: any ] of a ] is an ].
* ] on the ] of a set of forms.
* ] regarding the line bundle <math>K_{M} \otimes L^{\otimes m}</math> constructed from a ] ] <math>L</math> on a ] ] <math>M</math> and the ] <math>K_{M}</math> of <math>M</math>
* ]: do ]s have at most ]?
* Hartshorne's conjectures<ref>{{cite journal|title=On two conjectures of Hartshorne's |last1=Barlet |first1=Daniel |last2=Peternell |first2=Thomas |last3=Schneider |first3=Michael |doi=10.1007/BF01453563 |journal=] |year=1990 |volume=286 |issue=1–3 |pages=13–25|s2cid=122151259 }}</ref>
* ]: if a ] over a ]-0 field has a constant nonzero ], then it has a ] (i.e. with polynomial components) inverse function.
* ] on the distribution of ]s of bounded ] in certain subsets of ]
* ] on an equivalence between ] and ]<ref>{{citation
|last1=Maulik |first1=Davesh
|last2=Nekrasov |first2=Nikita |author-link2=Nikita Nekrasov
|last3=Okounov |first3=Andrei |author-link3=Andrei Okounov
|last4=Pandharipande |first4=Rahul |author-link4=Rahul Pandharipande
|title=Gromov–Witten theory and Donaldson–Thomas theory, I
|arxiv=math/0312059
|date=2004-06-05|bibcode=2003math.....12059M
}}</ref>
* ], specifically the minimal degree required for a ] to pass through a collection of very general points with prescribed ].
* ] that if <math>X</math> is a smooth ] and <math>L</math> is an ] on <math>X</math> of degree <math>d</math>, then for sufficiently large <math>r</math>, the ] satisfies <math>\varepsilon(p_1,\ldots,p_r;X,L) = d/\sqrt{r}</math>.
* ]: if a ] has a ring of ]s generated by its contained ], then it must be ].
* ]: the higher ] of any ] ] defined over a ] must vanish up to torsion.
* ] on splittings of ]s from ]s of complete ] over finitely-generated ] <math>k</math> to the ] of <math>k</math>.
* ] on algebraic cycles
* ] on the connection between ]s on ] and ] on ].
* ]: a certain ] encoding the ]s of a ] ] is fixed by an action of half of the ].
* Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of ] at ]<ref>{{cite journal|last=Zariski |first=Oscar |author-link=Oscar Zariski |title=Some open questions in the theory of singularities |journal=] |volume=77 |issue=4 |year=1971 |pages=481–491 |doi=10.1090/S0002-9904-1971-12729-5 |mr=0277533|doi-access=free }}</ref>

* Are infinite sequences of ] possible in dimensions greater than 3?
* ] in characteristic <math>p</math>

====Covering and packing====
* ] on upper and lower bounds for the number of smaller-diameter subsets needed to cover a ] ''n''-dimensional set.
* The ]: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?<ref>{{citation|last1=Bereg|first1=Sergey|last2=Dumitrescu|first2=Adrian|last3=Jiang|first3=Minghui|doi=10.1007/s00453-009-9298-z|issue=3|journal=Algorithmica|mr=2609053|pages=538–561|title=On covering problems of Rado|volume=57|year=2010|s2cid=6511998}}</ref>
* The ]: when <math>n</math> is a ], packing <math>n-1</math> circles in an equilateral triangle requires a triangle of the same size as packing <math>n</math> circles<ref>{{citation|last=Melissen|first=Hans|doi=10.2307/2324212|issue=10|journal=American Mathematical Monthly|mr=1252928|pages=916–925|title=Densest packings of congruent circles in an equilateral triangle|volume=100|year=1993|jstor=2324212}}</ref>
* The ] for dimensions other than 1, 2, 3, 4, 8 and 24<ref>{{citation |first=John H. |last=Conway |author-link=John Horton Conway |author2=Neil J.A. Sloane |author-link2=Neil Sloane |year=1999 |title=Sphere Packings, Lattices and Groups |edition=3rd |publisher=Springer-Verlag |location=New York |isbn=978-0-387-98585-5|pages=}}</ref>
* ]: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets<ref>{{citation
| last = Hales | first = Thomas | author-link = Thomas Callister Hales
| arxiv = 1703.01352
| title = The Reinhardt conjecture as an optimal control problem
| year = 2017}}</ref>
* ] problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
* ]: what is the asymptotic growth rate of wasted space?<ref>{{citation|last1=Brass|first1=Peter|last2=Moser|first2=William|last3=Pach|first3=János|author3-link=János Pach|isbn=978-0387-23815-9|mr=2163782|page=45|publisher=Springer|location=New York|title=Research Problems in Discrete Geometry|url=https://books.google.com/books?id=WehCspo0Qa0C&pg=PA45|year=2005}}</ref>
* ] about the identity of the worst-packing convex solid<ref>{{citation |last=Gardner |first=Martin |date=1995 |title=New Mathematical Diversions (Revised Edition) |location=Washington |publisher=Mathematical Association of America |page=251 }}</ref>
* The ] for numbers of nodes greater than 14 (except 24).<ref>{{cite journal |last1=Musin |first1=Oleg R. |last2=Tarasov |first2=Alexey S. |title=The Tammes Problem for N = 14 |journal=Experimental Mathematics |date=2015 |volume=24 |issue=4 |pages=460–468 |doi=10.1080/10586458.2015.1022842|s2cid=39429109 }}</ref>

==== Differential geometry ====
{{Main|Differential geometry}}
* The ], a generalization of ]
* ]: any convex, closed, and twice-differentiable surface in three-dimensional ] admits at least two ]s.
* ]: can the classical ] for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as ]?
* ] that the ] of a ] ] vanishes.
* ], a number of closely related conjectures.
* Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.<ref>{{citation
| last = Barros | first = Manuel
| jstor = 2162098
| journal = ]
| pages = 1503–1509
| title = General Helices and a Theorem of Lancret
| volume = 125
| issue = 5
| year = 1997| doi = 10.1090/S0002-9939-97-03692-7
| doi-access = free
}}</ref>
* The ], that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length<ref>{{citation
| last = Katz | first = Mikhail G.
| doi = 10.1090/surv/137
| isbn = 978-0-8218-4177-8
| mr = 2292367
| page = 57
| publisher = American Mathematical Society, Providence, RI
| series = Mathematical Surveys and Monographs
| title = Systolic geometry and topology
| url = https://books.google.com/books?id=R5_zBwAAQBAJ&pg=PA57
| volume = 137
| year = 2007}}</ref>
* The ]s relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds<ref>{{citation
| last = Rosenberg | first = Steven
| doi = 10.1017/CBO9780511623783
| isbn = 978-0-521-46300-3
| location = Cambridge
| mr = 1462892
| pages = 62–63
| publisher = Cambridge University Press
| series = London Mathematical Society Student Texts
| title = The Laplacian on a Riemannian Manifold: An introduction to analysis on manifolds
| url = https://books.google.com/books?id=gzJ6Vn0y7XQC&pg=PA62
| volume = 31
| year = 1997}}</ref>
* ] that the first ] for the ] on an embedded ] of <math>S^{n+1}</math> is <math>n</math>.

==== Discrete geometry ====
{{Main|Discrete geometry }}
] is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a ].) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.]]
* The ] on the existence of either many collinear points or many mutually visible points in large planar point sets<ref>{{citation
| last1 = Ghosh | first1 = Subir Kumar
| last2 = Goswami | first2 = Partha P.
| arxiv = 1012.5187
| doi = 10.1145/2543581.2543589
| issue = 2 | issue = 2
| journal = ] | journal = ACM Computing Surveys
| mr = 1139077 | pages = 22:1–22:29
| title = Unsolved problems in visibility graphs of points, segments, and polygons
| pages = 153–162
| volume = 46
| title = The worm problem of Leo Moser
| year = 2013| s2cid = 8747335
}}</ref>
* The ] on covering ''n''-dimensional convex bodies with at most 2<sup>''n''</sup> smaller copies<ref>{{citation|title=Results and Problems in Combinatorial Geometry|first1=V.|last1=Boltjansky|first2=I.|last2=Gohberg|publisher=Cambridge University Press|year=1985|contribution=11. Hadwiger's Conjecture|pages=44–46}}.</ref>
* Solving the ] for arbitrary <math>n</math><ref>{{citation
| last1 = Morris | first1 = Walter D.
| last2 = Soltan | first2 = Valeriu
| doi = 10.1090/S0273-0979-00-00877-6
| issue = 4
| journal = Bull. Amer. Math. Soc.
| mr = 1779413
| pages = 437–458
| title = The Erdős-Szekeres problem on points in convex position—a survey
| volume = 37
| year = 2000| doi-access = free
}}; {{citation
| last = Suk | first = Andrew
| arxiv = 1604.08657
| doi = 10.1090/jams/869
| journal = J. Amer. Math. Soc.
| title = On the Erdős–Szekeres convex polygon problem
| year = 2016
| volume=30
| issue = 4
| pages=1047–1053| s2cid = 15732134
}}</ref>
*Improving lower and upper bounds for the ].
* ] on the least possible number of faces of ] ].<ref name="kalai">{{citation
| last = Kalai | first = Gil | author-link = Gil Kalai
| doi = 10.1007/BF01788696
| issue = 1
| journal = ]
| mr = 1554357
| pages = 389–391
| title = The number of faces of centrally-symmetric polytopes
| volume = 5
| year = 1989| s2cid = 8917264 }}.</ref>
* The ] on triangles in line arrangements<ref>{{cite journal
| last1 = Moreno | first1 = José Pedro
| last2 = Prieto-Martínez | first2 = Luis Felipe
| hdl = 10486/705416
| issue = 1
| journal = La Gaceta de la Real Sociedad Matemática Española
| language = es
| mr = 4225268
| pages = 111–130
| title = El problema de los triángulos de Kobon
| trans-title = The Kobon triangles problem
| volume = 24
| year = 2021}}</ref>
* The ]: at most <math>2d</math> points can be equidistant in <math>L^1</math> spaces<ref>{{citation
| last = Guy | first = Richard K. | author-link = Richard K. Guy
| issue = 3
| journal = ]
| mr = 1540158
| pages = 196–200
| title = An olla-podrida of open problems, often oddly posed
| jstor = 2975549
| volume = 90
| year = 1983
| doi = 10.2307/2975549 }}</ref>
* The ] on projectively transforming sets of points into ]<ref>{{citation
| last = Matoušek | first = Jiří | author-link = Jiří Matoušek (mathematician)
| doi = 10.1007/978-1-4613-0039-7
| isbn = 978-0-387-95373-1
| mr = 1899299
| page = 206
| publisher = Springer-Verlag, New York
| series = Graduate Texts in Mathematics
| title = Lectures on discrete geometry
| volume = 212
| year = 2002}}</ref>
*] on finding ]s for various planar shapes
* ] can be determined by a set of {{mvar|n}} points in the Euclidean plane?<ref>{{citation
| last1 = Brass | first1 = Peter
| last2 = Moser | first2 = William
| last3 = Pach | first3 = János
| contribution = 5.1 The Maximum Number of Unit Distances in the Plane
| isbn = 978-0-387-23815-9
| mr = 2163782
| pages = 183–190
| publisher = Springer, New York
| title = Research problems in discrete geometry
| year = 2005}}</ref>
* Finding matching upper and lower bounds for ] and halving lines<ref>{{citation
| last = Dey | first = Tamal K. | author-link = Tamal Dey
| doi = 10.1007/PL00009354
| journal = ]
| mr = 1608878
| pages = 373–382
| title = Improved bounds for planar ''k''-sets and related problems
| volume = 19
| issue = 3
| year = 1998| doi-access = free
}}; {{citation
| last = Tóth | first = Gábor
| doi = 10.1007/s004540010022
| issue = 2
| journal = ]
| mr = 1843435
| pages = 187–194
| title = Point sets with many ''k''-sets
| volume = 26
| year = 2001| doi-access = free
}}.</ref>
* ]:<ref>{{citation|last1=Aronov|first1=Boris|author1-link=Boris Aronov|last2=Dujmović|first2=Vida|author2-link=Vida Dujmović|last3=Morin|first3=Pat|author3-link= Pat Morin |last4=Ooms|first4=Aurélien|last5=Schultz Xavier da Silveira |first5=Luís Fernando|issue=1|journal=]|page=P1.8|title=More Turán-type theorems for triangles in convex point sets |url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p8 |volume=26 |year=2019 |bibcode=2017arXiv170610193A |arxiv=1706.10193 |access-date=2019-02-18 |archive-url=https://web.archive.org/web/20190218082023/https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p8|archive-date=2019-02-18|url-status=live|doi-access=free|doi=10.37236/7224}}</ref> how many tripods can have their apexes packed into a given cube?

====Euclidean geometry====
{{Main|Euclidean geometry}}
* The ] on the invertibility of a certain <math>n</math>-by-<math>n</math> matrix depending on <math>n</math> points in <math>\mathbb{R}^{3}</math><ref>{{Citation | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | title=Configurations of points | doi=10.1098/rsta.2001.0840 | mr=1853626 | year=2001 | journal= Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences| issn=1364-503X | volume=359 | issue=1784 | pages=1375–1387| bibcode=2001RSPTA.359.1375A | s2cid=55833332 }}</ref>
* ] – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation<ref>{{citation |last1=Finch |first1=S. R. |last2=Wetzel |first2=J. E. |title=Lost in a forest |volume=11 |issue=8 |year=2004 |journal=] |pages=645–654 |mr=2091541 |doi=10.2307/4145038 |jstor=4145038}}</ref>
* ] — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?<ref>{{citation
| last = Howards | first = Hugh Nelson
| arxiv = 1406.3370
| doi = 10.1142/S0218216513500831
| issue = 14
| journal = Journal of Knot Theory and Its Ramifications
| mr = 3190121
| pages = 1350083, 15
| title = Forming the Borromean rings out of arbitrary polygonal unknots
| volume = 22
| year = 2013| s2cid = 119674622
}}</ref>
* Danzer's problem and Conway's dead fly problem – do ]s of bounded density or bounded separation exist?<ref>{{citation |last1=Solomon |first1=Yaar |last2=Weiss |first2=Barak |arxiv=1406.3807 |doi=10.24033/asens.2303 |issue=5 |journal=Annales Scientifiques de l'École Normale Supérieure |mr=3581810 |pages=1053–1074 |title=Dense forests and Danzer sets |volume=49 |year=2016 |s2cid=672315}}; {{citation |last=Conway |first=John H. |author-link=John Horton Conway |publisher=] |title=Five $1,000 Problems (Update 2017) |url=https://oeis.org/A248380/a248380.pdf |archive-url=https://web.archive.org/web/20190213123825/https://oeis.org/A248380/a248380.pdf |archive-date=2019-02-13 |access-date=2019-02-12 |url-status=live}}</ref>
* ] – is it possible for ] of every dimension?<ref>{{citation |last1=Brandts |first1=Jan |last2=Korotov |first2=Sergey |last3=Křížek |first3=Michal |last4=Šolc |first4=Jakub |doi=10.1137/060669073 |issue=2 |journal=SIAM Review |mr=2505583 |pages=317–335 |title=On nonobtuse simplicial partitions |volume=51 |year=2009 |url=https://pure.uva.nl/ws/files/836396/73198_315330.pdf |bibcode=2009SIAMR..51..317B |s2cid=216078793 |access-date=2018-11-22 |archive-date=2018-11-04 |archive-url=https://web.archive.org/web/20181104211116/https://pure.uva.nl/ws/files/836396/73198_315330.pdf |url-status=live}}. See in particular Conjecture 23, p. 327.</ref>
* ]: a convex body <math>K</math> in <math>n</math> dimensions containing a single lattice point in its interior as its ] cannot have volume greater than <math>(n+1)^{n}/n!</math>
* ]: sets of Hausdorff dimension greater than <math>d/2</math> in <math>\mathbb{R}^d</math> must have a distance set of nonzero ]<ref>{{citation |last1=Arutyunyants |first1=G. |last2=Iosevich |first2=A. |editor-last=Pach |editor-first=János |editor-link=János Pach |contribution=Falconer conjecture, spherical averages and discrete analogs |doi=10.1090/conm/342/06127 |mr=2065249 |pages=15–24 |publisher=Amer. Math. Soc., Providence, RI|series=Contemp. Math. |title=Towards a Theory of Geometric Graphs |volume=342 |year=2004 |isbn=978-0-8218-3484-8 |doi-access=free}}</ref>
* The values of the ]s for dimensions other than 1–8 and 24
* ], also known as ] and the square peg problem – does every ] have an inscribed square?<ref name="matschke">{{citation|last=Matschke|first=Benjamin|date=2014|title=A survey on the square peg problem|journal=]|volume=61|issue=4|pages=346–352|doi=10.1090/noti1100|doi-access=free}}</ref>
* The ] –&nbsp;do <math>n</math>-dimensional sets that contain a unit line segment in every direction necessarily have ] and ] equal to <math>n</math>?<ref>{{citation |last1=Katz |first1=Nets |author1-link=Nets Katz|last2=Tao|first2=Terence|author2-link=Terence Tao|title=Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000) |doi=10.5565/PUBLMAT_Esco02_07|series=Publicacions Matemàtiques|mr=1964819 |pages=161–179 |contribution=Recent progress on the Kakeya conjecture |year=2002 |citeseerx=10.1.1.241.5335 |s2cid=77088}}</ref>
* The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the ] as a solution to the Kelvin problem<ref>{{citation |title=The Kelvin Problem |editor-first=Denis |editor-last=Weaire |editor-link=Denis Weaire |publisher=CRC Press |year=1997 |isbn=978-0-7484-0632-6 |page=1 |url=https://books.google.com/books?id=otokU4KQnXIC&pg=PA1}}</ref>
* ] on the minimum-area convex shape in the plane that can cover any shape of diameter one<ref>{{citation |last1=Brass |first1=Peter |last2=Moser |first2=William |last3=Pach |first3=János |location=New York |mr=2163782 |page=457 |publisher=Springer |title=Research problems in discrete geometry|url=https://books.google.com/books?id=cT7TB20y3A8C&pg=PA457 |year=2005 |isbn=978-0-387-29929-7}}</ref>
* ] on the product of the volumes of a ] ] and its ].<ref>{{Cite journal|last1=Mahler|first1=Kurt|title=Ein Minimalproblem für konvexe Polygone |journal=Mathematica (Zutphen) B|pages=118–127|year=1939}}</ref>
* ] – what is the smallest area of a shape that can cover every unit-length curve in the plane?<ref>{{citation |last1=Norwood |first1=Rick |author1-link=Rick Norwood |last2=Poole |first2=George |last3=Laidacker |first3=Michael |doi=10.1007/BF02187832 |issue=2 |journal=] |mr=1139077 |pages=153–162 |title=The worm problem of Leo Moser |volume=7 |year=1992 |doi-access=free}}</ref>
* The ] – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?<ref>{{citation |last=Wagner |first=Neal R. |date=1976 |title=The Sofa Problem |journal=The American Mathematical Monthly |doi=10.2307/2977022 |jstor=2977022 |volume=83 |issue=3 |pages=188–189 |url=http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |access-date=2014-05-14 |archive-url=https://web.archive.org/web/20150420160001/http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |archive-date=2015-04-20 |url-status=live}}</ref>
* Does every convex polyhedron have ]?<ref name=cyz>{{citation |first1=Ying |last1=Chai |first2=Liping |last2=Yuan |first3=Tudor |last3=Zamfirescu |title=Rupert Property of Archimedean Solids |journal=] |volume=125 |issue=6 |pages=497–504 |date=June–July 2018 |doi=10.1080/00029890.2018.1449505| s2cid=125508192}}</ref><ref name=styu>{{citation|title=An algorithmic approach to Rupert's problem |first1=Jakob |last1=Steininger |first2=Sergey |last2=Yurkevich| date=December 27, 2021 |arxiv=2112.13754}}</ref>
* ] – does every ] have a ], or simple edge-unfolding?<ref>{{citation |last1=Demaine |first1=Erik D. |author1-link=Erik Demaine |last2=O'Rourke |first2=Joseph |author2-link=Joseph O'Rourke (professor) |date=2007 |title=Geometric Folding Algorithms: Linkages, Origami, Polyhedra |title-link=Geometric Folding Algorithms |publisher=Cambridge University Press |contribution=Chapter 22. Edge Unfolding of Polyhedra |pages=306–338}}</ref><ref>{{Cite journal |last=Ghomi |first=Mohammad |date=2018-01-01 |title=Dürer's Unfolding Problem for Convex Polyhedra |journal=Notices of the American Mathematical Society |volume=65 |issue=1 |pages=25–27 |doi=10.1090/noti1609 |issn=0002-9920 |doi-access=free}}</ref>
* Is there a non-convex polyhedron without self-intersections with ], all of which share an edge with each other?
* The ] – what is the minimum energy configuration of <math>n</math> mutually-repelling particles on a unit sphere?<ref>{{citation|last=Whyte|first=L. L.|doi=10.2307/2306764|journal=The American Mathematical Monthly|mr=0050303|pages=606–611|title=Unique arrangements of points on a sphere|volume=59|issue=9|year=1952|jstor=2306764}}</ref>
* Convex ]s – find and classify the complete set of these shapes<ref>{{citation |author=ACW |date=May 24, 2012 |title=Convex uniform 5-polytopes |url=http://www.openproblemgarden.org/op/convex_uniform_5_polytopes |work=Open Problem Garden |access-date=2016-10-04 |archive-url=https://web.archive.org/web/20161005164840/http://www.openproblemgarden.org/op/convex_uniform_5_polytopes |archive-date=October 5, 2016 |url-status=live}}.</ref>

=== Graph theory ===
{{Main|Graph theory}}

==== Algebraic graph theory ====
* ]: which groups are Babai invariant groups?
* ] on upper bounds for sums of ] of ] of graphs in terms of their number of edges

==== Games on graphs ====
* ] on the pebbling number of Cartesian products of graphs<ref>{{citation
| last = Pleanmani | first = Nopparat
| doi = 10.1142/s179383091950068x
| issue = 6
| journal = Discrete Mathematics, Algorithms and Applications
| mr = 4044549
| pages = 1950068, 7
| title = Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph
| volume = 11
| year = 2019| s2cid = 204207428
}}</ref>
* Meyniel's conjecture that ] is <math>O(\sqrt n)</math><ref>{{citation
| last1 = Baird | first1 = William
| last2 = Bonato | first2 = Anthony
| arxiv = 1308.3385
| doi = 10.4310/JOC.2012.v3.n2.a6
| issue = 2
| journal = Journal of Combinatorics
| mr = 2980752
| pages = 225–238
| title = Meyniel's conjecture on the cop number: a survey
| volume = 3
| year = 2012| s2cid = 18942362
}}</ref>

==== Graph coloring and labeling ====
]
* The ] that if <math>n</math> is odd or even and <math>k \geq n, n - 1</math> respectively, then a <math>k</math>-] with <math>2n</math> vertices is ].
** The ] that every ] on an even number of vertices admits a ].
* ] on the diameter of the space of colorings of degenerate graphs<ref>{{citation
| last1 = Bousquet | first1 = Nicolas
| last2 = Bartier | first2 = Valentin
| editor1-last = Bender | editor1-first = Michael A.
| editor2-last = Svensson | editor2-first = Ola
| editor3-last = Herman | editor3-first = Grzegorz
| contribution = Linear Transformations Between Colorings in Chordal Graphs
| doi = 10.4230/LIPIcs.ESA.2019.24
| pages = 24:1–24:15
| publisher = Schloss Dagstuhl - Leibniz-Zentrum für Informatik
| series = LIPIcs
| title = 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany
| volume = 144
| year = 2019| doi-access = free
| isbn = 978-3-95977-124-5
| s2cid = 195791634
}}</ref>
* The ]: what is the maximum chromatic number of biplanar graphs?<ref>{{citation
| last = Gethner | first = Ellen | author-link = Ellen Gethner
| editor1-last = Gera | editor1-first = Ralucca | editor1-link = Ralucca Gera
| editor2-last = Haynes | editor2-first = Teresa W. | editor2-link = Teresa W. Haynes
| editor3-last = Hedetniemi | editor3-first = Stephen T.
| contribution = To the Moon and beyond
| doi = 10.1007/978-3-319-97686-0_11
| mr = 3930641
| pages = 115–133
| publisher = Springer International Publishing
| series = Problem Books in Mathematics
| title = Graph Theory: Favorite Conjectures and Open Problems, II
| year = 2018| isbn = 978-3-319-97684-6 }}</ref>
* The ] on coloring unions of cliques<ref>{{citation
| last1 = Chung | first1 = Fan | author-link1 = Fan Chung
| last2 = Graham | first2 = Ron | author-link2 = Ronald Graham
| title = Erdős on Graphs: His Legacy of Unsolved Problems
| year = 1998
| publisher = A K Peters
| pages = 97–99}}.</ref>
* The ] that every tree admits a graceful labeling
** ] that all ] are graceful or nearly-graceful
* The ] on χ-boundedness of graphs with a forbidden induced tree<ref>{{citation
| last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky
| last2 = Seymour | first2 = Paul | author2-link = Paul Seymour (mathematician)
| doi = 10.1016/j.jctb.2013.11.002
| journal = ]
| mr = 3171779
| pages = 11–16
| series = Series B
| title = Extending the Gyárfás-Sumner conjecture
| volume = 105
| year = 2014| doi-access = free
}}</ref>
* The ] relating coloring to clique minors<ref>{{citation
| last = Toft | first = Bjarne
| journal = Congressus Numerantium
| mr = 1411244
| pages = 249–283
| title = A survey of Hadwiger's conjecture
| volume = 115
| year = 1996}}.</ref>
* The ] on the chromatic number of unit distance graphs<ref>{{citation
| last1 = Croft | first1 = Hallard T.
| last2 = Falconer | first2 = Kenneth J.
| last3 = Guy | first3 = Richard K. | author-link3 = Richard K. Guy
| title = Unsolved Problems in Geometry
| publisher = Springer-Verlag
| year = 1991}}, Problem G10.</ref>
* ]: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph<ref>{{citation
| last1 = Hägglund
| first1 = Jonas
| last2 = Steffen
| first2 = Eckhard
| issue = 1
| journal = Ars Mathematica Contemporanea
| mr = 3047618
| pages = 161–173
| title = Petersen-colorings and some families of snarks
| url = http://amc-journal.eu/index.php/amc/article/viewFile/288/247
| volume = 7 | volume = 7
| year = 1992}}</ref> | year = 2014
| doi = 10.26493/1855-3974.288.11a
* The ] – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?<ref>{{citation |last=Wagner |first=Neal R. |date=1976 |title=The Sofa Problem |journal=The American Mathematical Monthly |doi=10.2307/2977022 |jstor=2977022 |volume=83 |issue=3 |pages=188–189 |url=http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |format=PDF}}</ref>
| access-date = 2016-09-30
* ] – does every ] have a ]?<ref>{{citation |last1=Demaine |first1=Erik D. |author1-link=Erik Demaine |last2=O'Rourke |first2=Joseph |author2-link=Joseph O'Rourke (professor) |date=2007 |title=Geometric Folding Algorithms: Linkages, Origami, Polyhedra |publisher=Cambridge University Press |contribution=Chapter 22. Edge Unfolding of Polyhedra |pages=306–338}}</ref>
| archive-url = https://web.archive.org/web/20161003070647/http://amc-journal.eu/index.php/amc/article/viewFile/288/247
* The ] - what is the minimum energy configuration of N particles bound to the surface of a unit sphere that repel each other with a 1/r potential (or any potential in general)?
| archive-date = 2016-10-03
* ] - 15 types of convex pentagons are known to monohedrally tile the plane, and it is not known whether this list is complete.<ref>{{citation |title= Attack on the pentagon results in discovery of new mathematical tile |first=Alex |last=Bellos |newspaper=] |date=11 August 2015 |url=http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile}}</ref>
| url-status = live
* ]
| doi-access = free
* ]
}}.</ref>
* ]
* The ]: for every graph, the list chromatic index equals the chromatic index<ref>{{citation |last1=Jensen |first1=Tommy R. |last2=Toft |first2=Bjarne |year=1995 |title=Graph Coloring Problems |location=New York |publisher=Wiley-Interscience |isbn=978-0-471-02865-9 |chapter=12.20 List-Edge-Chromatic Numbers |pages=201–202}}.</ref>
* ]
* The ] that a graph with maximum degree <math>\Delta(G) \geq n/3</math> is ] if and only if it has an ] <math>S</math> satisfying <math>\Delta(S) = \Delta(G)</math>.
* ] - what is the convex shape in the plane of least area which provides an isometric cover for any shape of diameter one?
* The ] of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree<ref>{{citation
* ] - for a given shape of forest find the shortest escape path which will intersect the edge of the forest at some point for any given starting point and direction inside the forest.
| last1 = Molloy | first1 = Michael
| last2 = Reed | first2 = Bruce | author1-link = Bruce Reed (mathematician)
| doi = 10.1007/PL00009820
| issue = 2
| journal = ]
| mr = 1656544
| pages = 241–280
| title = A bound on the total chromatic number
| volume = 18
| year = 1998| citeseerx = 10.1.1.24.6514
| s2cid = 9600550
}}.</ref>


==== Graph drawing and embedding ====
===]s===
* The ]: the crossing number can be lower-bounded by the crossing number of a ] with the same ]<ref>{{citation|first1=János|last1=Barát|first2=Géza|last2=Tóth|year=2010|title=Towards the Albertson Conjecture|arxiv=0909.0413|journal=Electronic Journal of Combinatorics|volume=17|issue=1|page=R73|bibcode=2009arXiv0909.0413B|doi-access=free|doi=10.37236/345}}.</ref>
* ] conjecture – Is every invariant and ] measure for the <math>\times 2,\times 3</math> action on the circle either Lebesgue or atomic?
* ]<ref>{{citation |last1=Fulek |first1=Radoslav |last2=Pach |first2=János |author-link2=János Pach |title=A computational approach to Conway's thrackle conjecture|journal=] |volume=44 |year=2011|issue=6–7 |pages=345–355 |mr=2785903 |doi=10.1016/j.comgeo.2011.02.001|doi-access=free|arxiv=1002.3904 }}.</ref> that ]s cannot have more edges than vertices
* ] conjecture — Measure classification for diagonalizable actions in higher-rank groups
* The ] on whether minor-closed graph families have <math>\ell_1</math> embeddings with bounded distortion<ref>{{citation
* ] – Is the Mandelbrot set locally connected ?
| last1 = Gupta | first1 = Anupam
* ] - Does a regular compact ] ] of a ] on a ] carry at least one periodic orbit of the Hamiltonian flow?
| last2 = Newman | first2 = Ilan
* Is every ] in three or more dimensions locally reversible?<ref>{{citation
| last3 = Rabinovich | first3 = Yuri
| last = Kari | first = Jarkko | authorlink = Jarkko Kari
| last4 = Sinclair | first4 = Alistair | author4-link = Alistair Sinclair
| contribution = Structure of reversible cellular automata
| doi = 10.1007/978-3-642-03745-0_5 | doi = 10.1007/s00493-004-0015-x
| page = 6 | issue = 2
| journal = ]
| publisher = Springer
| mr = 2071334
| series = ]
| pages = 233–269
| title = Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7ÔÇô11, 2009, Proceedings
| title = Cuts, trees and <math>\ell_1</math>-embeddings of graphs
| volume = 5715
| year = 2009}}</ref> | volume = 24
| year = 2004| citeseerx = 10.1.1.698.8978
* Many problems concerning an ], for example show that outer billiards relative to almost every convex polygon has unbounded orbits.
| s2cid = 46133408
}}</ref>
* ]: every planar graph can be drawn with integer edge lengths<ref>{{citation|title=Pearls in Graph Theory: A Comprehensive Introduction|title-link= Pearls in Graph Theory |series=Dover Books on Mathematics|last1=Hartsfield|first1=Nora|last2=Ringel|first2=Gerhard|author2-link=Gerhard Ringel|publisher=Courier Dover Publications|year=2013|isbn=978-0-486-31552-2|at=|mr=2047103}}.</ref>
* ] on projective-plane embeddings of graphs with planar covers<ref>{{citation | last = Hliněný | first = Petr | doi = 10.1007/s00373-010-0934-9 | issue = 4 | journal = ] | mr = 2669457 | pages = 525–536 | title = 20 years of Negami's planar cover conjecture | url = http://www.fi.muni.cz/~hlineny/papers/plcover20-gc.pdf | volume = 26 | year = 2010 | citeseerx = 10.1.1.605.4932 | s2cid = 121645 | access-date = 2016-10-04 | archive-url = https://web.archive.org/web/20160304030722/http://www.fi.muni.cz/~hlineny/papers/plcover20-gc.pdf | archive-date = 2016-03-04 | url-status = live }}.</ref>
* The ]: every polyhedral graph has a convex greedy embedding<ref>{{citation | last1 = Nöllenburg | first1 = Martin | last2 = Prutkin | first2 = Roman | last3 = Rutter | first3 = Ignaz | doi = 10.20382/jocg.v7i1a3 | issue = 1 | journal = ] | mr = 3463906 | pages = 47–69 | title = On self-approaching and increasing-chord drawings of 3-connected planar graphs | volume = 7 | year = 2016| arxiv = 1409.0315 | s2cid = 1500695 }}</ref>
* ] – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?<ref>{{citation | last1 = Pach | first1 = János | author1-link = János Pach | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir | contribution = 5.1 Crossings—the Brick Factory Problem | pages = 126–127 | publisher = ] | series = Mathematical Surveys and Monographs | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures | volume = 152 | year = 2009}}.</ref>


* ]s of subquadratic size for planar graphs<ref>{{citation | last1 = Demaine | first1 = E. | author1-link = Erik Demaine | last2 = O'Rourke | first2 = J. | author2-link = Joseph O'Rourke (professor) | contribution = Problem 45: Smallest Universal Set of Points for Planar Graphs | title = The Open Problems Project | url = http://cs.smith.edu/~orourke/TOPP/P45.html | year = 2002–2012 | access-date = 2013-03-19 | archive-url = https://web.archive.org/web/20120814154255/http://cs.smith.edu/~orourke/TOPP/P45.html | archive-date = 2012-08-14 | url-status = live }}.</ref>
===]===
* ] that every cubic bipartite three-connected planar graph has a Hamiltonian cycle
* The ] on cycles with power-of-two lengths in cubic graphs
* The ] on finding large homogeneous sets in graphs with a forbidden induced subgraph
* The ] relating coloring to clique minors
* The ] on coloring unions of cliques
* ] that every planar graph can be drawn with integer edge lengths
* The ]
* ]
* The ]
* The ] on graceful labeling of trees
* How many unit distances can be determined by a set of n points? (see ])
* The ] on the chromatic number of unit distance graphs
* ]
* Deriving a closed-form expression for the ] values, especially <math>p_c</math> (square site)
* Tutte's conjectures that every bridgeless graph has a ] and every bridgeless graph without the ] as a ] has a nowhere-zero 4-flow
* ]
* The ] and ] concerning whether or not a graph is recognizable by the vertex deleted subgraphs.
* The ] that every bridgeless graph has a family of cycles that includes each edge twice.
* Does a ] with girth 5 and degree 57 exist?
* ]
* ] on the characterization of graphs with planar covers
* The ] on the book thickness of subdivisions
* ]
* ]
* ] - Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?


==== Restriction of graph parameters ====
===]===
* ]: does there exist a ] with parameters (99,14,1,2)?<ref>{{citation
* Is every ] ] finite?
| last = Conway
* The ]: is every finite group the Galois group of a Galois extension of the rationals?
| first = John H.
* For which positive integers ''m'', ''n'' is the ] {{nowrap|B(''m'',''n'')}} finite? In particular, is {{nowrap|B(2, 5)}} finite?
| author-link = John Horton Conway
* Is every group ]?
| access-date = 2019-02-12
* ]
| publisher = Online Encyclopedia of Integer Sequences
* ]
| title = Five $1,000 Problems (Update 2017)
* Does ] exist?
| url = https://oeis.org/A248380/a248380.pdf
| archive-url = https://web.archive.org/web/20190213123825/https://oeis.org/A248380/a248380.pdf
| archive-date = 2019-02-13
| url-status = live
}}</ref>
* ]: given two positive integers <math>d, k</math>, what is the largest graph of diameter <math>k</math> such that all vertices have degrees at most <math>d</math>?
* Jørgensen's conjecture that every 6-vertex-connected ''K''<sub>6</sub>-minor-free graph is an ]<ref>{{citation |last1=mdevos |title=Jorgensen's Conjecture |date=December 7, 2019 |url=http://www.openproblemgarden.org/op/jorgensens_conjecture |work=Open Problem Garden |archive-url=https://web.archive.org/web/20161114232136/http://www.openproblemgarden.org/op/jorgensens_conjecture |access-date=2016-11-13 |archive-date=2016-11-14 |last2=Wood |first2=David |url-status=live}}.</ref>
* Does a ] with girth 5 and degree 57 exist?<ref>{{citation
| last=Ducey
| first=Joshua E.
| doi=10.1016/j.disc.2016.10.001
| issue=5
| journal=]
| mr=3612450
| pages=1104–1109
| title=On the critical group of the missing Moore graph
| volume=340
| year=2017
| arxiv=1509.00327
| s2cid=28297244}}
</ref>
* Do there exist infinitely many ] ]s, or any strongly regular geodetic graphs that are not Moore graphs?<ref>{{citation
| last1 = Blokhuis | first1 = A.
| last2 = Brouwer | first2 = A. E. | author-link = Andries Brouwer
| doi = 10.1007/BF00191941
| issue = 1–3
| journal = ]
| mr = 925851
| pages = 527–533
| title = Geodetic graphs of diameter two
| volume = 25
| year = 1988| s2cid = 189890651
}}</ref>


===]=== ==== Subgraphs ====
* ]: every cubic bipartite three-connected planar graph has a Hamiltonian cycle<ref>{{citation
* ]
| last = Florek | first = Jan
| doi = 10.1016/j.disc.2010.01.018
| issue = 10–11
| journal = ]
| mr = 2601261
| pages = 1531–1535
| title = On Barnette's conjecture
| volume = 310
| year = 2010}}.</ref>
* ] that the Steiner ratio is <math>\sqrt{3}/2</math>
* ], that there is a number {{mvar|t}} such that every {{mvar|t}}-tough graph is Hamiltonian<ref>{{citation
| last1 = Broersma | first1 = Hajo
| last2 = Patel | first2 = Viresh
| last3 = Pyatkin | first3 = Artem
| doi = 10.1002/jgt.21734
| issue = 3
| journal = ]
| mr = 3153119
| pages = 244–255
| title = On toughness and Hamiltonicity of $2K_2$-free graphs
| volume = 75
| year = 2014| s2cid = 1377980
| url = https://ris.utwente.nl/ws/files/6416631/jgt21734.pdf
}}</ref>
* The ]: every bridgeless graph has a family of cycles that includes each edge twice<ref>{{citation
| last = Jaeger | first = F.
| contribution = A survey of the cycle double cover conjecture
| doi = 10.1016/S0304-0208(08)72993-1
| pages = 1–12
| series = North-Holland Mathematics Studies
| title = Annals of Discrete Mathematics 27 – Cycles in Graphs
| volume = 27
| year = 1985| isbn = 978-0-444-87803-8
}}.</ref>
* The ] on cycles with power-of-two lengths in cubic graphs<ref>{{citation|title=Erdös-Gyárfás conjecture for cubic planar graphs |first1=Christopher Carl |last1=Heckman |first2=Roi |last2=Krakovski |volume=20 |issue=2 |year=2013 |at=P7 |journal=Electronic Journal of Combinatorics |doi-access=free |doi=10.37236/3252}}.</ref>
* The ] on large cliques or independent sets in graphs with a forbidden induced subgraph<ref>{{citation
| last = Chudnovsky
| first = Maria
| author-link = Maria Chudnovsky
| arxiv = 1606.08827
| doi = 10.1002/jgt.21730
| issue = 2
| journal = ]
| mr = 3150572
| zbl = 1280.05086
| pages = 178–190
| title = The Erdös–Hajnal conjecture—a survey
| url = http://www.columbia.edu/~mc2775/EHsurvey.pdf
| volume = 75
| year = 2014
| s2cid = 985458
| access-date = 2016-09-22
| archive-url = https://web.archive.org/web/20160304102611/http://www.columbia.edu/~mc2775/EHsurvey.pdf
| archive-date = 2016-03-04
| url-status = live
}}.</ref>
* The ] conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree<ref>{{citation
| last1 = Akiyama | first1 = Jin | author1-link = Jin Akiyama
| last2 = Exoo | first2 = Geoffrey
| last3 = Harary | first3 = Frank
| doi = 10.1002/net.3230110108
| issue = 1
| journal = Networks
| mr = 608921
| pages = 69–72
| title = Covering and packing in graphs. IV. Linear arboricity
| volume = 11
| year = 1981}}.</ref>
* The ] on Hamiltonian paths in symmetric graphs<ref>{{Cite book |last=Babai |first=László |url=http://newtraell.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps |title=Handbook of Combinatorics |date=June 9, 1994 |chapter=Automorphism groups, isomorphism, reconstruction |format=PostScript |author-link=László Babai |archive-url=https://web.archive.org/web/20070613201449/http://www.cs.uchicago.edu/research/publications/techreports/TR-94-10 |archive-date=13 June 2007}}</ref>
* The ] on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.<ref>{{citation
| last1 = Lenz | first1 = Hanfried
| last2 = Ringel | first2 = Gerhard
| doi = 10.1016/0012-365X(91)90416-Y
| issue = 1–3
| journal = ]
| mr = 1140782
| pages = 3–16
| title = A brief review on Egmont Köhler's mathematical work
| volume = 97
| year = 1991}}</ref>
* What is the largest possible ] of an {{mvar|n}}-vertex ]?<ref>{{citation
| last1 = Fomin | first1 = Fedor V.
| last2 = Høie | first2 = Kjartan
| doi = 10.1016/j.ipl.2005.10.012
| issue = 5
| journal = Information Processing Letters
| mr = 2195217
| pages = 191–196
| title = Pathwidth of cubic graphs and exact algorithms
| volume = 97
| year = 2006}}
</ref>
* The ] and ] on whether a graph is uniquely determined by its vertex-deleted subgraphs.<ref>{{cite conference |last=Schwenk |first=Allen |year=2012 |title=Some History on the Reconstruction Conjecture |url=http://faculty.nps.edu/rgera/conjectures/jmm2012/Schwenk,%20%20Some%20History%20on%20the%20RC.pdf |conference=Joint Mathematics Meetings |archive-url=https://web.archive.org/web/20150409233306/http://faculty.nps.edu/rgera/Conjectures/jmm2012/Schwenk,%20%20Some%20History%20on%20the%20RC.pdf |archive-date=2015-04-09 |access-date=2018-11-26}}</ref><ref>{{citation
| last = Ramachandran | first = S.
| doi = 10.1016/S0095-8956(81)80019-6
| issue = 2
| journal = ]
| mr = 630977
| pages = 143–149
| series = Series B
| title = On a new digraph reconstruction conjecture
| volume = 31
| year = 1981| doi-access = free
}}</ref>
* The ] problem: what is the longest possible ] in an <math>n</math>-dimensional ] graph?
* ]: does every <math>(2n-2)</math>-vertex tournament contain as a subgraph every <math>n</math>-vertex oriented tree?<ref>{{citation
| last1 = Kühn | first1 = Daniela | author1-link = Daniela Kühn
| last2 = Mycroft | first2 = Richard
| last3 = Osthus | first3 = Deryk
| arxiv = 1010.4430
| doi = 10.1112/plms/pdq035
| issue = 4
| journal = Proceedings of the London Mathematical Society | series = Third Series
| mr = 2793448 | zbl=1218.05034
| pages = 731–766
| title = A proof of Sumner's universal tournament conjecture for large tournaments
| volume = 102
| year = 2011| s2cid = 119169562 }}.</ref>
* ]: every ] on the <math>n</math>-dimensional doubly-] ] can be routed with edge-disjoint ].
* ]: if the maximum number of disjoint triangles is <math>\nu</math>, can all triangles be hit by a set of at most <math>2\nu</math> edges?<ref>{{cite journal
| last = Tuza | first = Zsolt
| doi = 10.1007/BF01787705
| issue = 4
| journal = Graphs and Combinatorics
| mr = 1092587
| pages = 373–380
| title = A conjecture on triangles of graphs
| volume = 6
| year = 1990| s2cid = 38821128
}}</ref>
* ] on the ] of ]<ref>{{citation
| last1 = Brešar | first1 = Boštjan
| last2 = Dorbec | first2 = Paul
| last3 = Goddard | first3 = Wayne
| last4 = Hartnell | first4 = Bert L.
| last5 = Henning | first5 = Michael A.
| last6 = Klavžar | first6 = Sandi
| last7 = Rall | first7 = Douglas F.
| doi = 10.1002/jgt.20565
| issue = 1
| journal = ]
| mr = 2864622
| pages = 46–76
| title = Vizing's conjecture: a survey and recent results
| volume = 69
| year = 2012| citeseerx = 10.1.1.159.7029
| s2cid = 9120720
}}.</ref>
* ]: how many edges can there be in a ] on a given number of vertices with no ] of a given size?

==== Word-representation of graphs ====
*Are there any graphs on ''n'' vertices whose ] requires more than floor(''n''/2) copies of each letter?<ref name="KL15">{{Cite book |last1=Kitaev |first1=Sergey | author1-link = Sergey Kitaev|url=https://link.springer.com/book/10.1007/978-3-319-25859-1 |title=Words and Graphs |last2=Lozin |first2=Vadim |year=2015 |isbn=978-3-319-25857-7 |series=Monographs in Theoretical Computer Science. An EATCS Series |doi=10.1007/978-3-319-25859-1 |via=link.springer.com |s2cid=7727433}}</ref><ref name="K17">{{Cite conference |last=Kitaev |first=Sergey |date=2017-05-16 |title=A Comprehensive Introduction to the Theory of Word-Representable Graphs |conference=] |language=en |doi=10.1007/978-3-319-62809-7_2|arxiv=1705.05924v1 }}</ref><ref name="KP18">{{Cite journal|title=Word-Representable Graphs: a Survey|first1=S. V.|last1=Kitaev|first2=A. V.|last2=Pyatkin|date=April 1, 2018|journal=Journal of Applied and Industrial Mathematics|volume=12|issue=2|pages=278–296|via=Springer Link|doi=10.1134/S1990478918020084|s2cid=125814097 }}</ref><ref name="KP18-2">{{Cite journal |last1=Kitaev |first1=Sergey V. |last2=Pyatkin |first2=Artem V. |date=2018 |title=Графы, представимые в виде слов. Обзор результатов |trans-title=Word-representable graphs: A survey |url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus |journal=Дискретн. анализ и исслед. опер. |language=ru |volume=25 |issue=2 |pages=19–53 |doi=10.17377/daio.2018.25.588}}</ref>
*Characterise (non-)] ]s<ref name="KL15"/><ref name="K17"/><ref name="KP18"/><ref name="KP18-2"/>
*Characterise ]s in terms of (induced) forbidden subgraphs.<ref name="KL15"/><ref name="K17"/><ref name="KP18"/><ref name="KP18-2"/>
*Characterise ] near-triangulations containing the complete graph ''K''<sub>4</sub> (such a characterisation is known for ''K''<sub>4</sub>-free planar graphs<ref name="Glen2019">{{cite arXiv |eprint=1605.01688|author1=Marc Elliot Glen|title=Colourability and word-representability of near-triangulations|class=math.CO|year=2016}}</ref>)
*Classify graphs with representation number 3, that is, graphs that can be ] using 3 copies of each letter, but cannot be represented using 2 copies of each letter<ref name="Kit2013-3-repr">{{Cite arXiv|last=Kitaev |first=Sergey |date=2014-03-06 |title=On graphs with representation number 3 |class=math.CO |eprint=1403.1616v1 }}</ref>
*Is it true that out of all ]s, ]s require longest word-representants?<ref name="GKP18">{{cite journal|url = https://www.sciencedirect.com/science/article/pii/S0166218X18301045 | doi=10.1016/j.dam.2018.03.013 | volume=244 | title=On the representation number of a crown graph | year=2018 | journal=Discrete Applied Mathematics | pages=89–93 | last1 = Glen | first1 = Marc | last2 = Kitaev | first2 = Sergey | last3 = Pyatkin | first3 = Artem| arxiv=1609.00674 | s2cid=46925617 }}</ref>
*Is the ] of a non-] graph always non-]?<ref name="KL15"/><ref name="K17"/><ref name="KP18"/><ref name="KP18-2"/>
*Which (hard) problems on graphs can be translated to words ] them and solved on words (efficiently)?<ref name="KL15"/><ref name="K17"/><ref name="KP18"/><ref name="KP18-2"/>

==== Miscellaneous graph theory ====
* The ] on the existence of implicit representations for slowly-growing ]<ref>{{citation|first=Jeremy P.|last=Spinrad|title=Efficient Graph Representations|year=2003|isbn=978-0-8218-2815-1|chapter=2. Implicit graph representation|pages=17–30|publisher=American Mathematical Soc. |chapter-url=https://books.google.com/books?id=RrtXSKMAmWgC&pg=PA17}}.</ref>
* ] relating the maximum ] size and minimum ] size in ]s
* The ]: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?<ref>{{Cite web |title=Seymour's 2nd Neighborhood Conjecture |url=https://faculty.math.illinois.edu/~west/openp/2ndnbhd.html |url-status=live |archive-url=https://web.archive.org/web/20190111175310/https://faculty.math.illinois.edu/~west/openp/2ndnbhd.html |archive-date=11 January 2019 |access-date=17 August 2022 |website=faculty.math.illinois.edu}}</ref>
* ] on ] of graphs in ]s
* Tutte's conjectures:
** every bridgeless graph has a ]<ref>{{cite web |last=mdevos |date=May 4, 2007 |title=5-flow conjecture |url=http://www.openproblemgarden.org/op/5_flow_conjecture |url-status=live |archive-url=https://web.archive.org/web/20181126134833/http://www.openproblemgarden.org/op/5_flow_conjecture |archive-date=November 26, 2018 |website=Open Problem Garden}}</ref>
** every ]-]-free bridgeless graph has a nowhere-zero 4-flow<ref>{{cite web |last=mdevos |date=March 31, 2010 |title=4-flow conjecture |url=http://www.openproblemgarden.org/op/4_flow_conjecture |url-status=live |archive-url=https://web.archive.org/web/20181126134908/http://www.openproblemgarden.org/op/4_flow_conjecture |archive-date=November 26, 2018 |website=Open Problem Garden}}</ref>
* ] that the minimum number of edges in a ] of a ] is equal to the maximum number of disjoint ]s

=== Model theory and formal languages ===
{{Main|Model theory|formal languages}}
* The ]: A simple group whose first-order theory is ] in <math>\aleph_0</math> is a simple algebraic group over an algebraically closed field. * The ]: A simple group whose first-order theory is ] in <math>\aleph_0</math> is a simple algebraic group over an algebraically closed field.
* ]: can all ]s be expressed using ] with limited nesting depths of ]s?
* The Main Gap conjecture, e.g. for uncountable ], for ]s, and for <math>\aleph_1</math>-saturated models of a countable theory.<ref name=":0">Shelah S, ''Classification Theory'', North-Holland, 1990</ref>
* Determine the structure of Keisler's order<ref>Keisler, HJ, "Ultraproducts which are not saturated." J. Symb Logic 32 (1967) 23—46.</ref><ref>Malliaris M, Shelah S, "A dividing line in simple unstable theories." http://arxiv.org/abs/1208.2140</ref>
* The stable field conjecture: every infinite field with a ] first-order theory is separably closed.
* Is the theory of the field of Laurent series over <math>\mathbb{Z}_p</math> decidable? of the field of polynomials over <math>\mathbb{C}</math>?
* (BMTO) Is the Borel monadic theory of the real order ]? (MTWO) Is the monadic theory of well-ordering consistently decidable?<ref>Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., ''Model-Theoretic Logics'' (New York: Springer-Verlag, 1985), 479–506.</ref>
* The Stable Forking Conjecture for simple theories<ref>Peretz, Assaf, "Geometry of forking in simple theories." J. Symbolic Logic Volume 71, Issue 1 (2006), 347–359.</ref>
* For which number fields does ] hold? * For which number fields does ] hold?
* Kueker's conjecture<ref>{{cite journal |last1=Hrushovski |first1=Ehud |year=1989 |title=Kueker's conjecture for stable theories |journal=Journal of Symbolic Logic |volume=54 |issue=1| pages=207–220 |doi=10.2307/2275025| jstor=2275025 |s2cid=41940041}}</ref>
* Assume K is the class of models of a countable first order theory omitting countably many ]. If K has a model of cardinality <math>\aleph_{\omega_1}</math> does it have a model of cardinality continuum?<ref>{{cite journal |last=Shelah |first=Saharon |authorlink=Saharon Shelah |date=1999 |title=Borel sets with large squares |journal=] |arxiv=math/9802134 |volume=159 |issue=1 |pages=1–50}}</ref>
* The main gap conjecture, e.g. for uncountable ], for ], and for <math>\aleph_1</math>-saturated models of a countable theory.<ref name=":0">{{cite book |vauthors=Shelah S |title=Classification Theory |publisher=North-Holland |year=1990}}</ref>
* Shelah's eventual Categority conjecture: For every cardinal \lambda there exists a cardinal \mu(\lambda) such that If an ] K with LS(K)<= \lambda is categorical in a cardinal above \mu(\lambda) then it is categorical in all cardinals above \mu(\lambda).<ref name=":0" /><ref>{{Cite book
* Shelah's categoricity conjecture for <math>L_{\omega_1,\omega}</math>: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.<ref name=":0" />
* Shelah's eventual categoricity conjecture: For every cardinal <math>\lambda</math> there exists a cardinal <math>\mu(\lambda)</math> such that if an ] K with LS(K)<= <math>\lambda</math> is categorical in a cardinal above <math>\mu(\lambda)</math> then it is categorical in all cardinals above <math>\mu(\lambda)</math>.<ref name=":0" /><ref>{{Cite book
| title = Classification theory for abstract elementary classes | title = Classification theory for abstract elementary classes
| last = Shelah | last = Shelah
Line 248: Line 917:
| year = 2009 | year = 2009
| isbn = 978-1-904987-71-0 | isbn = 978-1-904987-71-0
}}</ref>
| location =
* The stable field conjecture: every infinite field with a ] first-order theory is separably closed.
| pages =
* The stable forking conjecture for simple theories<ref>{{cite journal | last1 = Peretz | first1 = Assaf | year = 2006 | title = Geometry of forking in simple theories | journal = Journal of Symbolic Logic| volume = 71 | issue = 1| pages = 347–359 | doi=10.2178/jsl/1140641179| arxiv = math/0412356| s2cid = 9380215 }}</ref>
}}</ref>
* ]: is the ] of the ]s with the ] ]?
* Shelah's categoricity conjecture for L_{\omega_1,\omega}: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.<ref name=":0" />
* The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?<ref>{{cite journal |last1=Cherlin |first1=Gregory |last2=Shelah |first2=Saharon | author-link2=Saharon Shelah|date=May 2007 |title=Universal graphs with a forbidden subtree |journal=] | series=Series B |arxiv=math/0512218 |doi=10.1016/j.jctb.2006.05.008 | doi-access=free |volume=97 |issue=3 |pages=293–333|s2cid=10425739 }}</ref>
* Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?<ref>Makowsky J, "Compactness, embeddings and definability," in ''Model-Theoretic Logics'', eds Barwise and Feferman, Springer 1985 pps. 645–715.</ref>
* If the class of atomic models of a complete first order theory is ] in the <math>\aleph_n</math>, is it categorical in every cardinal?<ref>{{cite book |last=Baldwin |first=John T. |date=July 24, 2009 |title=Categoricity |publisher=] |isbn=978-0821848937 |url=http://www.math.uic.edu/~jbaldwin/pub/AEClec.pdf |format=PDF |accessdate=February 20, 2014}}</ref><ref>{{cite journal |last=Shelah |first=Saharon |title=Introduction to classification theory for abstract elementary classes |url=http://front.math.ucdavis.edu/0903.3428}}</ref>
* Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
* Kueker's conjecture<ref>Hrushovski, Ehud, "Kueker's conjecture for stable theories." Journal of Symbolic Logic Vol. 54, No. 1 (Mar., 1989), pp. 207–220.</ref>
* Does there exist an ] first order theory with a trans-exponential (rapid growth) function?
* Lachlan's decision problem
* Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
* Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)
* The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?<ref>{{cite journal |last1=Cherlin |first1=G. |last2=Shelah |first2=S. |date=May 2007 |title=Universal graphs with a forbidden subtree |journal=] |arxiv=math/0512218 |doi=10.1016/j.jctb.2006.05.008 |volume=97 |issue=3 |pages=293–333}}</ref>
* The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?<ref>Džamonja, Mirna, "Club guessing and the universal models." ''On PCF'', ed. M. Foreman, (Banff, Alberta, 2004).</ref> * The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?<ref>Džamonja, Mirna, "Club guessing and the universal models." ''On PCF'', ed. M. Foreman, (Banff, Alberta, 2004).</ref>
* ]: the number of ] models of a ] ] in a countable ] is either finite, <math>\aleph_{0}</math>, or <math>2^{\aleph_{0}}</math>.


* Assume K is the class of models of a countable first order theory omitting countably many ]. If K has a model of cardinality <math>\aleph_{\omega_1}</math> does it have a model of cardinality continuum?<ref>{{cite journal |last=Shelah |first=Saharon |author-link=Saharon Shelah |date=1999 |title=Borel sets with large squares |journal=] |arxiv=math/9802134 |volume=159 |issue=1 |pages=1–50|bibcode=1998math......2134S |doi=10.4064/fm-159-1-1-50 |s2cid=8846429 }}</ref>
===]===
* Do the ]s have the ]?
*]
* Does a finitely presented homogeneous structure for a finite relational language have finitely many ]s?
**]
* Does there exist an ] first order theory with a trans-exponential (rapid growth) function?
***]
* If the class of atomic models of a complete first order theory is ] in the <math>\aleph_n</math>, is it categorical in every cardinal?<ref>{{cite book |last=Baldwin |first=John T. |date=July 24, 2009 |title=Categoricity |publisher=] |isbn=978-0-8218-4893-7 |url=http://www.math.uic.edu/~jbaldwin/pub/AEClec.pdf |access-date=February 20, 2014 |archive-url=https://web.archive.org/web/20100729073738/http://www.math.uic.edu/%7Ejbaldwin/pub/AEClec.pdf |archive-date=July 29, 2010 |url-status=live }}</ref><ref>{{cite arXiv |last=Shelah |first=Saharon |title=Introduction to classification theory for abstract elementary classes |year=2009 |class=math.LO |eprint=0903.3428 }}</ref>
* ]
* Is every infinite, minimal field of characteristic zero ]? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
** ] (Proof claimed in 2012, currently under review.)
* Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?<ref>Gurevich, Yuri, "Monadic Second-Order Theories," in ], ], eds., ''Model-Theoretic Logics'' (New York: Springer-Verlag, 1985), 479–506.</ref>
* ]
* Is the theory of the field of Laurent series over <math>\mathbb{Z}_p</math> ]? of the field of polynomials over <math>\mathbb{C}</math>?
* ]
* Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?<ref>Makowsky J, "Compactness, embeddings and definability," in ''Model-Theoretic Logics'', eds Barwise and Feferman, Springer 1985 pps. 645–715.</ref>
* ]

* ]
* Determine the structure of Keisler's order.<ref>{{cite journal | last1 = Keisler | first1 = HJ | year = 1967 | title = Ultraproducts which are not saturated | journal = J. Symb. Log. | volume = 32 | issue = 1| pages = 23–46 | doi=10.2307/2271240| jstor = 2271240 | s2cid = 250345806 }}</ref><ref>{{Cite arXiv |eprint=1208.2140 |class=math.LO |first1=Maryanthe |last1=Malliaris |first2=Saharon |last2=Shelah |author-link=Maryanthe Malliaris |author-link2=Saharon Shelah |title=A Dividing Line Within Simple Unstable Theories |date=10 August 2012}} {{Cite arXiv |title=A Dividing Line within Simple Unstable Theories |eprint=1208.2140 |last1=Malliaris |first1=M. |last2=Shelah |first2=S. |date=2012 |class=math.LO }}</ref>
* ]

* ]
=== Probability theory ===
* ]
{{Main|Probability theory}}
* ]
* ]
* ]

* ]
=== Number theory ===
* ]
{{Main articles|Category:Unsolved problems in number theory}}
* ]
{{See also|Number theory }}
* Do any ] exist?

* Are there infinitely many ]?
==== General ====
* Do ]s exist?
] because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them is odd.]]
* Do any odd ]s exist?
*]
* Do any ]s exist?
* ]: are there any integer solutions to <math>n! + 1 = m^{2}</math> other than <math>n = 4, 5, 7</math>?
* Is 10 a ]?
* ] on sufficiently large sequences of square numbers with constant second difference.
* ]
* ]: do all values of ] have ] greater than <math>1</math>?
* Do any ] exist for ''n''>1?
* ]: if a polynomial of degree <math>d</math> defined over a ] <math>K</math> of ] <math>0</math> has a factor in common with its first through <math>d - 1</math>-th derivative, then must <math>f</math> be the <math>d</math>-th power of a linear polynomial?
* ]: existence of integers, (''n'',''m''), such that ''n''!+1 = ''m''<sup>2</sup> other than ''n''=4, 5, 7
* ]: no ]s are infinite but non-repeating.
* ]
* ]: is there a ] of points in the plane all at rational distances from one-another?
* ]
* ]: for all <math>\epsilon > 0</math>, is the pair <math>(\epsilon, 1/2 + \epsilon)</math> an ]?
* ]
* The ]: how far can the number of integer points in a circle centered at the origin be from the area of the circle?
* ]
*]: do the nontrivial zeros of all ]s lie on the critical line <math>1/2 + it</math> with real <math>t</math>?
* ]
**]: do the nontrivial zeros of all ]s lie on the critical line <math>1/2 + it</math> with real <math>t</math>?
* ] (a corollary to ], per ])
***]: do the nontrivial zeros of the ] lie on the critical line <math>1/2 + it</math> with real <math>t</math>?
* ]: if φ(''n'') divides ''n''&nbsp;−&nbsp;1, must ''n'' be prime?
* ]: each element of a set of consecutive ]s can be assigned a distinct ] that divides it.
* Are there infinitely many ]?
* ]: for any <math>\epsilon > 0</math>, there is some constant <math>c(\epsilon)</math> such that either <math>y^{2} = x^{3}</math> or <math>|y^{2} - x^{3}| > c(\epsilon)x^{1/2 - \epsilon}</math>.
* ]
* ]: the nontrivial zeros of the ] correspond to ] of a ].
* ]: classify ]s over ]s.
* ]: find the most general ] for the ] of <math>k</math>-th order in a general ], where <math>k</math> is a power of a prime.
* ]: extend the ] on ]s of <math>\mathbb{Q}</math> to any base number field.
* Keating–Snaith conjecture concerning the asymptotics of an integral involving the ]<ref>{{citation
|last=Conrey |first=Brian |author-link=Brian Conrey
|doi=10.1090/bull/1525
|title=Lectures on the Riemann zeta function (book review)
|journal=]
|volume=53 |number=3 |pages=507–512 |year=2016|doi-access=free}}</ref>
*]: if <math>\phi(n)</math> divides <math>n - 1</math>, must <math>n</math> be prime?
* ]: a ] analogue of the ] of an ] does not vanish.
* ] that for all <math>\epsilon > 0</math>, <math>\zeta(1/2 + it) = o(t^{\epsilon})</math>
** The ] for zeroes of the Riemann zeta function
* ]: for any two real numbers <math>\alpha, \beta</math>, <math>\liminf_{n \rightarrow \infty} n\,\Vert n\alpha\Vert\,\Vert n\beta\Vert = 0</math>, where <math>\Vert x\Vert</math> is the distance from <math>x</math> to the nearest integer.
* ] that no real number <math>x</math> has the property that the fractional parts of <math>x(3/2)^{n}</math> are less than <math>1/2</math> for all positive integers <math>n</math>.
* ]: the normalized pair ] between pairs of zeros of the ] is the same as the pair correlation function of ].
* ]: a generalization of the ''abc'' conjecture to more than three integers.
** ]: for any <math>\epsilon > 0</math>, <math>\text{rad}(abc)^{1+\epsilon} < c</math> is true for only finitely many positive <math>a, b, c</math> such that <math>a + b = c</math>.
** ]: for any <math>\epsilon > 0</math>, there is some constant <math>C(\epsilon)</math> such that, for any elliptic curve <math>E</math> defined over <math>\mathbb{Q}</math> with minimal discriminant <math>\Delta</math> and conductor <math>f</math>, we have <math>|\Delta| \leq C(\epsilon) \cdot f^{6+\epsilon}</math>.
* ]: the ] satisfies any arbitrary congruence infinitely often.
* ] on bounding <math>\Delta_{k}(x) = D_{k}(x) - xP_{k}(\log(x))</math>
** ]: the specific case of the Piltz divisor problem for <math>k = 1</math>
* ]: a number of related conjectures that are generalizations of the original conjecture.
* ]: also a number of related conjectures that are generalizations of the original conjecture.
* ]: the length of the shortest ] producing <math>2^{n} - 1</math> is at most <math>n - 1</math> plus the length of the shortest addition chain producing <math>n</math>.
* Do ]s exist?
* ]: is there a finite upper bound on the multiplicities of the entries greater than 1 in ]?<ref>{{citation |last=Singmaster |first=David |title=Research Problems: How often does an integer occur as a binomial coefficient? |journal=] |volume=78 |issue=4 |pages=385–386 |year=1971 |doi=10.2307/2316907 |jstor=2316907 |mr=1536288 |author-link=David Singmaster}}.</ref>
* ] on ] of points on ] over ]s.

* Are there infinitely many ]s?
*Do any ]s exist?
*Do ]s exist?
*Do any non-power of 2 ]s exist?
*Are there 65, 66, or 67 ]s?
* Are there any pairs of ] which have opposite parity? * Are there any pairs of ] which have opposite parity?
* Are there any pairs of ] which have same parity?
* Are there any pairs of ] ]? * Are there any pairs of ] ]?
* Are there infinitely many ]?
* Are there infinitely many ]? * Are there infinitely many ]?
* Are there any pairs of ] which have same parity? * Are there infinitely many ]s?
* Does every ] with an odd denominator have an ]?
* The ] – how far can the number of integer points in a circle centered at the origin be from the area of the circle?
* Do any ]s exist?
* Is π a ] (its digits are "random")?<ref>{{cite web|url=http://www2.lbl.gov/Science-Articles/Archive/pi-random.html|title=Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key|publisher=}}</ref>
* Do any odd ]s exist?
* ]
* Do any odd ]s exist?
* ]
* Do any ] exist?
* Find value of ]
* Do any ] exist for ''n''&nbsp;>&nbsp;1?
* Which integers can be written as the sum of three ]?<ref>http://arxiv.org/pdf/1604.07746v1.pdf</ref>
* Is there a ] with odd distinct moduli?<ref>{{citation
| last1 = Guo | first1 = Song
| last2 = Sun | first2 = Zhi-Wei
| doi = 10.1016/j.aam.2005.01.004
| issue = 2
| journal = Advances in Applied Mathematics
| mr = 2152886
| pages = 182–187
| title = On odd covering systems with distinct moduli
| volume = 35
| year = 2005| arxiv = math/0412217
| s2cid = 835158
}}</ref>
* Is <math>\pi</math> a ] (i.e., is each digit 0–9 equally frequent)?<ref>{{cite web|url=http://www2.lbl.gov/Science-Articles/Archive/pi-random.html|title=Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key|access-date=2016-03-18|archive-url=https://web.archive.org/web/20160327035021/http://www2.lbl.gov/Science-Articles/Archive/pi-random.html|archive-date=2016-03-27|url-status=live}}</ref>
* Are all ] ] numbers normal?
* Is 10 a ]?
* Can a 3×3 ] be constructed from 9 distinct perfect square numbers?<ref>{{Cite journal |last=Robertson |first=John P. |date=1996-10-01 |title=Magic Squares of Squares |journal=Mathematics Magazine |volume=69 |issue=4 |pages=289–293 |doi=10.1080/0025570X.1996.11996457 |issn=0025-570X}}</ref>
* Find the value of the ].

==== Additive number theory ====
{{Main|Additive number theory }}
{{See also|Problems involving arithmetic progressions}}
* ] that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long ]s.
* ]: if <math>B</math> is an ] of order <math>2</math>, then the number of ways that positive integers <math>n</math> can be expressed as the sum of two numbers in <math>B</math> must tend to infinity as <math>n</math> tends to infinity.
* ] on consecutive applications of the unsigned ] operator to the sequence of ]s.
* ]: every even natural number greater than <math>2</math> is the sum of two ]s.
* ]: if the sum of <math>m</math> <math>k</math>-th powers of positive integers is equal to a different sum of <math>n</math> <math>k</math>-th powers of positive integers, then <math>m + n \geq k</math>.
* ]: all odd integers greater than <math>5</math> can be represented as the sum of an odd ] and an even ].
* ] of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set <math>\{1, \ldots, 2n\}</math>
* ]
* Does every nonnegative integer appear in ]?
* ]: can an algorithm determine if a ] contains a zero?
* The values of ''g''(''k'') and ''G''(''k'') in ]

* Do the ]s have a positive density?

* Determine growth rate of ''r''<sub>''k''</sub>(''N'') (see ])

==== Algebraic number theory ====
{{Main|Algebraic number theory }}
* ]: are there infinitely many ] with ]?
* ]: actually numerous conjectures, all proposed by ] and ].
* ]: a ] problem in ].
* ]
* ]: is it possible, for any natural number <math>n</math>, to assign a sequence of ]s to each ] such that the sequence for <math>x</math> is eventually ] if and only if <math>x</math> is ] of degree <math>n</math>?
* ]: primes <math>p</math> do not divide the ] of the maximal real ] of the <math>p</math>-th ].
* Lang and Trotter's conjecture on ] that the number of ] less than a constant <math>X</math> is within a constant multiple of <math>\sqrt{X}/\ln{X}</math>
* ]: the ] of the ] on ]s of ]s are at least <math>1/4</math>.
* ] (including ])

* Characterize all algebraic number fields that have some ].

====Computational number theory====
{{Main|Computational number theory}}
* Can ] be done in ]?

==== Diophantine approximation and transcendental number theory ====
{{Further|Diophantine approximation|Transcendental number theory}}
], which may or may not be a rational number.]]
* ] on the ] of certain ] of the rational numbers.<ref name="waldschmidt">{{citation |last=Waldschmidt |first=Michel |title=Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables |pages=14, 16 |year=2013 |url=https://books.google.com/books?id=Wrj0CAAAQBAJ&pg=PA14 |publisher=Springer |isbn=978-3-662-11569-5}}</ref> In particular: Are <math>\pi</math> and <math>e</math> ]? Which nontrivial combinations of ] (such as <math>e + \pi, e\pi, \pi^e, \pi^{\pi}, e^e</math>) are themselves transcendental?<ref>{{Cite conference |last=Waldschmidt |first=Michel |date=2008 |title=An introduction to irrationality and transcendence methods. |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture5.pdf |conference=2008 Arizona Winter School |archive-url=https://web.archive.org/web/20141216004531/http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture5.pdf |archive-date=16 December 2014 |access-date=15 December 2014}}</ref><ref>{{Citation |last=Albert |first=John |title=Some unsolved problems in number theory |url=http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf |access-date=15 December 2014 |archive-url=https://web.archive.org/web/20140117150133/http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf |archive-date=17 January 2014}}</ref>
* The ]: the transcendence of at least one of four exponentials of combinations of irrationals<ref name="waldschmidt" />
* Are ] <math>\gamma</math> and ] <math>G</math> irrational? Are they transcendental? Is ] <math>\zeta(3)</math> transcendental?<ref>For some background on the numbers in this problem, see articles by ] at ] (all articles accessed 22 August 2024):

*
*
*
* ({{Webarchive|url=https://web.archive.org/web/20150327024040/http://mathworld.wolfram.com/IrrationalNumber.html|date=2015-03-27}})
* ({{Webarchive|url=https://web.archive.org/web/20141113174913/http://mathworld.wolfram.com/TranscendentalNumber.html|date=2014-11-13}})
* ({{Webarchive|url=https://web.archive.org/web/20150421203736/http://mathworld.wolfram.com/IrrationalityMeasure.html|date=2015-04-21}})</ref><ref name=":1">{{Cite arXiv |last=Waldschmidt |first=Michel |date=2003-12-24 |title=Open Diophantine Problems |eprint=math/0312440 |language=en}}</ref>
* Which transcendental numbers are ]?<ref>{{Citation |last1=Kontsevich |first1=Maxim |title=Periods |date=2001 |work=Mathematics Unlimited — 2001 and Beyond |pages=771–808 |editor-last=Engquist |editor-first=Björn |url=https://link.springer.com/chapter/10.1007/978-3-642-56478-9_39 |access-date=2024-08-22 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-56478-9_39 |isbn=978-3-642-56478-9 |last2=Zagier |first2=Don |editor2-last=Schmid |editor2-first=Wilfried}}</ref>
* How well can ] irrational numbers be approximated? What is the ] of specific (suspected) transcendental numbers such as <math>\pi</math> and <math>\gamma</math>?<ref name=":1" />
* Which irrational numbers have ] terms whose ] converges to ]?<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Khinchin's Constant |url=https://mathworld.wolfram.com/KhinchinsConstant.html |access-date=2024-09-22 |website=mathworld.wolfram.com |language=en}}</ref>

==== Diophantine equations ====
{{Further|Diophantine equation}}
* ]: for all integral solutions to <math>A^{x} + B^{y} = C^{z}</math> where <math>x, y, z > 2</math>, all three numbers <math>A, B, C</math> must share some prime factor.
* ] (a corollary to ], per ]): determine precisely what rational numbers are ]s.
* Erdős–Moser problem: is <math>1^{1} + 2^{1} = 3^{1}</math> the only solution to the ]?
* ]: for every <math>n \geq 2</math>, there are positive integers <math>x, y, z</math> such that <math>4/n = 1/x + 1/y + 1/z</math>.
* ]: there are finitely many distinct solutions <math>(a^{m}, b^{n}, c^{k})</math> to the equation <math>a^{m} + b^{n} = c^{k}</math> with <math>a, b, c</math> being positive ] and <math>m, n, k</math> being positive integers satisfying <math>1/m + 1/n + 1/k < 1</math>.
* ] on solutions to <math>(x^{m} - 1)/(x - 1) = (y^{n} - 1)/(y - 1)</math> where <math>x > y > 1</math> and <math>m, n > 2</math>.
* The ]<ref>{{citation
| last = Aigner | first = Martin
| doi = 10.1007/978-3-319-00888-2
| isbn = 978-3-319-00887-5
| location = Cham
| mr = 3098784
| publisher = Springer
| title = Markov's theorem and 100 years of the uniqueness conjecture
| year = 2013}}</ref> that every ] is the largest number in exactly one normalized solution to the Markov ].
* ]: for any <math>A, B, C</math>, the equation <math>Ax^{m} - By^{n} = C</math> has finitely many solutions when <math>m, n</math> are not both <math>2</math>.
* Which integers can be written as the ]?<ref>{{Cite arXiv |eprint = 1604.07746|last1 = Huisman |first1 = Sander G.|title = Newer sums of three cubes|class = math.NT|year = 2016}}</ref>
* ]


==== Prime numbers ====
===]===
{{Main|Prime numbers}}
{{Prime number conjectures}} {{Prime number conjectures}}
] states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.]]
* ]
* ] * ] on the ]s that <math>p</math> is prime if and only if <math>pB_{p-1} \equiv -1 \pmod p</math>
* ] that given ] <math>n</math> and <math>r</math>, if <math>(X - 1)^{n} \equiv X^{n} - 1 \pmod{n, X^{r} - 1}</math>, then either <math>n</math> is prime or <math>n^{2} \equiv 1 \pmod{r}</math>
* The ] problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
* ] that if an integer is neither a perfect square nor <math>-1</math>, then it is a ] modulo infinitely many ]s <math>p</math>
* ]
* ]: there are always at least <math>4</math> ]s between consecutive squares of prime numbers, aside from <math>2^{2}</math> and <math>3^{2}</math>.
* ]
* ]: if an integer-coefficient polynomial <math>f</math> has a positive leading coefficient, is irreducible over the integers, and has no common factors over all <math>f(x)</math> where <math>x</math> is a positive integer, then <math>f(x)</math> is prime infinitely often.
* Are there infinitely many ]s?
* ]: some ] is composite and thus all Catalan–Mersenne numbers are composite after some point.
* ]: for a finite set of linear forms <math>a_{1} + b_{1}n, \ldots, a_{k} + b_{k}n</math> with each <math>b_{i} \geq 1</math>, there are infinitely many <math>n</math> for which all forms are ], unless there is some ] condition preventing it.
* Dubner's conjecture: every even number greater than <math>4208</math> is the sum of two ] which both have a ].
* ] on the distribution of ]s in ]s.
* ]: no three consecutive numbers are all ].
* ]: for all distinct ]s <math>p</math> and <math>q</math>, <math>(p^{q} - 1)/(p - 1)</math> does not divide <math>(q^{p} - 1)/(q - 1)</math>
* Fortune's conjecture that no ] is composite.
* The ] problem: is it possible to find an infinite sequence of distinct ]s such that the difference between consecutive numbers in the sequence is bounded?
* ] on the distribution of ] divisors of ].
* ]
** ]: all even ]s greater than <math>2</math> are the sum of two ]s.
** ]: for every positive integer <math>n</math>, there is a prime between <math>n^{2}</math> and <math>(n+1)^{2}</math>.
** ]: there are infinitely many ]s.
** Are there infinitely many primes of the form <math>n^{2} + 1</math>?
* Problems associated to ]
* ]: for any odd ] <math>p</math>, if any two of the three conditions <math>p = 2^{k} \pm 1</math> or <math>p = 4^{k} \pm 3</math>, <math>2^{p} - 1</math> is prime, and <math>(2^{p} + 1)/3</math> is prime are true, then the third condition is also true.
* ]: for all positive even numbers <math>n</math>, there are infinitely many ]s of size <math>n</math>.
* ] that for every finite collection <math>\{f_{1}, \ldots, f_{k}\}</math> of nonconstant ]s over the integers with positive leading coefficients, either there are infinitely many positive integers <math>n</math> for which <math>f_{1}(n), \ldots, f_{k}(n)</math> are all ], or there is some fixed divisor <math>m > 1</math> which, for all <math>n</math>, divides some <math>f_{i}(n)</math>.
* ]: is 78,557 the lowest ]?
* Does the ] hold for all natural numbers?

* Are all ]s ]?
* Are all ]s ]?
* Are all ]s of prime index ]?
* Are there any composite ''c'' satisfying 2<sup>''c'' − 1</sup> ≡ 1 (mod ''c''<sup>2</sup>)?
* Are there any ]s?
* Are there any ]s in base 47?
* Are there infinitely many ]s?
* Are there infinitely many Carol primes?
* Are there infinitely many ]s?
* Are there infinitely many ]s? * Are there infinitely many ]s?
* Are there infinitely many ]s? * Are there infinitely many ]s?
* Are there infinitely many ]s?
* Are there infinitely many ]s?
* Are there infinitely many ]s?
* Are there infinitely many Kynea primes?
* Are there infinitely many ]s?
* Are there infinitely many ]s (]); equivalently, infinitely many even ]s? * Are there infinitely many ]s (]); equivalently, infinitely many even ]s?
* Are there infinitely many ]s? * Are there infinitely many ]s?
* Are there infinitely many ]s? * Are there infinitely many ]s to every base?
* Are there infinitely many ]?
* Are there infinitely many ]s? * Are there infinitely many ]s?
* Are there infinitely many ]s?
* Are there infinitely many ]s?
* Are there infinitely many ]s, and if so is their relative density <math>e^{-1/2}</math>? * Are there infinitely many ]s, and if so is their relative density <math>e^{-1/2}</math>?
* Are there infinitely many ] primes to every base except perfect power and numbers of the form -4''k''<sup>4</sup>? * Are there infinitely many ]s?
* Are there infinitely many ]s? * Are there infinitely many ]?
* Are there infinitely many ]s? * Are there infinitely many ]s?
* Are there infinitely many ]s to every base?
* Are there infinitely many ]s?
* Are there infinitely many ]s?
* Are there infinitely many ]s?
* Are there infinitely many ]s?
* Are all ]s of prime index ]?
* Are there infinitely many ]s? * Are there infinitely many ]s?
* Are there any Wieferich primes in base 47?
* Are there any composite ''c'' satisfying 2<sup>''c'' - 1</sup> ≡ 1 (mod ''c''<sup>2</sup>)?
* For any given integer ''a'' > 0, are there infinitely many primes ''p'' such that ''a''<sup>''p'' - 1</sup> ≡ 1 (mod ''p''<sup>2</sup>)?<ref>{{cite book |last=Ribenboim |first=P. |authorlink=Paulo Ribenboim |date=2006 |title=Die Welt der Primzahlen |edition=2nd |language=German |publisher=Springer |doi=10.1007/978-3-642-18079-8 |isbn=978-3-642-18078-1 |pages=242–243 |url=http://books.google.de/books?id=XMyzh-2SClUC&lpg=PR5&dq=die%20welt%20der%20primzahlen&hl=de&pg=PA242#v=snippet&q=die%20folgenden%20probleme%20sind%20ungel%C3%B6st&f=false}}</ref>
* Can a prime ''p'' satisfy 2<sup>''p''&nbsp;−&nbsp;1</sup> ≡ 1 (mod ''p''<sup>2</sup>) and 3<sup>''p''&nbsp;−&nbsp;1</sup>&nbsp;≡&nbsp;1&nbsp;(mod&nbsp;''p''<sup>2</sup>) simultaneously?<ref>{{Citation |last=Dobson |first= J. B. |date=June 2012 |title=On Lerch's formula for the Fermat quotient |origyear= 2011 |arxiv=1103.3907|page=15}}</ref>
* Are there infinitely many ]s? * Are there infinitely many ]s?
* Are there infinitely many ]s? * Are there infinitely many ]s?
* Are there any ]s? * Are there infinitely many ]s?
* Can a prime ''p'' satisfy <math>2^{p-1}\equiv 1\pmod{p^2}</math> and <math>3^{p-1}\equiv 1\pmod{p^2}</math> simultaneously?<ref>{{cite arXiv |last=Dobson |first= J. B. |date=1 April 2017 |title=On Lerch's formula for the Fermat quotient |eprint=1103.3907v6|page=23|mode=cs2|class= math.NT }}</ref>
* Is every ] 2<sup>2<sup>''n''</sup></sup>&nbsp;+&nbsp;1 composite for <math>n > 4</math>?
* Does every prime number appear in the ]?
* Are all Fermat numbers ]?
* What is the smallest ]?
* For any given integer ''a'' which is not a square and does not equal to -1, are there infinitely many primes with ''a'' as a primitive root?
* For any given integer ''a'' > 0, are there infinitely many ]s associated with the pair (''a'', −1)? (Specially, when ''a'' = 1, this is the Fibonacci-Wieferich primes, and when ''a'' = 2, this is the Pell-Wieferich primes)
* ]
* For any given integer ''a'' > 0, are there infinitely many primes ''p'' such that ''a''<sup>''p'' − 1</sup> ≡ 1 (mod ''p''<sup>2</sup>)?<ref>{{cite book |last=Ribenboim |first=P. |author-link=Paulo Ribenboim |date=2006 |title=Die Welt der Primzahlen |edition=2nd |language=de |publisher=Springer |doi=10.1007/978-3-642-18079-8 |isbn=978-3-642-18078-1 |pages=242–243 |url=https://books.google.com/books?id=XMyzh-2SClUC&q=die+folgenden+probleme+sind+ungel%C3%B6st&pg=PA242|series=Springer-Lehrbuch }}</ref>
* Is 78,557 the lowest ] (so-called ])?
* For any given integer ''a'' which is not a square and does not equal to −1, are there infinitely many primes with ''a'' as a primitive root?
* For any given integer ''b'' which is not a perfect power and not of the form −4''k''<sup>4</sup> for integer ''k'', are there infinitely many ] primes to base ''b''?
* For any given integers <math>k\geq 1, b\geq 2, c\neq 0</math>, with {{nowrap|1=gcd(''k'', ''c'') = 1}} and {{nowrap|1=gcd(''b'', ''c'') = 1,}} are there infinitely many primes of the form <math>(k\times b^n+c)/\text{gcd}(k+c,b-1)</math> with integer ''n'' ≥ 1?
* Is every ] <math>2^{2^n} + 1</math> composite for <math>n > 4</math>?
* Is 509,203 the lowest ]? * Is 509,203 the lowest ]?
* Fortune's conjecture (that no ] is composite)
* ]
* ]
* Does every prime number appear in the ]?
* Does the ] hold for all natural numbers?
* ]
* Problems associated to ]


=== Set theory ===
===]===
{{Main|Set theory}}
* Regularity of solutions of ]
* Regularity of solutions of ]

===]===
* The values of the ], particularly <math>R(5, 5)</math>
* The values of the ]s
* ]


Note: These conjectures are about ] of ] with ], and may not be able to be expressed in models of other set theories such as the various ] or ].
===]===
* (]) Does the ] below a ] imply the ] everywhere?
* Does the ] entail ] for every ] <math>\lambda</math>?
* Does the ] imply the existence of an ]?
* If ℵ<sub>ω</sub> is a strong limit cardinal, is <math>2^{\aleph_\omega} < \aleph_{\omega_1}</math> (see ])? The best bound, ℵ<sub>ω<sub>4</sub></sub>, was obtained by ] using his ].
* The problem of finding the ultimate ], one that contains all ]. * The problem of finding the ultimate ], one that contains all ].
* ] ]: if there is a ] of ]s, then ] satisfies an analogue of ].
* If ℵ<sub>ω</sub> is a strong limit cardinal, then 2<sup>ℵ<sub>ω</sub></sup> < ℵ<sub>ω<sub>1</sub></sub> (see ]). The best bound, ℵ<sub>ω<sub>4</sub></sub>, was obtained by ] using his ].
* ] ].
* Does the ] of the existence of a ] imply the consistent existence of a ]? * Does the ] of the existence of a ] imply the consistent existence of a ]?
* (]) Does the ] below a ] imply the ] everywhere?
* Does there exist a ] on ℵ<sub>ω</sub>? * Does there exist a ] on ℵ<sub>ω</sub>?
* Is OCA (the ]) consistent with <math>2^{\aleph_{0}}>\aleph_{2}</math>?
* Without assuming the ], can a ] ''V''→''V'' exist?
* ]s: Without assuming the ], can a ] ''V''→''V'' exist?
* Does the ] entail ] for every ] <math>\lambda</math>?
* Does the ] imply the existence of an ]?


===Other=== ===Topology===
{{Main|Topology}}
{{see also|List of conjectures}}
] asks whether there is an efficient algorithm to identify when the shape presented in a ] is actually the ].]]
* ]: the ] is an ].
* ] that the only ] in the ] which admit ] ] are ]s.
* ]: every <math>n</math>-dimensional ] ] is a ].
* ]: ] ]s are determined up to ] by their ]s.
* ] on rational ]s of certain ]s.
* ]: if a ] ] has a ], ] on a ], then the group must be a ].
* Mazur's conjectures<ref>{{citation |last=Mazur |first=Barry |author-link=Barry Mazur |title=The topology of rational points |journal=] |volume=1 |number=1 |year=1992 |pages=35–45 |doi=10.1080/10586458.1992.10504244 |s2cid=17372107 |url=https://projecteuclid.org/euclid.em/1048709114 |access-date=2019-04-07 |archive-url=https://web.archive.org/web/20190407161124/https://projecteuclid.org/euclid.em/1048709114 |archive-date=2019-04-07 |url-status=live }}</ref>
* ] on the ] of certain ]s in the ]es of a ], arising from the ].
* ]s of ]s: it has been conjectured that wild knots always have infinitely many quadrisecants.<ref>{{citation
| last = Kuperberg | first = Greg | author-link = Greg Kuperberg
| arxiv = math/9712205
| doi = 10.1142/S021821659400006X
| journal = ]
| mr = 1265452
| pages = 41–50
| title = Quadrisecants of knots and links
| volume = 3
| year = 1994| s2cid = 6103528 }}</ref>
* ]: the last of ] in ] to be resolved.{{efn|A disproof has been announced, with a preprint made available on ].<ref>{{cite arXiv |last1=Burklund |first1=Robert |last2=Hahn |first2=Jeremy |last3=Levy |first3=Ishan |last4=Schlank |first4=Tomer |title=K-theoretic counterexamples to Ravenel's telescope conjecture |date=2023 |class=math.AT |eprint=2310.17459 }}</ref>}}
* ]: can ]s be recognized in ]?
* ] relating ]s of ] to the ] of their ]s.
* ]: every ] ] of a two-dimensional ] ] is aspherical.
* ]: given a finite ] two-dimensional ] <math>K</math>, is the space <math>K \times </math> ]?

== Problems solved since 1995 ==
], here illustrated with a 2D manifold, was the key tool in ]'s ].]]

===Algebra===
* ] (Vessilin Dimitrov, Ziyang Gao, and Philipp Habegger, 2020)<ref>{{cite journal
|first1=Vessilin
|last1=Dimitrov
|first2=Ziyang
|last2=Gao
|first3=Philipp
|last3=Habegger
|title=Uniformity in Mordell–Lang for curves
|journal = ]
|volume = 194
|year=2021
|pages=237–298
|doi=10.4007/annals.2021.194.1.4
|arxiv=2001.10276
|s2cid=210932420
|url=https://hal.sorbonne-universite.fr/hal-03374335/file/Dimitrov%20et%20al.%20-%202021%20-%20Uniformity%20in%20Mordell%E2%80%93Lang%20for%20curves.pdf}}
</ref>
* ] (Qi'an Guan and ], 2015) <ref>{{cite journal
| jstor=24523356
| last1=Guan
| first1=Qi'an
| last2=Zhou
| first2=Xiangyu
| author2-link=Xiangyu Zhou
| title=A solution of an <math>L^2</math> extension problem with optimal estimate and applications
| journal=Annals of Mathematics
| year=2015
| volume=181
| issue=3
| pages=1139–1208
| doi=10.4007/annals.2015.181.3.6
| s2cid=56205818
| arxiv=1310.7169}}
</ref>
* ] (], 1996)<ref>{{cite journal
| last1 = Merel
| first1 = Loïc
| year = 1996
| title = "Bornes pour la torsion des courbes elliptiques sur les corps de nombres"
| journal = Inventiones Mathematicae
| volume = 124
| issue = 1
| pages = 437–449
| doi = 10.1007/s002220050059
| mr = 1369424
| bibcode = 1996InMat.124..437M
| s2cid = 3590991 }}
</ref>
* ] (], 1995)<ref>{{citation
| last1=Cohen
| first1=Stephen D.
| last2=Fried
| first2=Michael D.
| author2-link=Michael D. Fried
| doi=10.1006/ffta.1995.1027
| issue=3
| journal=Finite Fields and Their Applications
| mr=1341953
| pages=372–375
| title=Lenstra's proof of the Carlitz–Wan conjecture on exceptional polynomials: an elementary version
| volume=1
| year=1995
| doi-access=free}}
</ref>
* ] (], 1995)

===Analysis===
* ] (], ] and ], 2013)<ref name=Casazza2006>{{cite book|last1=Casazza|first1=Peter G.|last2=Fickus|first2=Matthew|last3=Tremain|first3=Janet C.|last4=Weber|first4=Eric|editor1-last=Han|editor1-first=Deguang|editor2-last=Jorgensen|editor2-first=Palle E. T.|editor3-last=Larson|editor3-first=David Royal|contribution=The Kadison-Singer problem in mathematics and engineering: A detailed account|series=Contemporary Mathematics|date=2006|volume=414|pages=299–355|contribution-url=https://books.google.com/books?id=9b-4uqEGJdoC&pg=PA299|access-date=24 April 2015|title=Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida|publisher=American Mathematical Society.|isbn=978-0-8218-3923-2|doi=10.1090/conm/414/07820}}</ref><ref name=SIAM02.2014>{{cite news|last1=Mackenzie|first1=Dana|title=Kadison–Singer Problem Solved|url=https://www.siam.org/pdf/news/2123.pdf|access-date=24 April 2015|work=SIAM News|issue=January/February 2014|publisher=]|archive-url=https://web.archive.org/web/20141023120958/http://www.siam.org/pdf/news/2123.pdf|archive-date=23 October 2014|url-status=live}}</ref> (and the ], Anderson's paving conjectures, Weaver's discrepancy theoretic <math>KS_r</math> and <math>KS'_r</math> conjectures, Bourgain-Tzafriri conjecture and <math>R_\epsilon</math>-conjecture)
* ] (], 2004)<ref name="Agol">{{cite arXiv | eprint = math/0405568|last1 = Agol |first1 = Ian|title = Tameness of hyperbolic 3-manifolds|year = 2004}}</ref>
* ] (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)<ref>{{Cite journal
| arxiv=math/9906212
| last1=Kurdyka | first1=Krzysztof
| last2=Mostowski | first2=Tadeusz
| last3=Parusiński | first3=Adam
| title = Proof of the gradient conjecture of R. Thom
| journal=Annals of Mathematics
| pages=763–792
| volume=152
| date=2000
| issue=3
| doi=10.2307/2661354| jstor=2661354 | s2cid=119137528 }}</ref>

===Combinatorics===
* ] (Joel Moreira, Florian Richter, Donald Robertson, 2018)<ref>{{Cite journal |last1=Moreira |first1=Joel |last2=Richter |first2=Florian K. |last3=Robertson |first3=Donald |title=A proof of a sumset conjecture of Erdős |journal=] |doi=10.4007/annals.2019.189.2.4 |volume=189 |number=2 |pages=605–652 |language=en-US|year=2019 |arxiv=1803.00498 |s2cid=119158401 }}</ref>
* ] on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)<ref>{{citation|last=Stanley|first=Richard P. |editor1-last=Bisztriczky|editor1-first=T.|editor2-last=McMullen|editor2-first=P.|editor3-last=Schneider|editor3-first=R.|editor4-last=Weiss|editor4-first=A. Ivić|contribution=A survey of Eulerian posets|location=Dordrecht|mr=1322068|pages=301–333 |publisher=Kluwer Academic Publishers|series=NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences |title=Polytopes: abstract, convex and computational (Scarborough, ON, 1993)|volume=440|year=1994}}. See in particular .</ref><ref>{{cite web |last1=Kalai |first1=Gil |title=Amazing: Karim Adiprasito proved the g-conjecture for spheres! |url=https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/ |access-date=2019-02-15 |archive-url=https://web.archive.org/web/20190216031650/https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/ |archive-date=2019-02-16 |url-status=live |date=2018-12-25 }}</ref>
* ] (], 2010)<ref>{{cite journal |last=Santos |first=Franciscos |date=2012 |title=A counterexample to the Hirsch conjecture |journal=Annals of Mathematics |volume=176 |issue=1 |pages=383–412 |doi=10.4007/annals.2012.176.1.7 |arxiv=1006.2814 |s2cid=15325169 }}</ref><ref>{{cite journal |last=Ziegler |first=Günter M. |date=2012 |title=Who solved the Hirsch conjecture? |journal=Documenta Mathematica |series=Documenta Mathematica Series |volume=6 |issue=Extra Volume "Optimization Stories" |pages=75–85 |doi=10.4171/dms/6/13 |isbn=978-3-936609-58-5 | url=https://www.math.uni-bielefeld.de/documenta/vol-ismp/22_ziegler-guenter.html}}</ref>
* ] (], ], and ], 2009)<ref>{{cite journal | last1=Kauers | first1=Manuel | author1-link=Manuel Kauers | last2=Koutschan | first2=Christoph | author2-link=Christoph Koutschan | last3=Zeilberger | first3=Doron | author3-link=Doron Zeilberger | title=Proof of Ira Gessel's lattice path conjecture | journal=Proceedings of the National Academy of Sciences | volume=106 | issue=28 | date=2009-07-14 | issn=0027-8424 | doi=10.1073/pnas.0901678106 | pages=11502–11505 | pmc=2710637 | arxiv=0806.4300 | bibcode=2009PNAS..10611502K | doi-access=free }}</ref>
* ] (] and ], 2004)<ref>{{cite journal |last1=Chung |first1=Fan |last2=Greene |first2=Curtis |last3=Hutchinson |first3=Joan |date=April 2015 |title=Herbert S. Wilf (1931–2012) |journal=] |volume=62 |issue=4 |page=358 |issn=1088-9477 |oclc=34550461 |quote=The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004. |doi=10.1090/noti1247 |doi-access=free }}</ref> (and also the Alon–Friedgut conjecture)
* ] (], 2003, Carlos di Fiore, 2003)<ref>{{cite journal|title=Kemnitz' conjecture revisited | doi=10.1016/j.disc.2005.02.018 |doi-access=free| volume=297|issue=1–3 |journal=Discrete Mathematics|pages=196–201|year=2005 | last1 = Savchev | first1 = Svetoslav}}</ref>
* ] (], 2003, Alexander Sapozhenko, 2003)<ref>{{cite journal | last = Green | first = Ben | author-link = Ben J. Green | arxiv = math.NT/0304058 | doi = 10.1112/S0024609304003650 | issue = 6 | journal = The Bulletin of the London Mathematical Society | mr = 2083752 | pages = 769–778 | title = The Cameron–Erdős conjecture | volume = 36 | year = 2004| s2cid = 119615076 }}</ref><ref>{{cite web |url=https://www.ams.org/news?news_id=155 |title=News from 2007 |author=<!--Staff writer(s); no by-line.--> |date=31 December 2007 |website=American Mathematical Society |publisher=AMS |access-date=2015-11-13 |quote=The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..." |archive-url=https://web.archive.org/web/20151117030726/http://www.ams.org/news?news_id=155 |archive-date=17 November 2015 |url-status=live }}</ref>

===Dynamical systems===
* ] (Aaron Brown, David Fisher, and Sebastián Hurtado-Salazar, 2017)<ref>{{cite arXiv
| last1=Brown
| first1=Aaron
| last2=Fisher
| first2=David
| last3=Hurtado
| first3=Sebastian
| date=2017-10-07
| title=Zimmer's conjecture for actions of {{not a typo|SL(𝑚,ℤ)}}
| eprint=1710.02735
| class=math.DS}}
</ref>
* ] (Jinxin Xue, 2014)<ref name="Xue1">{{Cite arXiv|title=Noncollision Singularities in a Planar Four-body Problem|last=Xue|first=Jinxin|date=2014|class=math.DS |eprint = 1409.0048}}</ref><ref name="Xue2">{{Cite journal|title=Non-collision singularities in a planar 4-body problem|last=Xue|first=Jinxin|date=2020|journal=]|volume=224|issue=2|pages=253–388|doi=10.4310/ACTA.2020.v224.n2.a2|s2cid=226420221}}</ref>

===Game theory===
* Existence of a non-terminating game of ] (Brayden Casella, 2024)<ref>
{{cite web | url= https://richardpmann.com/beggar-my-neighbour-records.html | title= Known Historical Beggar-My-Neighbour Records |author= Richard P Mann |access-date= 2024-02-10 }}</ref>
* The ] (Various independent proofs, 2006)<ref>{{Cite web |url=http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf |title=The angel game in the plane |first=Brian H. |last=Bowditch|date=2006|location=School of Mathematics, ] |publisher=warwick.ac.uk ]|access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160304185616/http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf |archive-date=2016-03-04 |url-status=live }}</ref><ref>{{Cite web |url=http://home.broadpark.no/~oddvark/angel/Angel.pdf |title=A Solution to the Angel Problem |first=Oddvar |last=Kloster |publisher=SINTEF ICT |location=Oslo, Norway|access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160107125925/http://home.broadpark.no/~oddvark/angel/Angel.pdf |archive-date=2016-01-07 }}</ref><ref>{{Cite journal |url=http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf |title=The Angel of power 2 wins |first=Andras |last=Mathe |date=2007|journal=] |volume=16 |number=3|pages= 363–374|doi=10.1017/S0963548306008303 |doi-broken-date=1 November 2024 |s2cid=16892955 |access-date=2016-03-18 |archive-url=https://web.archive.org/web/20161013034302/http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf |archive-date=2016-10-13 |url-status=live }}</ref><ref>{{Cite web |last=Gacs |first=Peter |date=June 19, 2007 |title=THE ANGEL WINS |url=http://www.cs.bu.edu/~gacs/papers/angel.pdf |archive-url=https://web.archive.org/web/20160304030433/http://www.cs.bu.edu/~gacs/papers/angel.pdf |archive-date=2016-03-04 |access-date=2016-03-18}}</ref>

===Geometry===
====21st century====
* ] (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2024)<ref>{{Cite journal |last1=Smith |first1=David |last2=Myers |first2=Joseph Samuel |last3=Kaplan |first3=Craig S. |last4=Goodman-Strauss |first4=Chaim |date=2024 |title=An aperiodic monotile |url=https://escholarship.org/uc/item/3317z9z9 |journal=Combinatorial Theory |language=en |volume=4 |issue=1 |doi=10.5070/C64163843 |issn=2766-1334}}</ref>
* ] (Eric Larson, 2018)<ref>{{Cite arXiv| eprint=1711.04906 | last1=Larson | first1=Eric | title=The Maximal Rank Conjecture | year=2017 | class=math.AG }}</ref>
* ] (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018)<ref>{{citation
| first1=Moritz
| last1=Kerz
| first2=Florian
| last2=Strunk
| first3=Georg
| last3=Tamme
| title=Algebraic ''K''-theory and descent for blow-ups
| journal=]
| volume=211
| year=2018
| issue=2
| pages=523–577
| mr=3748313
| doi=10.1007/s00222-017-0752-2
| arxiv=1611.08466| bibcode=2018InMat.211..523K
| s2cid=253741858
}}
</ref>
* ] (], 2018)<ref>{{cite web
| url = https://www.ams.org/amsmtgs/2251_abstracts/1147-53-499.pdf
| title = Existence of infinitely many minimal hypersurfaces in closed manifolds.
| author = Song, Antoine
| work = www.ams.org
| quote = "..I will present a solution of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves.."
| access-date = 19 June 2021}}
</ref><ref>{{Cite web
| url=https://www.claymath.org/people/antoine-song
| title = Antoine Song &#124; Clay Mathematics Institute
| quote="...Building on work of Codá Marques and Neves, in 2018 Song proved Yau's conjecture in complete generality"}}
</ref>
* ] (Michaël Rao, 2017)<ref>{{citation|url=https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/|magazine=]|title=Pentagon Tiling Proof Solves Century-Old Math Problem|first=Natalie|last=Wolchover|date=July 11, 2017|access-date=July 18, 2017|archive-url=https://web.archive.org/web/20170806093353/https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/|archive-date=August 6, 2017}}</ref>
* ] (] and ], 2012)<ref>{{cite journal|last1=Marques |first1=Fernando C.|first2=André|last2=Neves|title=Min-max theory and the Willmore conjecture|journal=Annals of Mathematics |year=2013|arxiv=1202.6036|doi=10.4007/annals.2014.179.2.6|volume=179|issue=2|pages=683–782|s2cid=50742102}}</ref>
* ] (], ], 2011)<ref>{{cite journal
| arxiv=1011.4105
| last1=Guth | first1=Larry
| last2=Katz | first2=Nets Hawk
| title=On the Erdos distinct distance problem in the plane
| journal=Annals of Mathematics
| pages=155–190
| volume=181
| date=2015
| issue=1
| doi=10.4007/annals.2015.181.1.2 | doi-access=free}}</ref>
* ] (Frederick V. Henle and James M. Henle, 2008)<ref>{{Cite web |url=http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf |title=Squaring the Plane |first1=Frederick V. |last1=Henle |first2=James M. |last2=Henle |access-date=2016-03-18 |publisher=www.maa.org ]|archive-url=https://web.archive.org/web/20160324074609/http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf |archive-date=2016-03-24 |url-status=live }}</ref>
* ] (], 2004)<ref name="Agol"/>
* ] (], ], ], 2004)<ref>{{Cite journal
| arxiv=math/0412006
| last1=Brock | first1=Jeffrey F.
| last2=Canary | first2=Richard D.
| last3=Minsky | first3=Yair N. | author-link3=Yair Minsky
| title=The classification of Kleinian surface groups, II: The Ending Lamination Conjecture
| date=2012
| journal=Annals of Mathematics
| volume=176
| issue=1
| pages=1–149
| doi=10.4007/annals.2012.176.1.1 | doi-access=free}}</ref>
* ] (], ], Günter Rote, 2003)<ref>{{citation
| last1 = Connelly | first1 = Robert | author1-link = Robert Connelly
| last2 = Demaine | first2 = Erik D. | author2-link = Erik Demaine
| last3 = Rote | first3 = Günter
| doi = 10.1007/s00454-003-0006-7 | doi-access = free
| issue = 2
| journal = ]
| mr = 1931840
| pages = 205–239
| title = Straightening polygonal arcs and convexifying polygonal cycles
| url = http://page.mi.fu-berlin.de/~rote/Papers/pdf/Straightening+polygonal+arcs+and+convexifying+polygonal+cycles-DCG.pdf
| volume = 30
| year = 2003| s2cid = 40382145 }}</ref>
* ] (Carel Faber and ], 2003)<ref>{{Citation
| first1=C.
| last1=Faber
| first2=R.
| last2=Pandharipande
| author2-link=Rahul Pandharipande
| title=Hodge integrals, partition matrices, and the <math>\lambda_g</math> conjecture
| journal=Ann. of Math.
| series= 2
| volume=157
| issue=1
| pages=97–124
| year=2003
| arxiv=math.AG/9908052
| doi=10.4007/annals.2003.157.97}}
</ref>
* ] (Ivan Shestakov, Ualbai Umirbaev, 2003)<ref>{{cite journal
| last1 = Shestakov | first1 = Ivan P.
| last2 = Umirbaev | first2 = Ualbai U.
| doi = 10.1090/S0894-0347-03-00440-5
| issue = 1
| journal = Journal of the American Mathematical Society
| mr = 2015334
| pages = 197–227
| title = The tame and the wild automorphisms of polynomial rings in three variables
| volume = 17
| year = 2004}}</ref>
* ] (], ], Manuel Ritoré, Antonio Ros, 2002)<ref>{{cite journal
| last1 = Hutchings | first1 = Michael
| last2 = Morgan | first2 = Frank
| last3 = Ritoré | first3 = Manuel
| last4 = Ros | first4 = Antonio
| doi = 10.2307/3062123
| issue = 2
| journal = Annals of Mathematics
| mr = 1906593
| pages = 459–489
| series = Second Series
| title = Proof of the double bubble conjecture
| volume = 155
| year = 2002| jstor = 3062123
| arxiv = math/0406017
| hdl = 10481/32449
}}</ref>

====20th century====
* ] (], 1999)<ref>{{Cite journal
| arxiv=math/9906042
| last1=Hales | first1=Thomas C. | author-link1=Thomas Callister Hales
| title=The Honeycomb Conjecture
| journal=]
| volume=25
| pages=1–22
| date=2001
| doi=10.1007/s004540010071 | doi-access=free}}</ref>
* ] (] and Barbara Russo, 1999)<ref>{{cite journal
| last1=Teixidor i Bigas
| first1=Montserrat
| author1-link=Montserrat Teixidor i Bigas
| first2=Barbara
| last2=Russo
| title=On a conjecture of Lange
| arxiv=alg-geom/9710019
| mr=1689352
| year=1999
| journal=Journal of Algebraic Geometry
| issn=1056-3911
| volume=8
| issue=3
| pages=483–496
| bibcode=1997alg.geom.10019R }}
</ref>
* ] (], 1998, ], 1998)<ref>{{cite journal | last1 = Ullmo | first1 = E | year = 1998 | title = Positivité et Discrétion des Points Algébriques des Courbes | journal = Annals of Mathematics | volume = 147 | issue = 1| pages = 167–179 | doi = 10.2307/120987 | zbl= 0934.14013| jstor = 120987 | arxiv = alg-geom/9606017 | s2cid = 119717506 }}</ref><ref>{{cite journal | last1 = Zhang | first1 = S.-W. | year = 1998 | title = Equidistribution of small points on abelian varieties | journal = Annals of Mathematics | volume = 147 | issue = 1| pages = 159–165 | doi = 10.2307/120986 | jstor = 120986 }}</ref>
* ] (Samuel Ferguson, ], 1998)<ref>{{cite journal
| arxiv=1501.02155
| last1=Hales | first1=Thomas
| last2=Adams | first2=Mark
| last3=Bauer | first3=Gertrud
| last4=Dang | first4=Dat Tat
| last5=Harrison | first5=John
| last6=Hoang | first6=Le Truong
| last7=Kaliszyk | first7=Cezary
| last8=Magron | first8=Victor
| last9=McLaughlin | first9=Sean
| last10=Nguyen | first10=Tat Thang
| last11=Nguyen | first11=Quang Truong
| last12=Nipkow | first12=Tobias
| last13=Obua | first13=Steven
| last14=Pleso | first14=Joseph
| last15=Rute | first15=Jason
| last16=Solovyev | first16=Alexey
| last17=Ta | first17=Thi Hoai An
| last18=Tran | first18=Nam Trung
| last19=Trieu | first19=Thi Diep
| last20=Urban | first20=Josef
| last21=Ky | first21=Vu
| last22=Zumkeller | first22=Roland
| title=A formal proof of the Kepler conjecture
| journal=Forum of Mathematics, Pi
| volume=5
| date=2017
| pages=e2
| doi=10.1017/fmp.2017.1 | doi-access=free}}</ref>
* ] (], Sean McLaughlin, 1998)<ref>{{Cite journal
| arxiv=math/9811079
| last1=Hales | first1=Thomas C.
| last2=McLaughlin | first2=Sean
| title=The dodecahedral conjecture
| journal=Journal of the American Mathematical Society
| volume=23
| date=2010
| issue=2 | pages=299–344
| doi=10.1090/S0894-0347-09-00647-X | bibcode=2010JAMS...23..299H | doi-access=free}}</ref>

===Graph theory===
* ] (] and Huy Tuan Pham, 2022)<ref>{{cite arXiv |last1=Park |first1=Jinyoung |last2=Pham |first2=Huy Tuan |date=2022-03-31 |title=A Proof of the Kahn-Kalai Conjecture |class=math.CO |eprint=2203.17207 }}</ref>
* ] on the book thickness of subdivisions (], ], Robert Hickingbotham, ], and ], 2021)<ref>{{cite journal
| last1 = Dujmović | first1 = Vida | author1-link = Vida Dujmović
| last2 = Eppstein | first2 = David | author2-link = David Eppstein
| last3 = Hickingbotham | first3 = Robert
| last4 = Morin | first4 = Pat | author4-link = Pat Morin
| last5 = Wood | first5 = David R. | author5-link = David Wood (mathematician)
| arxiv = 2011.04195
| date = August 2021
| doi = 10.1007/s00493-021-4585-7
| journal = ]
| title = Stack-number is not bounded by queue-number| volume = 42 | issue = 2 | pages = 151–164 | s2cid = 226281691 }}</ref>
*] that the complete graph <math>K_{2n+1}</math> can be decomposed into <math>2n+1</math> copies of any tree with <math>n</math> edges (Richard Montgomery, ], Alexey Pokrovskiy, 2020)<ref>{{cite journal|last1=Huang |first1=C.|title=Further results on tree labellings |journal=Utilitas Mathematica |volume=21 |pages=31–48 |year=1982|mr=668845|last2=Kotzig|first2=A.|last3=Rosa|first3=A.|author2-link=Anton Kotzig}}.</ref><ref>{{Cite web |url=https://www.quantamagazine.org/mathematicians-prove-ringels-graph-theory-conjecture-20200219/|title=Rainbow Proof Shows Graphs Have Uniform Parts|last=Hartnett |first=Kevin|website=Quanta Magazine|date=19 February 2020|language=en|access-date=2020-02-29}}</ref>
*Disproof of ] on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)<ref>{{cite journal |last1=Shitov |first1=Yaroslav |date=2019-09-01 |df=dmy-all |title=Counterexamples to Hedetniemi's conjecture |journal=Annals of Mathematics |volume=190 |issue=2 |pages=663–667 |arxiv=1905.02167 |doi=10.4007/annals.2019.190.2.6 |jstor=10.4007/annals.2019.190.2.6 |mr= 3997132 |zbl=1451.05087 |s2cid=146120733 |url=https://annals.math.princeton.edu/2019/190-2/p06 |access-date=2021-07-19}}</ref>
* ] (Dawei He, Yan Wang, and Xingxing Yu, 2020)<ref>{{Cite journal
| last1=He
| first1=Dawei
| last2=Wang
| first2=Yan
| last3=Yu
| first3=Xingxing
| date=2019-12-11
| title=The Kelmans-Seymour conjecture I: Special separations
| url=http://www.sciencedirect.com/science/article/pii/S0095895619301224
| journal=Journal of Combinatorial Theory, Series B
| volume=144
| pages=197–224
| doi=10.1016/j.jctb.2019.11.008
| issn=0095-8956
| arxiv=1511.05020
| s2cid=29791394}}
</ref><ref>{{Cite journal
| last1=He
| first1=Dawei
| last2=Wang
| first2=Yan
| last3=Yu
| first3=Xingxing
| date=2019-12-11
| title=The Kelmans-Seymour conjecture II: 2-Vertices in K4−
| url=http://www.sciencedirect.com/science/article/pii/S0095895619301212
| journal=Journal of Combinatorial Theory, Series B
| volume=144
| pages=225–264
| doi=10.1016/j.jctb.2019.11.007
| issn=0095-8956
| arxiv=1602.07557| s2cid=220369443
}}
</ref><ref>{{Cite journal
| last1=He
| first1=Dawei
| last2=Wang
| first2=Yan
| last3=Yu
| first3=Xingxing
| date=2019-12-09
| title=The Kelmans-Seymour conjecture III: 3-vertices in K4−
| url=http://www.sciencedirect.com/science/article/pii/S0095895619301200
| journal=Journal of Combinatorial Theory, Series B
| volume=144
| pages=265–308
| doi=10.1016/j.jctb.2019.11.006
| issn=0095-8956
| arxiv=1609.05747
| s2cid=119625722}}
</ref><ref>{{Cite journal
| last1=He
| first1=Dawei
| last2=Wang
| first2=Yan
| last3=Yu
| first3=Xingxing
| date=2019-12-19
| title=The Kelmans-Seymour conjecture IV: A proof
| url=http://www.sciencedirect.com/science/article/pii/S0095895619301248
| journal=Journal of Combinatorial Theory, Series B
| volume=144
| pages=309–358
| doi=10.1016/j.jctb.2019.12.002
| issn=0095-8956
| arxiv=1612.07189
| s2cid=119175309}}
</ref>
* ] (Guantao Chen, Guangming Jing, and Wenan Zang, 2019)<ref>{{Cite arXiv
| last1=Zang
| first1=Wenan
| last2=Jing
| first2=Guangming
| last3=Chen
| first3=Guantao
| date=2019-01-29
| title=Proof of the Goldberg–Seymour Conjecture on Edge-Colorings of Multigraphs
| class=math.CO
| language=en
| eprint=1901.10316v1}}
</ref>
* ] (Alireza Abdollahi, Maysam Zallaghi, 2015)<ref>{{cite journal | first= Zallaghi M.|last= Abdollahi A. | year = 2015 | journal = Communications in Algebra | title = Character sums for Cayley graphs | volume = 43| issue = 12| pages = 5159–5167 | doi = 10.1080/00927872.2014.967398 |s2cid= 117651702 }}</ref>
* ] (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
* ] (Hao Huang, ], 2012)
* ] (], 2009)<ref>{{cite journal
| last=Huh
| first=June
| author-link=June Huh
| title=Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
| arxiv=1008.4749
| journal=Journal of the American Mathematical Society
| volume=25
| date=2012
| issue=3
| pages=907–927
| doi=10.1090/S0894-0347-2012-00731-0
| doi-access=free}}</ref>
* ] (Jeremie Chalopin and Daniel Gonçalves, 2009)<ref>{{cite conference
| last1 = Chalopin | first1 = Jérémie
| last2 = Gonçalves | first2 = Daniel
| editor-last = Mitzenmacher | editor-first = Michael
| contribution = Every planar graph is the intersection graph of segments in the plane: extended abstract
| doi = 10.1145/1536414.1536500
| pages = 631–638
| publisher = ACM
| title = Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009
| year = 2009}}</ref>
* ] (], Eli Berger 2007)<ref>{{Cite journal
| arxiv=math/0509397
| last1=Aharoni | first1=Ron | author1-link=Ron Aharoni
| last2=Berger | first2=Eli
| title = Menger's theorem for infinite graphs
| journal=Inventiones Mathematicae
| volume=176
| pages=1–62
| date=2009
| issue=1 | doi=10.1007/s00222-008-0157-3 | bibcode=2009InMat.176....1A | doi-access=free}}</ref>
* ] (], 2007)<ref>{{cite news |last=Seigel-Itzkovich |first=Judy |title=Russian immigrant solves math puzzle |newspaper=The Jerusalem Post |date=2008-02-08
|url=http://www.jpost.com/Home/Article.aspx?id=91431 |access-date=2015-11-12}}</ref>
* ] (], ], 2004)<ref>{{cite book |last=Diestel |first=Reinhard |year=2005 |chapter=Minors, Trees, and WQO |edition=Electronic Edition 2005 |pages=326–367 |publisher=Springer |title=Graph Theory |chapter-url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/preview/Ch12.pdf}}</ref>
* ] (], ], ] and ], 2002)<ref>{{cite journal |url=https://annals.math.princeton.edu/2006/164-1/p02 |title=The strong perfect graph theorem |last1=Chudnovsky |first1=Maria |last2=Robertson |first2=Neil |last3=Seymour |first3=Paul |last4=Thomas |first4=Robin |journal=Annals of Mathematics |year=2002 |volume=164 |pages=51–229 |arxiv=math/0212070 |doi=10.4007/annals.2006.164.51 |bibcode=2002math.....12070C |s2cid=119151552}}</ref>
* ] (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001)<ref>Klin, M. H., M. Muzychuk and R. Poschel: The isomorphism problem for circulant graphs via Schur ring theory, Codes and Association Schemes, American Math. Society, 2001.</ref>
* Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen, 1996)<ref>{{Cite journal
| url=https://www.researchgate.net/publication/220188021
| doi=10.1016/0012-365X(95)00163-Q
| doi-access=free
| title=Harary's conjectures on integral sum graphs
| journal=]
| volume=160
| issue=1–3
| pages=241–244
| year=1996
| last1=Chen
| first1=Zhibo}}
</ref>

===Group theory===
* ] (Joel Friedman, 2011, Igor Mineyev, 2011)<ref>{{Cite journal |last=Friedman |first=Joel |date=January 2015 |title=Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: with an Appendix by Warren Dicks |url=https://www.cs.ubc.ca/~jf/pubs/web_stuff/shnc_memoirs.pdf |journal=Memoirs of the American Mathematical Society |language=en |volume=233 |issue=1100 |page=0 |doi=10.1090/memo/1100 |s2cid=117941803 |issn=0065-9266}}</ref><ref>{{cite journal
| last = Mineyev | first = Igor
| doi = 10.4007/annals.2012.175.1.11
| issue = 1
| journal = Annals of Mathematics
| mr = 2874647
| pages = 393–414
| series = Second Series
| title = Submultiplicativity and the Hanna Neumann conjecture
| volume = 175
| year = 2012}}</ref>
* ] (Hossein Namazi, Juan Souto, 2010)<ref>{{Cite journal |url=https://www.researchgate.net/publication/228365532 |doi=10.1007/s11511-012-0088-0|title=Non-realizability and ending laminations: Proof of the density conjecture|journal=Acta Mathematica|volume=209|issue=2|pages=323–395|year=2012|last1=Namazi|first1=Hossein|last2=Souto|first2=Juan|doi-access=free}}</ref>
* Full ] (], ], 2008)

===Number theory===
====21st century====
*] (], Ananth Shankar, ], 2021)<ref>{{cite arXiv |last1=Pila |first1=Jonathan |last2=Shankar |first2=Ananth |last3=Tsimerman |first3=Jacob |last4=Esnault |first4=Hélène |last5=Groechenig |first5=Michael |date=2021-09-17 |title=Canonical Heights on Shimura Varieties and the André-Oort Conjecture |class=math.NT |eprint=2109.08788}}</ref>
*] (], ], 2019)
* ] (], Ciprian Demeter, ], 2015)<ref>{{cite journal|last1=Bourgain |first1=Jean|first2=Demeter|last2=Ciprian|first3=Guth|last3=Larry|title=Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three|journal=Annals of Mathematics |year=2015|doi=10.4007/annals.2016.184.2.7|volume=184|issue=2|pages=633–682|hdl=1721.1/115568|bibcode=2015arXiv151201565B|arxiv=1512.01565|s2cid=43929329}}</ref>
* ] (], 2013)<ref>{{cite arXiv |eprint=1305.2897 |title = Major arcs for Goldbach's theorem|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}</ref><ref>{{cite arXiv |eprint=1205.5252 |title = Minor arcs for Goldbach's problem |last = Helfgott|first = Harald A.|class=math.NT |year=2012}}</ref><ref>{{cite arXiv |eprint=1312.7748 |title = The ternary Goldbach conjecture is true|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}</ref>
*] (], ], ], 2013)<ref>{{Cite journal|last=Zhang|first=Yitang|date=2014-05-01|title=Bounded gaps between primes|journal=Annals of Mathematics|volume=179|issue=3|pages=1121–1174|doi=10.4007/annals.2014.179.3.7|issn=0003-486X}}</ref><ref>{{Cite web|title=Bounded gaps between primes - Polymath Wiki|url=https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes|access-date=2021-08-27|website=asone.ai|archive-date=2020-12-08|archive-url=https://web.archive.org/web/20201208045925/https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes}}</ref><ref>{{Cite journal|last=Maynard|first=James|date=2015-01-01|title=Small gaps between primes|journal=Annals of Mathematics|pages=383–413|doi=10.4007/annals.2015.181.1.7|arxiv=1311.4600|s2cid=55175056|issn=0003-486X}}</ref>
* ] (Javier Cilleruelo, ], and Carlos Vinuesa, 2010)<ref>{{cite journal|title=Generalized Sidon sets|doi=10.1016/j.aim.2010.05.010 | volume=225|issue=5|journal=]|pages=2786–2807|year=2010 | last1 = Cilleruelo | first1 = Javier|hdl=10261/31032|s2cid=7385280|doi-access=free|hdl-access=free}}</ref>
* ] (] and ], 2008)<ref>{{Citation |last1=Khare |first1=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre's modularity conjecture (I) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=485–504 |doi=10.1007/s00222-009-0205-7 |bibcode=2009InMat.178..485K |citeseerx=10.1.1.518.4611 |s2cid=14846347 }}</ref><ref>{{Citation |last1=Khare |first1=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre's modularity conjecture (II) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=505–586 |doi=10.1007/s00222-009-0206-6 |bibcode=2009InMat.178..505K |citeseerx=10.1.1.228.8022 |s2cid=189820189 }}</ref><ref>{{cite journal |author=<!--Staff writer(s); no by-line.--> |title=2011 Cole Prize in Number Theory |url=https://www.ams.org/notices/201104/rtx110400610p.pdf |journal=] |volume=58 |issue=4 |pages=610–611 |issn=1088-9477 |oclc=34550461 |access-date=2015-11-12 |archive-url=https://web.archive.org/web/20151106051835/http://www.ams.org/notices/201104/rtx110400610p.pdf |archive-date=2015-11-06 |url-status=live }}</ref>
* ] (] and ], 2004)<ref>{{cite journal |author=<!--Staff writer(s); no by-line.--> |date=May 2010 |title=Bombieri and Tao Receive King Faisal Prize |url=https://www.ams.org/notices/201005/rtx100500642p.pdf |journal=] |volume=57 |issue=5 |pages=642–643 |issn=1088-9477 |oclc=34550461 |quote=Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem. |access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160304063504/http://www.ams.org/notices/201005/rtx100500642p.pdf |archive-date=2016-03-04 |url-status=live }}</ref>
* ] (], 2002)<ref>{{cite journal |last=Metsänkylä |first=Tauno |date=5 September 2003 |title=Catalan's conjecture: another old diophantine problem solved |url=https://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf |journal=] |volume=41 |issue=1 |pages=43–57 |issn=0273-0979 |quote=The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu. |doi=10.1090/s0273-0979-03-00993-5 |access-date=13 November 2015 |archive-url=https://web.archive.org/web/20160304082755/http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf |archive-date=4 March 2016 |url-status=live }}</ref>
* ] (], 2000)<ref>{{cite book | last = Croot | first = Ernest S. III | author-link = Ernest S. Croot III | publisher = ], Athens | series = Ph.D. thesis | title = Unit Fractions | year = 2000}} {{cite journal | last = Croot | first = Ernest S. III | author-link = Ernest S. Croot III | arxiv = math.NT/0311421 | doi = 10.4007/annals.2003.157.545 | issue = 2 | journal = ] | pages = 545–556 | title = On a coloring conjecture about unit fractions | volume = 157 | year = 2003| bibcode = 2003math.....11421C | s2cid = 13514070 }}</ref>

====20th century====
* ] (], 1998)<ref>{{Citation | last1=Lafforgue | first1=Laurent | title=Chtoucas de Drinfeld et applications | language=fr | trans-title=Drinfelʹd shtukas and applications | url=http://www.math.uni-bielefeld.de/documenta/xvol-icm/07/Lafforgue.MAN.html | mr=1648105 | year=1998 | journal=Documenta Mathematica | issn=1431-0635 | volume=II | pages=563–570 | access-date=2016-03-18 | archive-url=https://web.archive.org/web/20180427200241/https://www.math.uni-bielefeld.de/documenta/xvol-icm/07/Lafforgue.MAN.html | archive-date=2018-04-27 | url-status=live }}</ref>
* ] (] and ], 1995)<ref>{{cite journal|last=Wiles|first=Andrew|author-link=Andrew Wiles|year=1995|title=Modular elliptic curves and Fermat's Last Theorem|url=http://math.stanford.edu/~lekheng/flt/wiles.pdf|journal=Annals of Mathematics|volume=141|issue=3|pages=443–551|oclc=37032255|doi=10.2307/2118559|jstor=2118559|citeseerx=10.1.1.169.9076|access-date=2016-03-06|archive-url=https://web.archive.org/web/20110510062158/http://math.stanford.edu/%7Elekheng/flt/wiles.pdf|archive-date=2011-05-10|url-status=live}}</ref><ref>{{cite journal |author=], ] |year=1995 |title=Ring theoretic properties of certain Hecke algebras |url=http://www.math.harvard.edu/~rtaylor/hecke.ps |journal=Annals of Mathematics |volume=141 |issue=3 |pages=553–572 |citeseerx=10.1.1.128.531 |doi=10.2307/2118560 |jstor=2118560 |oclc=37032255 |archive-url=https://web.archive.org/web/20000916161311/http://www.math.harvard.edu/~rtaylor/hecke.ps |archive-date=16 September 2000}}</ref>

===Ramsey theory===
* ] (Choongbum Lee, 2017)<ref>{{cite journal | last1 = Lee | first1 = Choongbum | year = 2017 | title = Ramsey numbers of degenerate graphs | journal = Annals of Mathematics | volume = 185 | issue = 3| pages = 791–829 | doi = 10.4007/annals.2017.185.3.2 | arxiv = 1505.04773 | s2cid = 7974973 }}</ref>
* ] (], Oliver Kullmann, ], 2016)<ref>{{cite journal |last=Lamb |first=Evelyn |date=26 May 2016 |title=Two-hundred-terabyte maths proof is largest ever |journal=Nature |doi=10.1038/nature.2016.19990 |volume=534 |issue=7605 |pages=17–18 |pmid=27251254 |bibcode=2016Natur.534...17L|doi-access=free }}</ref><ref>{{cite book
| last1 = Heule | first1 = Marijn J. H. | author1-link=Marijn Heule
| last2 = Kullmann | first2 = Oliver
| last3 = Marek | first3 = Victor W. | author3-link=Victor W. Marek
| editor-last1 = Creignou | editor-first1 = N.
| editor-last2 = Le Berre | editor-first2 = D.
| arxiv = 1605.00723
| chapter = Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer
| doi = 10.1007/978-3-319-40970-2_15
| mr = 3534782
| pages = 228–245
| publisher = Springer,
| series = Lecture Notes in Computer Science
| title = Theory and Applications of Satisfiability Testing – SAT 2016
| volume = 9710
| year = 2016| isbn = 978-3-319-40969-6
| s2cid = 7912943
}}</ref>

===Theoretical computer science===
*] for Boolean functions (], 2019)<ref>{{cite web |author=Linkletter, David |date=27 December 2019 |title=The 10 Biggest Math Breakthroughs of 2019 |url=https://www.popularmechanics.com/science/math/g30346822/biggest-math-breakthroughs-2019/ |access-date=20 June 2021 |work=]}}</ref>

===Topology===
*Deciding whether the ] is a ] (], 2020)<ref>{{Cite journal |last=Piccirillo |first=Lisa |date=2020 |title=The Conway knot is not slice |url=https://annals.math.princeton.edu/2020/191-2/p05 |journal=] |volume=191 |issue=2 |pages=581–591 |doi=10.4007/annals.2020.191.2.5|s2cid=52398890 }}</ref><ref>{{Cite web |last=Klarreich |first=Erica |author-link=Erica Klarreich |date=2020-05-19 |title=Graduate Student Solves Decades-Old Conway Knot Problem |url=https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/ |access-date=2022-08-17 |website=] |language=en}}</ref>
* ] (], Daniel Groves, Jason Manning, 2012)<ref>{{Cite journal
| arxiv = 1204.2810v1
| last1 = Agol | first1 = Ian
| title = The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves, and Jason Manning)
| journal=Documenta Mathematica
| volume=18
| date=2013
| pages=1045–1087
| doi = 10.4171/dm/421 | s2cid = 255586740 | url=https://www.math.uni-bielefeld.de/documenta/vol-18/33.pdf}}</ref> (and by work of ] also ])
* ] (], 2012)<ref>{{Cite journal
| arxiv=1203.6597
| last1 = Brendle | first1 = Simon | author1-link=Simon Brendle
| title = Embedded minimal tori in <math>S^3</math> and the Lawson conjecture
| journal=Acta Mathematica
| volume=211
| issue=2
| pages=177–190
| date=2013
| doi=10.1007/s11511-013-0101-2 | doi-access=free}}</ref>
* ] (], ], 2011)<ref>{{Cite journal
| arxiv=1101.1330
| last1=Kahn | first1=Jeremy | author1-link=Jeremy Kahn
| last2=Markovic | first2=Vladimir | author2-link=Vladimir Markovic
| title=The good pants homology and the Ehrenpreis conjecture
| journal=Annals of Mathematics
| pages=1–72
| volume=182
| date=2015
| issue=1
| doi=10.4007/annals.2015.182.1.1 | doi-access=free}}</ref>
* ] for groups with finite subgroups of unbounded order (Austin, 2009)<ref>{{cite journal
| arxiv = 0909.2360
| last1 = Austin |first1 = Tim
| title = Rational group ring elements with kernels having irrational dimension
| journal = Proceedings of the London Mathematical Society
| volume = 107
| issue = 6
| pages = 1424–1448
| date = December 2013
| doi = 10.1112/plms/pdt029 | bibcode = 2009arXiv0909.2360A|s2cid = 115160094}}</ref>
* ] (], 2008)<ref>{{cite journal | last1 = Lurie | first1 = Jacob | year = 2009 | title = On the classification of topological field theories | journal = Current Developments in Mathematics | volume = 2008 | pages = 129–280 | doi=10.4310/cdm.2008.v2008.n1.a3| bibcode = 2009arXiv0905.0465L | arxiv = 0905.0465 | s2cid = 115162503 }}</ref>
* ] (], 2006)
* ] (], 2002)<ref name="auto">{{cite press release | publisher=] | date=March 18, 2010 | title=Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman | url=http://www.claymath.org/sites/default/files/millenniumprizefull.pdf | access-date=November 13, 2015 | quote=The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. | archive-url=https://web.archive.org/web/20100322192115/http://www.claymath.org/poincare/ | archive-date=March 22, 2010 | url-status=live }}</ref>
* ], (],<ref name="auto" /> series of preprints in 2002–2003)<ref>{{Cite arXiv |eprint = 0809.4040|last1 = Morgan |first1 = John |title = Completion of the Proof of the Geometrization Conjecture|last2 = Tian|first2 = Gang|class = math.DG|year = 2008}}</ref>
* ] (], 1999)<ref>{{cite journal
| first1=M.E. | last1=Rudin | author-link1=Mary Ellen Rudin
| title=Nikiel's Conjecture
| journal=Topology and Its Applications
| volume=116
| year=2001
| issue=3 | pages=305–331
| doi=10.1016/S0166-8641(01)00218-8 | doi-access=free}}</ref>
* Disproof of the ] (Iwase, 1997)<ref>{{cite web|url=https://www.researchgate.net/publication/220032558|title=Ganea's Conjecture on Lusternik-Schnirelmann Category|author=Norio Iwase|date=1 November 1998|work=ResearchGate}}</ref>

===Uncategorised===
====2010s====
* ] (], 2015)<ref>{{Cite arXiv |eprint = 1509.05363v5|last1 = Tao|first1 = Terence | author-link1=Terence Tao|title = The Erdős discrepancy problem|class = math.CO|year = 2015}}</ref>
* ] conjecture (John F. R. Duncan, Michael J. Griffin, ], 2015)<ref>{{cite journal|title=Proof of the umbral moonshine conjecture|first1=John F. R.|last1=Duncan|first2=Michael J.|last2=Griffin|first3=Ken|last3=Ono|date=1 December 2015|journal=Research in the Mathematical Sciences|volume=2|issue=1|page=26|doi=10.1186/s40687-015-0044-7|bibcode=2015arXiv150301472D|arxiv=1503.01472|s2cid=43589605 |doi-access=free }}</ref>
* Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (], Aaron Naber, 2014)<ref>{{cite journal
| arxiv=1406.6534
| last1=Cheeger | first1=Jeff
| last2=Naber | first2=Aaron
| title=Regularity of Einstein Manifolds and the Codimension 4 Conjecture
| journal=Annals of Mathematics
| pages=1093–1165
| volume=182
| issue=3
| date=2015
| doi=10.4007/annals.2015.182.3.5 | doi-access=free}}</ref>
* ] (], 2014)<ref>{{Cite magazine |last=Wolchover |first=Natalie |date=March 28, 2017 |title=A Long-Sought Proof, Found and Almost Lost |url=https://www.quantamagazine.org/20170328-statistician-proves-gaussian-correlation-inequality/ |url-status=live |magazine=] |archive-url=https://web.archive.org/web/20170424133433/https://www.quantamagazine.org/20170328-statistician-proves-gaussian-correlation-inequality/ |archive-date=April 24, 2017 |access-date=May 2, 2017}}</ref>
* Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman, ], 2011)<ref>{{Cite arXiv |eprint = 1104.2922|last1=Newman |first1=Alantha | last2=Nikolov | first2=Aleksandar |title = A counterexample to Beck's conjecture on the discrepancy of three permutations|class = cs.DM|year = 2011}}</ref>
* ] (], 2011)<ref>{{Cite web |url=https://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf |title=On motivic cohomology with Z/''l''-coefficients |last=Voevodsky |first=Vladimir |access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160327035457/http://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf |location=Princeton, NJ |website=annals.math.princeton.edu |publisher=] |date=1 July 2011|volume=174|issue=1|pages=401–438|archive-date=2016-03-27 |url-status=live }}</ref> (and ] and by work of ] and ] (2001) also ]<ref>{{cite journal
| last1 = Geisser | first1 = Thomas
| last2 = Levine | first2 = Marc
| doi = 10.1515/crll.2001.006
| journal = Journal für die Reine und Angewandte Mathematik
| mr = 1807268
| pages = 55–103
| title = The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky
| volume = 2001
| year = 2001| issue = 530
}}</ref><ref>{{cite web |last=Kahn |first=Bruno |title=Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry |url=https://webusers.imj-prg.fr/~bruno.kahn/preprints/kcag.pdf |url-status=live |archive-url=https://web.archive.org/web/20160327035553/https://webusers.imj-prg.fr/~bruno.kahn/preprints/kcag.pdf |archive-date=2016-03-27 |access-date=2016-03-18 |website=webusers.imj-prg.fr}}</ref>{{Rp|page=359}}<ref>{{cite web|url=https://mathoverflow.net/q/87162 |title=motivic cohomology – Milnor–Bloch–Kato conjecture implies the Beilinson-Lichtenbaum conjecture – MathOverflow|access-date=2016-03-18 }}</ref>)

====2000s====
* ] (Thomas Mattman, Pablo Solis, 2009)<ref>{{Cite journal
| arxiv = 0906.1612
| last1 = Mattman |first1 = Thomas W.
| last2 = Solis | first2 = Pablo
| title = A proof of the Kauffman-Harary Conjecture
| journal = Algebraic & Geometric Topology
| volume = 9
| issue = 4
| pages = 2027–2039
| year = 2009
| doi = 10.2140/agt.2009.9.2027 | bibcode = 2009arXiv0906.1612M | s2cid = 8447495}}</ref>
* ] (], ], 2009)<ref>{{cite journal
| arxiv=0910.5501
| last1 = Kahn | first1 = Jeremy
| last2 = Markovic | first2 = Vladimir
| title = Immersing almost geodesic surfaces in a closed hyperbolic three manifold
| journal = Annals of Mathematics
| pages=1127–1190
| volume=175
| issue=3
| year=2012
| doi=10.4007/annals.2012.175.3.4 | doi-access=free}}</ref>
* ] and the ] (Zhiqin Lu, 2007)<ref>{{cite journal
| first=Zhiqin | last=Lu
| orig-date=2007
| title=Normal Scalar Curvature Conjecture and its applications
| arxiv=0711.3510
| journal=Journal of Functional Analysis
| volume=261
| issue=5
| date=September 2011
| pages=1284–1308
| doi=10.1016/j.jfa.2011.05.002 | doi-access=free}}</ref>
* ] (], 2005)<ref>{{citation |last=Dencker |first=Nils |author-link=Nils Dencker |title=The resolution of the Nirenberg–Treves conjecture |journal=] |volume=163 |issue=2 |year=2006 |pages=405–444 |url=https://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf |doi=10.4007/annals.2006.163.405 |s2cid=16630732 |access-date=2019-04-07 |archive-url=https://web.archive.org/web/20180720145723/http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf |archive-date=2018-07-20 |url-status=live }}</ref><ref>{{cite web |url=https://www.claymath.org/research |title=Research Awards |website=] |access-date=2019-04-07 |archive-url=https://web.archive.org/web/20190407160116/https://www.claymath.org/research |archive-date=2019-04-07 |url-status=live }}</ref>
* ] (], ], Motakuri Ramana, 2005)<ref>{{cite journal
| last1 = Lewis | first1 = A. S.
| last2 = Parrilo | first2 = P. A.
| last3 = Ramana | first3 = M. V.
| doi = 10.1090/S0002-9939-05-07752-X
| issue = 9
| journal = Proceedings of the American Mathematical Society
| mr = 2146191
| pages = 2495–2499
| title = The Lax conjecture is true
| volume = 133
| year = 2005| s2cid = 17436983
}}</ref>
* The ] (] and ], 2004)<ref>{{cite web |url=http://www.icm2010.in/prize-winners-2010/fields-medal-ngo-bao-chau |title=Fields Medal – Ngô Bảo Châu |author=<!--Staff writer(s); no by-line.--> |date=19 August 2010 |website=International Congress of Mathematicians 2010 |publisher=ICM |access-date=2015-11-12 |quote=Ngô Bảo Châu is being awarded the 2010 Fields Medal for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods. |archive-url=https://web.archive.org/web/20150924032610/http://www.icm2010.in/prize-winners-2010/fields-medal-ngo-bao-chau |archive-date=24 September 2015 |url-status=live }}</ref>
* ] (], 2003)<ref>{{cite journal |title=Reduced power operations in motivic cohomology |pages=1–57|journal=Publications Mathématiques de l'IHÉS |volume=98 |year=2003 |last1=Voevodsky |first1=Vladimir |doi=10.1007/s10240-003-0009-z |citeseerx=10.1.1.170.4427 |url=http://archive.numdam.org/item/PMIHES_2003__98__1_0/ |access-date=2016-03-18 |url-status=live |archive-url=https://web.archive.org/web/20170728114725/http://archive.numdam.org/item/PMIHES_2003__98__1_0 |archive-date=2017-07-28 |arxiv=math/0107109 |s2cid=8172797}}</ref>
* ] (Ehud Baruch, 2003)<ref>{{cite journal
| last = Baruch | first = Ehud Moshe
| doi = 10.4007/annals.2003.158.207
| issue = 1
| journal = Annals of Mathematics
| mr = 1999922
| pages = 207–252
| series = Second Series
| title = A proof of Kirillov's conjecture
| volume = 158
| year = 2003}}</ref>
* ] (Bertrand Haas, 2002)<ref>{{Cite journal |last=Haas |first=Bertrand |date=2002 |title=A Simple Counterexample to Kouchnirenko's Conjecture |url=https://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf |url-status=live |journal=Beiträge zur Algebra und Geometrie |volume=43 |issue=1 |pages=1–8 |archive-url=https://web.archive.org/web/20161007091417/http://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf |archive-date=2016-10-07 |access-date=2016-03-18}}</ref>
* ] (], 2001)<ref>{{cite journal
| last = Haiman | first = Mark
| doi = 10.1090/S0894-0347-01-00373-3
| issue = 4
| journal = Journal of the American Mathematical Society
| mr = 1839919
| pages = 941–1006
| title = Hilbert schemes, polygraphs and the Macdonald positivity conjecture
| volume = 14
| year = 2001| s2cid = 9253880
}}</ref> (and also ])
* ] (], ], ], ], and Philipp Tchamitchian, 2001)<ref>{{cite journal
| last1 = Auscher | first1 = Pascal
| last2 = Hofmann | first2 = Steve
| last3 = Lacey | first3 = Michael
| last4 = McIntosh | first4 = Alan
| last5 = Tchamitchian | first5 = Ph.
| doi = 10.2307/3597201
| issue = 2
| journal = Annals of Mathematics
| mr = 1933726
| pages = 633–654
| series = Second Series
| title = The solution of the Kato square root problem for second order elliptic operators on <math>\mathbb{R}^n</math>
| volume = 156
| year = 2002| jstor = 3597201
}}</ref>
* ] (Luca Barbieri-Viale, Andreas Rosenschon, ], 2001)<ref>{{cite journal
| arxiv=math/0102150
| last1=Barbieri-Viale |first1=Luca
| last2=Rosenschon | first2=Andreas
| last3=Saito | first3=Morihiko
| title = Deligne's Conjecture on 1-Motives
| journal=Annals of Mathematics
| pages=593–633
| volume=158
| date=2003
| issue=2
| doi=10.4007/annals.2003.158.593 | doi-access=free}}</ref>
* ] (], ], ], and ], 2001)<ref>{{Citation | last1=Breuil | first1=Christophe | last2=Conrad | first2=Brian | last3=Diamond | first3=Fred | last4=Taylor | first4=Richard | title=On the modularity of elliptic curves over '''Q''': wild 3-adic exercises | doi=10.1090/S0894-0347-01-00370-8 | mr=1839918 | year=2001 | journal=] | issn=0894-0347 | volume=14 | issue=4 | pages=843–939| doi-access=free }}</ref>
* ] (], 2001)<ref>{{Cite journal|url=https://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf|doi=10.1090/s0025-5718-00-01178-9|title=On a conjecture of Erdős and Stewart|journal=Mathematics of Computation|volume=70|issue=234|pages=893–897|year=2000|last1=Luca|first1=Florian|access-date=2016-03-18|archive-url=https://web.archive.org/web/20160402030443/http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf|archive-date=2016-04-02|url-status=live|bibcode=2001MaCom..70..893L}}</ref>
* ] (], 2000)<ref>{{cite book
| last = Atiyah | first = Michael | author-link = Michael Atiyah
| editor-last = Yau | editor-first = Shing-Tung | editor-link = Shing-Tung Yau
| contribution = The geometry of classical particles
| doi = 10.4310/SDG.2002.v7.n1.a1
| mr = 1919420
| pages = 1–15
| publisher = International Press | location = Somerville, Massachusetts
| series = Surveys in Differential Geometry
| title = Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer
| volume = 7
| year = 2000}}
</ref>

== See also ==
* ]
* ] * ]
* ] * ]
* ] * ]
* ]
* ]
* ] * '']''
* '']''
* ]
*]
* ]
* ]
* ]
* ]
* ]
* Prove ] for all unique ]
* ]
* Assorted ] problems, e.g. the densest irregular hypersphere packings
* ]: Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.<ref>{{citation
| last = Barros | first = Manuel
| jstor = 2162098
| journal = American Mathematical Society
| pages = 1503–1509
| title = General Helices and a Theorem of Lancret
| volume = 125
| year = 1997}}</ref>
* ]
* ]
* ]
* ] (open since 1911)
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]
* ]


==Notes==
==Problems solved since 1995==
{{notelist}}
* ] (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)<ref>http://arxiv.org/pdf/1605.00723v1.pdf</ref>
* ] (], Ciprian Demeter, ], 2015)<ref>{{cite journal|last=Bourgain |first=Jean|first2=Demeter|last2=Ciprian|first3=Guth|last3=Larry|title=Proof of the main conjecture in Vinogradov’s Mean Value Theorem for degrees higher than three|journal=Annals of Mathematics |year=2015}}</ref>
* ] (], 2015)<ref>http://arxiv.org/pdf/1509.05363v5.pdf</ref>
* ] (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)<ref></ref>
* ] (Cheeger, Naber, 2014)<ref>http://arxiv.org/pdf/1406.6534v10.pdf</ref>
* ] (], 2013)<ref>{{cite arXiv |eprint=1305.2897 |title = Major arcs for Goldbach's theorem|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}</ref><ref>{{cite arXiv |eprint=1205.5252 |title = Minor arcs for Goldbach's problem |last = Helfgott|first = Harald A.|class=math.NT |year=2012}}</ref><ref>{{cite arXiv |eprint=1312.7748 |title = The ternary Goldbach conjecture is true|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}</ref>
* ] (], ] and ], 2013)<ref name=Casazza2006>{{cite journal|last1=Casazza|first1=Peter G.|last2=Fickus|first2=Matthew|last3=Tremain|first3=Janet C.|last4=Weber|first4=Eric|editor1-last=Han|editor1-first=Deguang|editor2-last=Jorgensen|editor2-first=Palle E. T.|editor3-last=Larson|editor3-first=David Royal|title=The Kadison-Singer problem in mathematics and engineering: A detailed account|journal=Contemporary Mathematics|date=2006|volume=414|pages=299–355|url=http://books.google.com/books?id=9b-4uqEGJdoC&pg=PA299|accessdate=24 April 2015|series=Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida|publisher=American Mathematical Society.|isbn=978-0-8218-3923-2|doi=10.1090/conm/414/07820}}</ref><ref name=SIAM02.2014>{{cite news|last1=Mackenzie|first1=Dana|title=Kadison–Singer Problem Solved|url=https://www.siam.org/pdf/news/2123.pdf|accessdate=24 April 2015|work=SIAM News|issue=January/February 2014|publisher=]}}</ref> (and the ], Anderson’s paving conjectures, Weaver’s discrepancy theoretic <math>KS_r</math> and <math>KS'_r</math> conjectures, Bourgain-Tzafriri conjecture and <math>R_\epsilon</math>-conjecture)
* ] (Agol, Groves, Manning, 2012)<ref>http://arxiv.org/pdf/1204.2810v1.pdf</ref> (and by work of Wise also ])
* ] (Brendle, 2012)<ref>http://www.math.jhu.edu/~js/Math646/brendle.lawson.pdf</ref>
* ] (] and ], 2012)<ref>{{cite journal|last=Marques |first=Fernando C.|first2=André|last2=Neves|title=Min-max theory and the Willmore conjecture|journal=Annals of Mathematics |year=2013|id=|arxiv=1202.6036|doi=10.4007/annals.2014.179.2.6}}</ref>
* ] (Kahn, Markovic, 2011)<ref>http://arxiv.org/pdf/1101.1330v4.pdf</ref>
* ] (Mineyev, 2011)<ref>http://www.math.uiuc.edu/~mineyev/math/art/submult-shnc.pdf</ref>
* ] (Voevodsky, 2011)<ref>http://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf</ref> (and ] and by work of Geisser and Levine (2001) also ]<ref>https://www.uni-due.de/~bm0032/publ/BlochKato.pdf</ref><ref></ref><ref>{{cite web|url=http://mathoverflow.net/questions/87162/milnor-bloch-kato-conjecture-implies-the-beilinson-lichtenbaum-conjecture|title=motivic cohomology - Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture - MathOverflow|publisher=}}</ref>)
* ] (Larry Guth, Netz Hawk Katz, 2011)<ref>http://arxiv.org/pdf/1011.4105v3.pdf</ref>
* ] (Namazi, Souto, 2010)<ref>https://www.researchgate.net/profile/Juan_Souto3/publication/228365532_Non-realizability_and_ending_laminations_Proof_of_the_Density_Conjecture/links/541d85a10cf2218008d1d2e5.pdf</ref>
* ] (], 2010)<ref>{{cite journal |last=Santos |first=Franciscos |date=2012 |title=A counterexample to the Hirsch conjecture |journal=Annals of Mathematics |volume=176 |issue=1 |pages=383–412 |publisher=Princeton University and Institute for Advanced Study |doi=10.4007/annals.2012.176.1.7 |url=http://annals.math.princeton.edu/2012/176-1/p07}}</ref><ref>{{cite journal |last=Ziegler |first=Günter M. |date=2012 |title=Who solved the Hirsch conjecture? |journal=Documenta Mathematica |volume=Extra Volume "Optimization Stories" |pages=75–85 |url=http://www.math.uiuc.edu/documenta/vol-ismp/22_ziegler-guenter.html}}</ref>
* ] (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)<ref>{{cite web|url=http://www.sciencedirect.com/science/article/pii/S0001870810001945|title=Generalized Sidon sets|publisher=}}</ref>
* ] (Austin, 2009)<ref>http://arxiv.org/pdf/0909.2360v3.pdf</ref>
* ] (Matmann, Solis, 2009)<ref>http://arxiv.org/pdf/0906.1612v2.pdf</ref>
* ] (Kahn, Markovic, 2009)<ref>http://arxiv.org/pdf/0910.5501v5.pdf</ref>
* ] (Jeremie Chalopin and Daniel Gonçalves, 2009)<ref>http://www.csie.ntu.edu.tw/~hil/bib/ChalopinG09.pdf</ref>
* ] (proof was completed by Morgan and Tian in 2008<ref>http://arxiv.org/pdf/0809.4040.pdf</ref> and it is based mostly on work of ], 2002)<ref name="auto">{{cite press release|publisher=]| date=March 18, 2010 | format=PDF | title = Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman|url = http://www.claymath.org/sites/default/files/millenniumprizefull.pdf | accessdate=November 13, 2015 | quote = The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.}}</ref>
* ] (] and ], 2008)<ref>{{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre’s modularity conjecture (I) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=485–504 |doi=10.1007/s00222-009-0205-7 }}</ref><ref>{{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre’s modularity conjecture (II) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=505–586 |doi=10.1007/s00222-009-0206-6 }}</ref><ref>{{cite journal |author=<!--Staff writer(s); no by-line.--> |title=2011 Cole Prize in Number Theory |url=http://www.ams.org/notices/201104/rtx110400610p.pdf |format=PDF |journal=] |location=Providence, Rhode Island, United States |publisher=American Mathematical Society |volume=58 |issue=4 |pages=610–611 |issn=1088-9477 |oclc=34550461}}</ref>
* ] (Frederick V. Henle and James M. Henle, 2008)<ref>http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf</ref>
* ] (Aharoni, Berger 2007)<ref>http://arxiv.org/pdf/math/0509397.pdf</ref>
* ] (], 2007)<ref>{{cite news | last =Seigel-Itzkovich | first =Judy | title =Russian immigrant solves math puzzle | pages = | publisher =The Jerusalem Post | date = 2008-02-08 |
url =http://www.jpost.com/Home/Article.aspx?id=91431 | accessdate = 2015-11-12}}</ref>
* The ] (Various independent proofs, 2006)<ref>http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf</ref><ref>http://home.broadpark.no/~oddvark/angel/Angel.pdf</ref><ref>http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf</ref><ref>http://www.cs.bu.edu/~gacs/papers/angel.pdf</ref>
* ] (Lewis, Parrilo, Ramana, 2005)<ref>http://www.ams.org/journals/proc/2005-133-09/S0002-9939-05-07752-X/S0002-9939-05-07752-X.pdf</ref>
* The ] (] and ], 2004)<ref>{{cite web |url=http://www.icm2010.in/prize-winners-2010/fields-medal-ngo-bao-chau |title=Fields Medal – Ngô Bảo Châu |author=<!--Staff writer(s); no by-line.--> |date=19 August 2010 |website=International Congress of Mathematicians 2010 |publisher=ICM |access-date=2015-11-12 |quote=Ngô Bảo Châu is being awarded the 2010 Fields Medal for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods.}}</ref>
* ] and ] (], 2004)<ref>http://arxiv.org/pdf/math/0405568v1.pdf</ref>
* ] (Robertson, Seymour, 2004)<ref>{{cite web|url=http://www.flooved.com/reader/3447#348|title=Graph Theory|publisher=}}</ref>
* ] (] and ], 2004)<ref>{{cite journal |last1=Chung |first1=Fan|last2=Greene |first2=Curtis|last3=Hutchinson |first3=Joan |date=April 2015 |title=Herbert S. Wilf (1931–2012) |url=http://www.ams.org/notices/201504/rnoti-p346.pdf |format=PDF |journal=] |location=Providence, Rhode Island, United States |publisher=American Mathematical Society |volume=62 |issue=4 |page=358 |issn=1088-9477 |oclc=34550461 |quote=The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004.}}</ref> (and also ])
* ] (] and ], 2004)<ref>{{cite journal |author=<!--Staff writer(s); no by-line.--> |date=May 2010 |title=Bombieri and Tao Receive King Faisal Prize |url=http://www.ams.org/notices/201005/rtx100500642p.pdf |format=PDF |journal=] |location=Providence, Rhode Island, United States |publisher=American Mathematical Society |volume=57 |issue=5 |pages=642–643 |issn=1088-9477 |oclc=34550461 |quote=Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem.}}</ref>
* ] (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)<ref>http://arxiv.org/pdf/math/0412006v2.pdf</ref>
* ] (], 2003, Alexander Sapozhenko, 2003)<ref>{{citation | last = Green | first = Ben | author-link = Ben J. Green | arxiv = math.NT/0304058 | doi = 10.1112/S0024609304003650
| issue = 6 | journal = The Bulletin of the London Mathematical Society | mr = 2083752 | pages = 769–778 | title = The Cameron–Erdős conjecture | volume = 36 | year = 2004}}</ref><ref>{{cite web |url=http://www.ams.org/news?news_id=155 |title=News from 2007 |author=<!--Staff writer(s); no by-line.--> |date=31 December 2007 |website=American Mathematical Society |publisher=AMS |access-date=2015-11-13 |quote=The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..."}}</ref>
* ] (], 2003)<ref>http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_2003__98_/PMIHES_2003__98__1_0/PMIHES_2003__98__1_0.pdf</ref>
* ] (Reiher, 2003, di Fiore, 2003)<ref>{{cite web|url=http://www.sciencedirect.com/science/article/pii/S0012365X05002281|title=Kemnitz’ conjecture revisited|publisher=}}</ref>
* ] (Shestakov, Umirbaev, 2003)<ref>http://www.ams.org/journals/jams/2004-17-01/S0894-0347-03-00440-5/S0894-0347-03-00440-5.pdf</ref>
* ] (Baruch, 2003)<ref>http://annals.math.princeton.edu/wp-content/uploads/annals-v158-n1-p04.pdf</ref>
* ] (], 2002)<ref name="auto"/>
* ] (], ], ] and ], 2002)<ref>{{cite web|url=http://annals.math.princeton.edu/2006/164-1/p02|title=The strong perfect graph theorem|publisher=}}</ref>
* ] (Haas, 2002)<ref>http://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf</ref>
* ] (Knight, 2002)<ref>Knight, R. W. (2002), The Vaught Conjecture: A Counterexample, manuscript</ref>
* ] (Hutchings, Morgan, Ritoré, Ros, 2002)<ref>http://www.ugr.es/~ritore/preprints/0406017.pdf</ref>
* ] (], 2002)<ref>{{cite journal |last=Metsänkylä |first=Tauno |date=5 September 2003 |title=Catalan's conjecture: another old diophantine problem solved |url=http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf |format=PDF |journal=] |publisher=American Mathematical Society |volume=41 |issue=1 |pages=43–57 |issn=0273-0979 |quote=The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu.}}</ref>
* ] (Haiman, 2001)<ref>http://www.ams.org/journals/jams/2001-14-04/S0894-0347-01-00373-3/S0894-0347-01-00373-3.pdf</ref> (and also ])
* ] (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)<ref>http://junon.u-3mrs.fr/monniaux/AHLMT02.pdf</ref>
* ] (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)<ref>http://arxiv.org/pdf/math/0102150v4.pdf</ref>
* ] (Breuil, Conrad, Diamond and Taylor, 2001)<ref>{{Citation | last1=Breuil | first1=Christophe | last2=Conrad | first2=Brian | last3=Diamond | first3=Fred | last4=Taylor | first4=Richard | title=On the modularity of elliptic curves over '''Q''': wild 3-adic exercises | doi=10.1090/S0894-0347-01-00370-8 | mr=1839918 | year=2001 | journal=] | issn=0894-0347 | volume=14 | issue=4 | pages=843–939}}</ref>
* ] (Florian Luca, 2001)<ref>http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf</ref>
* ] (Atiyah, 2000)<ref>http://intlpress.com/site/pub/files/_fulltext/journals/sdg/2002/0007/0001/SDG-2002-0007-0001-a001.pdf</ref>
* ] (Croot, 2000)<ref>{{citation | last = Croot | first = Ernest S., III | author-link = Ernest S. Croot III | publisher = ], Athens | series = Ph.D. thesis | title = Unit Fractions | year = 2000}}. {{citation | last = Croot | first = Ernest S., III | author-link = Ernest S. Croot III | arxiv = math.NT/0311421 | doi = 10.4007/annals.2003.157.545 | issue = 2 | journal = ] | pages = 545–556 | title = On a coloring conjecture about unit fractions | volume = 157 | year = 2003}}</ref>
* ] (Thomas Hales, 1999)<ref>http://arxiv.org/pdf/math/9906042v2.pdf</ref>
* ] (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)<ref>http://arxiv.org/pdf/math/9906212v2.pdf</ref>
* ] (], 1998, ], 1998)<ref>Ullmo, E. (1998), "Positivité et Discrétion des Points Algébriques des Courbes", Annals of Mathematics 147 (1): 167–179, doi:10.2307/120987, Zbl 0934.14013</ref><ref>Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics 147 (1): 159–165, doi:10.2307/120986</ref>
* ] (Laurent Lafforgue, 1998)<ref>{{Citation | last1=Lafforgue | first1=Laurent | title=Chtoucas de Drinfeld et applications | language=fr | trans-title=Drinfelʹd shtukas and applications | url=http://www.math.uni-bielefeld.de/documenta/xvol-icm/07/Lafforgue.MAN.html | mr=1648105 | year=1998 | journal=Documenta Mathematica | issn=1431-0635 | volume=II | pages=563–570}}</ref>
* ] (Ferguson, Hales, 1998)<ref>http://arxiv.org/pdf/1501.02155.pdf</ref>
* ] (Hales, McLaughlin, 1998)<ref>http://arxiv.org/pdf/math/9811079v3.pdf</ref>
* ] (Iwase, 1997)<ref>{{cite web|url=https://www.researchgate.net/publication/220032558_Ganea's_Conjecture_on_Lusternik-Schnirelmann_Category|title=Ganea's Conjecture on Lusternik-Schnirelmann Category|author=Norio Iwase|date=1 November 1998|work=ResearchGate}}</ref>
* ] (Merel, 1996)<ref>Merel, Loïc (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" . Inventiones Mathematicae (in French) 124 (1): 437–449. doi:10.1007/s002220050059. MR 1369424</ref>
* ] (Chen, 1996)<ref>https://www.researchgate.net/profile/Zhibo_Chen/publication/220188021_Harary's_conjectures_on_integral_sum_graphs/links/5422b2490cf290c9e3aac7fe.pdf</ref>
* ] (] and ], 1995)<ref>{{cite journal|last=Wiles|first=Andrew|authorlink=Andrew Wiles|year=1995|title=Modular elliptic curves and Fermat's Last Theorem|url=http://math.stanford.edu/~lekheng/flt/wiles.pdf|journal=Annals of Mathematics|volume=141|issue=3|pages=443–551|oclc=37032255|format=PDF|doi=10.2307/2118559|jstor=2118559|publisher=Annals of Mathematics}}</ref><ref>{{cite journal | author = ], ] | year = 1995 | journal = Annals of Mathematics | title = Ring theoretic properties of certain Hecke algebras | volume = 141 | issue = 3| pages = 553–572 | oclc = 37032255 | url = http://wayback.archive.org/web/20050301000000*/http://www.math.harvard.edu/~rtaylor/hecke.ps | doi = 10.2307/2118560 | jstor = 2118560 | publisher = Annals of Mathematics}}</ref>


==References== == References ==
{{reflist|colwidth=30em}} {{reflist|colwidth=30em}}


==Further reading== == Further reading ==


===Books discussing recently solved problems{{Update after|2014|12|15}}=== === Books discussing problems solved since 1995 ===


* {{cite book |last=Singh |first=Simon |author-link=Simon Singh |date=2002 |title=Fermat's Last Theorem |publisher=Fourth Estate |isbn=978-1-84115-791-7|title-link=Fermat's Last Theorem (book) }}
* {{cite book |last=O'Shea |first=Donal |author-link=Donal O'Shea| date=2007 |title=The Poincaré Conjecture |publisher=Penguin |isbn=978-1-84614-012-9}}
* {{cite book |last=Szpiro |first=George G. |author-link=George Szpiro| date=2003 |title=Kepler's Conjecture |publisher=Wiley |isbn=978-0-471-08601-7}}
* {{cite book |last=Ronan |first=Mark |author-link=Mark Ronan| date=2006 |title=Symmetry and the Monster |publisher=Oxford |isbn=978-0-19-280722-9}}


=== Books discussing unsolved problems ===
* {{cite book |last=Singh |first=Simon |authorlink=Simon Singh |date=2002 |title=] |publisher=Fourth Estate |isbn=1-84115-791-0}}
* {{cite book |last=O'Shea |first=Donal |date=2007 |title=The Poincaré Conjecture |publisher=Penguin |isbn=978-1-84614-012-9}} * {{cite book |first1=Fan|last1= Chung|author-link1=Fan Chung |last2=Graham |first2=Ron |author-link2=Ronald Graham| title=Erdös on Graphs: His Legacy of Unsolved Problems|title-link= Erdős on Graphs |publisher=AK Peters |year=1999 |isbn=978-1-56881-111-6}}
* {{cite book |last1=Croft |first1=Hallard T. |last2=Falconer |first2=Kenneth J. |last3=Guy |first3=Richard K. |author-link2=Kenneth Falconer (mathematician) |author-link3=Richard K. Guy |date=1994 |title=Unsolved Problems in Geometry |publisher=Springer |isbn=978-0-387-97506-1 |url-access=registration |url=https://archive.org/details/unsolvedproblems0000crof }}
* {{cite book |last=Szpiro |first=George G. |date=2003 |title=Kepler's Conjecture |publisher=Wiley |isbn=0-471-08601-0}}
* {{cite book |last=Ronan |first=Mark |date=2006 |title=Symmetry and the Monster |publisher=Oxford |isbn=0-19-280722-6}} * {{cite book |last=Guy |first=Richard K. |author-link=Richard K. Guy |date=2004 |title=Unsolved Problems in Number Theory |publisher=Springer |isbn=978-0-387-20860-2}}
* {{cite book |last1=Klee |first1=Victor |author-link1=Victor Klee |last2=Wagon |first2=Stan |author-link2=Stan Wagon |date=1996 |title=Old and New Unsolved Problems in Plane Geometry and Number Theory |url=https://archive.org/details/oldnewunsolvedpr0000klee |url-access=registration |publisher=The Mathematical Association of America |isbn=978-0-88385-315-3}}
* {{cite book |last=du Sautoy |first=Marcus |author-link=Marcus du Sautoy |date=2003 |title=The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics |publisher=Harper Collins |isbn=978-0-06-093558-0 |url-access=registration |url=https://archive.org/details/musicofprimes00marc }}
* {{cite book |last=Derbyshire |first=John |author-link=John Derbyshire |date=2003 |title=Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics |publisher=Joseph Henry Press |isbn=978-0-309-08549-6 |url-access=registration |url=https://archive.org/details/primeobsessionbe00derb_0 }}
* {{cite book |last=Devlin |first=Keith |author-link=Keith Devlin |date=2006 |title=The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time |publisher=Barnes & Noble |isbn=978-0-7607-8659-8}}
* {{cite book |last1=Blondel |first1=Vincent D. |last2=Megrestski |first2=Alexandre |author-link1=Vincent Blondel |date=2004 |title=Unsolved problems in mathematical systems and control theory |publisher=Princeton University Press |isbn=978-0-691-11748-5}}
* {{cite book |first1=Lizhen|last1= Ji|author-link1=Lizhen Ji |first2=Yat-Sun|last2= Poon |first3=Shing-Tung|last3= Yau|author-link3=Shing-Tung Yau |date=2013 |title=Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics) |publisher=International Press of Boston |isbn=978-1-57146-278-7}}
* {{cite journal |last=Waldschmidt |first=Michel |author-link=Michel Waldschmidt |date=2004 |title=Open Diophantine Problems |journal=Moscow Mathematical Journal |issn=1609-3321 |zbl=1066.11030 |volume=4 |number=1 |pages=245–305 |url=http://www.math.jussieu.fr/~miw/articles/pdf/odp.pdf |doi=10.17323/1609-4514-2004-4-1-245-305 |arxiv=math/0312440 |s2cid=11845578 }}
* {{cite arXiv |last1=Mazurov |first1=V. D. |author-link1=Victor Mazurov |last2=Khukhro |first2=E. I. |eprint=1401.0300v6 |title= Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version) |date= 1 Jun 2015|class=math.GR }}


== External links ==
===Books discussing unsolved problems===
* {{cite book |author1=Fan Chung |last2=Graham |first2=Ron |title=Erdos on Graphs: His Legacy of Unsolved Problems |publisher=AK Peters |year=1999 |isbn=1-56881-111-X}}
* {{cite book |last1=Croft |first1=Hallard T. |last2=Falconer |first2=Kenneth J. |last3=Guy |first3=Richard K. |date=1994 |title=Unsolved Problems in Geometry |publisher=Springer |isbn=0-387-97506-3}}
* {{cite book |last=Guy |first=Richard K. |date=2004 |title=Unsolved Problems in Number Theory |publisher=Springer |isbn=0-387-20860-7}}
* {{cite book |last1=Klee |first1=Victor |authorlink1=Victor Klee |last2=Wagon |first2=Stan |authorlink2=Stan Wagon |date=1996 |title=Old and New Unsolved Problems in Plane Geometry and Number Theory |publisher=The Mathematical Association of America |isbn=0-88385-315-9}}
* {{cite book |last=Du Sautoy |first=Marcus |date=2003 |title=The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics |publisher=Harper Collins |isbn=0-06-093558-8}}
* {{cite book |last=Derbyshire |first=John |date=2003 |title=Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics |publisher=Joseph Henry Press |isbn=0-309-08549-7}}
* {{cite book |last=Devlin |first=Keith |date=2006 |title=The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time |publisher=Barnes & Noble |isbn=978-0-7607-8659-8}}
* {{cite book |last1=Blondel |first1=Vincent D. |last2=Megrestski |first2=Alexandre |date=2004 |title=Unsolved problems in mathematical systems and control theory |publisher=Princeton University Press |isbn=0-691-11748-9}}
* {{cite book |last1=Lizhen Ji |first1= |last2=Yat-Sun Poon |first2=Shing-Tung Yau |date=2013 |title=Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics) |publisher=International Press of Boston |isbn=1-571-46278-3}}
* {{cite journal |last=Waldschmidt |first=Michel |authorlink=Michel Waldschmidt |date=2004 |title=Open Diophantine Problems |journal=Moscow Mathematical Journal |issn=1609-3321 |zbl=1066.11030 |volume=4 |number=1 |pages=245–305 |url=http://www.math.jussieu.fr/~miw/articles/pdf/odp.pdf |format=PDF}}
* {{cite arXiv |last1=Mazurov |first1=V. D. |last2=Khukhro |first2=E. I. |eprint=1401.0300v6 |title=
Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version) |date= 1 Jun 2015}}
* {{cite book |last1=Derbyshire |first1=John |date=2003 |title=Prime Obsession |publisher=The Joseph Henry Press |isbn=0-309-08549-7}}

==External links==
* *
* *
* The collection of open problems in mathematics build on the principle of user editable ("wiki") site *
* *
* . MathPro Press. * . MathPro Press.
* {{cite web|last1=Ball|first1=John M.|authorlink=John M. Ball|title=Some Open Problems in Elasticity|url=https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf}} * {{cite web|last1=Ball|first1=John M.|author-link=John M. Ball|title=Some Open Problems in Elasticity|url=https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf}}
* {{cite web|last1=Constantin|first1=Peter|authorlink=Peter Constantin|title=Some open problems and research directions in the mathematical study of fluid dynamics|url=https://web.math.princeton.edu/~const/2k.pdf}} * {{cite web|last1=Constantin|first1=Peter|author-link=Peter Constantin|title=Some open problems and research directions in the mathematical study of fluid dynamics|url=https://web.math.princeton.edu/~const/2k.pdf}}
* {{cite web|last1=Serre|first1=Denis|authorlink=Denis Serre|title=Five Open Problems in Compressible Mathematical Fluid Dynamics|url=http://www.umpa.ens-lyon.fr/~serre/DPF/Ouverts.pdf}} * {{cite web|last1=Serre|first1=Denis|author-link=Denis Serre|title=Five Open Problems in Compressible Mathematical Fluid Dynamics|url=http://www.umpa.ens-lyon.fr/~serre/DPF/Ouverts.pdf}}
* *
* * {{Webarchive|url=https://web.archive.org/web/20170515145908/http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html |date=2017-05-15 }}
* , discrete and computational geometry problems * , discrete and computational geometry problems
* *
* *
* *
*
* *
* *
* {{cite web|last1=Aizenman|first1=Michael|authorlink=Michael Aizenman|title=Open Problems in Mathematical Physics|url=http://web.math.princeton.edu/~aizenman/OpenProblems.iamp/}} * {{cite web|last1=Aizenman|first1=Michael|author-link=Michael Aizenman|title=Open Problems in Mathematical Physics|url=https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/OPlist.html}}
* * ]'s
* ].

{{unsolved problems}} {{unsolved problems}}


{{DEFAULTSORT:Unsolved problems in mathematics}} {{DEFAULTSORT:Unsolved problems in mathematics}}
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Latest revision as of 10:14, 23 December 2024

This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.

Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention.

This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance.

Lists of unsolved problems in mathematics

Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.

List Number of
problems		 Number unsolved
or incompletely solved		 Proposed by Proposed
in
Hilbert's problems 23 15 David Hilbert 1900
Landau's problems 4 4 Edmund Landau 1912
Taniyama's problems 36 - Yutaka Taniyama 1955
Thurston's 24 questions 24 - William Thurston 1982
Smale's problems 18 14 Stephen Smale 1998
Millennium Prize Problems 7 6 Clay Mathematics Institute 2000
Simon problems 15 <12 Barry Simon 2000
Unsolved Problems on Mathematics for the 21st Century 22 - Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges 23 - DARPA 2007
Erdős's problems >893 603 Paul Erdős Over six decades of Erdős' career, from the 1930s to 1990s
The Riemann zeta function, subject of the Riemann hypothesis

Millennium Prize Problems

Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:

The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman in 2003. However, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is unsolved.

Notebooks

  • The Kourovka Notebook (Russian: Коуровская тетрадь) is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.
  • The Sverdlovsk Notebook (Russian: Свердловская тетрадь) is a collection of unsolved problems in semigroup theory, first published in 1965 and updated every 2 to 4 years since.
  • The Dniester Notebook (Russian: Днестровская тетрадь) lists several hundred unsolved problems in algebra, particularly ring theory and modulus theory.
  • The Erlagol Notebook (Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory.

Unsolved problems

Algebra

Main article: Algebra
In the Bloch sphere representation of a qubit, a SIC-POVM forms a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.

Group theory

Main article: Group theory
The free Burnside group B ( 2 , 3 ) {\displaystyle B(2,3)} is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups B ( m , n ) {\displaystyle B(m,n)} are finite remains open.

Representation theory

Analysis

Main article: Mathematical analysis

Combinatorics

Main article: Combinatorics
  • The 1/3–2/3 conjecture – does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?
  • The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
  • Problems in Latin squares – open questions concerning Latin squares
  • The lonely runner conjecture – if k {\displaystyle k} runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance 1 / k {\displaystyle 1/k} from each other runner) at some time?
  • Map folding – various problems in map folding and stamp folding.
  • No-three-in-line problem – how many points can be placed in the n × n {\displaystyle n\times n} grid so that no three of them lie on a line?
  • Rudin's conjecture on the number of squares in finite arithmetic progressions
  • The sunflower conjecture – can the number of k {\displaystyle k} size sets required for the existence of a sunflower of r {\displaystyle r} sets be bounded by an exponential function in k {\displaystyle k} for every fixed r > 2 {\displaystyle r>2} ?
  • Frankl's union-closed sets conjecture – for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets

Dynamical systems

Main article: Dynamical system
A detail of the Mandelbrot set. It is not known whether the Mandelbrot set is locally connected or not.

Games and puzzles

Main article: Game theory

Combinatorial games

Main article: Combinatorial game theory

Games with imperfect information

Geometry

Main article: Geometry

Algebraic geometry

Main article: Algebraic geometry

Covering and packing

  • Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set.
  • The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?
  • The Erdős–Oler conjecture: when n {\displaystyle n} is a triangular number, packing n − 1 {\displaystyle n-1} circles in an equilateral triangle requires a triangle of the same size as packing n {\displaystyle n} circles
  • The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24
  • Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets
  • Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
  • Square packing in a square: what is the asymptotic growth rate of wasted space?
  • Ulam's packing conjecture about the identity of the worst-packing convex solid
  • The Tammes problem for numbers of nodes greater than 14 (except 24).

Differential geometry

Main article: Differential geometry

Discrete geometry

Main article: Discrete geometry
In three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.

Euclidean geometry

Main article: Euclidean geometry

Graph theory

Main article: Graph theory

Algebraic graph theory

Games on graphs

Graph coloring and labeling

An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.

Graph drawing and embedding

Restriction of graph parameters

Subgraphs

Word-representation of graphs

Miscellaneous graph theory

Model theory and formal languages

Main articles: Model theory and formal languages
  • The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ℵ 0 {\displaystyle \aleph _{0}} is a simple algebraic group over an algebraically closed field.
  • Generalized star height problem: can all regular languages be expressed using generalized regular expressions with limited nesting depths of Kleene stars?
  • For which number fields does Hilbert's tenth problem hold?
  • Kueker's conjecture
  • The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for ℵ 1 {\displaystyle \aleph _{1}} -saturated models of a countable theory.
  • Shelah's categoricity conjecture for L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.
  • Shelah's eventual categoricity conjecture: For every cardinal λ {\displaystyle \lambda } there exists a cardinal μ ( λ ) {\displaystyle \mu (\lambda )} such that if an AEC K with LS(K)<= λ {\displaystyle \lambda } is categorical in a cardinal above μ ( λ ) {\displaystyle \mu (\lambda )} then it is categorical in all cardinals above μ ( λ ) {\displaystyle \mu (\lambda )} .
  • The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
  • The stable forking conjecture for simple theories
  • Tarski's exponential function problem: is the theory of the real numbers with the exponential function decidable?
  • The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?
  • The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?
  • Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is either finite, ℵ 0 {\displaystyle \aleph _{0}} , or 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} .
  • Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ℵ ω 1 {\displaystyle \aleph _{\omega _{1}}} does it have a model of cardinality continuum?
  • Do the Henson graphs have the finite model property?
  • Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
  • Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
  • If the class of atomic models of a complete first order theory is categorical in the ℵ n {\displaystyle \aleph _{n}} , is it categorical in every cardinal?
  • Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
  • Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?
  • Is the theory of the field of Laurent series over Z p {\displaystyle \mathbb {Z} _{p}} decidable? of the field of polynomials over C {\displaystyle \mathbb {C} } ?
  • Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?
  • Determine the structure of Keisler's order.

Probability theory

Main article: Probability theory

Number theory

Main page: Category:Unsolved problems in number theory See also: Number theory

General

6 is a perfect number because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them is odd.

Additive number theory

Main article: Additive number theory See also: Problems involving arithmetic progressions

Algebraic number theory

Main article: Algebraic number theory
  • Characterize all algebraic number fields that have some power basis.

Computational number theory

Main article: Computational number theory

Diophantine approximation and transcendental number theory

Further information: Diophantine approximation and Transcendental number theory
The area of the blue region converges to the Euler–Mascheroni constant, which may or may not be a rational number.

Diophantine equations

Further information: Diophantine equation

Prime numbers

Main article: Prime numbers
Prime number conjectures
Goldbach's conjecture states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.

Set theory

Main article: Set theory

Note: These conjectures are about models of Zermelo-Frankel set theory with choice, and may not be able to be expressed in models of other set theories such as the various constructive set theories or non-wellfounded set theory.

Topology

Main article: Topology
The unknotting problem asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot.

Problems solved since 1995

Ricci flow, here illustrated with a 2D manifold, was the key tool in Grigori Perelman's solution of the Poincaré conjecture.

Algebra

Analysis

Combinatorics

Dynamical systems

Game theory

Geometry

21st century

20th century

Graph theory

Group theory

Number theory

21st century

20th century

Ramsey theory

Theoretical computer science

Topology

Uncategorised

2010s

2000s

See also

Notes

  1. A disproof has been announced, with a preprint made available on arXiv.

References

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