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{{Short description|None}} {{Short description|List article of unsolved mathematical problems}}

{{Dynamic list}} {{Dynamic list}}


Many ] have not yet been solved. These ] occur in multiple domains, including ], ], ], ], ], ], ], ] and ], ], ], ], ], ] and ] theories, ]s, and ]s. Some problems may belong to more than one discipline of ] and be studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems, such as the list of ], receive considerable attention. 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.


This article is a composite of notable unsolved problems derived from many sources, including but not limited to lists considered authoritative. The list is not comprehensive, for at least the reason that entries may not be updated at the time of viewing. This list includes problems which are considered by the mathematical community to be widely varying in both difficulty and centrality to the science as a whole. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative. Although this list may never be comprehensive, the problems listed here vary widely in both difficulty and importance.


== Lists of unsolved problems in mathematics == == Lists of unsolved problems in mathematics ==
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{| class="wikitable sortable" {| class="wikitable sortable"
|- |-
! List !! Number of<br />problems !! Number unresolved <br /> or incompletely resolved !! Proposed by !! Proposed<br />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 | ]<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
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| ] || 18 || 14 || ] || 1998 | ] || 18 || 14 || ] || 1998
|- |-
| ] || 7 || 6<ref name="auto1">{{cite web|url=http://claymath.org/millennium-problems|title=Millennium Problems|access-date=2015-01-20|archive-url=https://web.archive.org/web/20170606121331/http://claymath.org/millennium-problems|archive-date=2017-06-06|url-status=dead}}</ref> || ] || 2000 | ] || 7 || 6<ref name="auto1">{{cite web |title=Millennium Problems |url=http://claymath.org/millennium-problems |url-status=dead |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
|- |-
| ] || 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 |url-status=dead }}</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 | ] || 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 |url-status=dead }}</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
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| Unsolved Problems on Mathematics for the 21st Century<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 = 978-9051994902}}</ref> || 22 || - || Jair Minoro Abe, Shotaro Tanaka || 2001 | Unsolved Problems on Mathematics for the 21st Century<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 = 978-9051994902}}</ref> || 22 || - || Jair Minoro Abe, Shotaro Tanaka || 2001
|- |-
| DARPA'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 | ]'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 | archive-date = 2012-10-01}}</ref> || 23 || - || ] || 2007
|} |}
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=== Millennium Prize Problems === === Millennium Prize Problems ===
Of the original seven ] set by the ] in 2000, six have yet to be solved as of August, 2021:<ref name="auto1"/> Of the original seven ] set by the ] in 2000, six remain unsolved:<ref name="auto1"/>


* ] * ]
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* ] * ]


The seventh problem, the ], has been solved;<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 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|access-date=2019-08-06|archive-url=https://web.archive.org/web/20180125203721/http://www.openproblemgarden.org/?q=op%2Fsmooth_4_dimensional_poincare_conjecture|archive-date=2018-01-25|url-status=live}}</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 topological sphere can have two or more inequivalent ]s—is unsolved.<ref>{{cite web |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 == == Unsolved problems ==
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] representation of a ], a ] forms a ]. Zauner conjectured that analogous structures exist in complex ]s of all finite dimensions.]] ] representation of a ], a ] forms a ]. Zauner conjectured that analogous structures exist in complex ]s of all finite dimensions.]]


====Notebook problems====
* The Dniester Notebook (''Dnestrovskaya Tetrad'') collects several hundred unresolved problems in algebra, particularly ] and ].<ref>{{citation|year=1993|title=Dnestrovskaya notebook|publisher=The Russian Academy of Sciences|language=ru |url=http://math.nsc.ru/LBRT/a1/files/dnestr93.pdf}}<br />{{citation |url=https://math.usask.ca/~bremner/research/publications/dniester.pdf |title=Dniester Notebook: Unsolved Problems in the Theory of Rings and Modules |website=] |access-date=2019-08-15}}</ref>
* The Erlagol Notebook (''Erlagolskaya Tetrad'') collects unresolved problems in algebra and model theory.<ref>{{citation|year=2018|title=Erlagol notebook|publisher=The Novosibirsk State University|language=ru |url=http://uamt.conf.nstu.ru/erl_note.pdf}}</ref>

====Conjectures and problems====
* ] on the relation between the order of the ] of the ] of the ] of a ] to the field's ]. * ] 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. * ]s on densities of rational points of ]s and ] defined on ] and their ]s.
* ] in ] theory. * ] in ] theory
* ] that 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>. * ]: 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 ]s of ]s over the integers. * ] on ]s of ]s over the integers.
* ] that a group with ] 2 also has a 2-dimensional ] <math>K(G, 1)</math>. * ]: a group with ] 2 also has a 2-dimensional ] <math>K(G, 1)</math>.
* ] on whether certain ]s are ]. * ] on whether certain ]s are ].
** ]: a specific case of the Farrell–Jones conjecture. ** ]. a specific case of the Farrell–Jones conjecture
* ]:<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> is every finite ] isomorphic to the ] of some finite ]? * ]: 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>
* ] that the ] of a non-] is determined by the extent to which it, as a ], has ]. * ]: the ] of a non-] is determined by the extent to which it, as a ], has ].
* ] * ]
* ]: for every positive integer <math>k</math>, a ] of order <math>4k</math> exists. * ]: for every positive integer <math>k</math>, a ] of order <math>4k</math> exists.
* ]: what is the largest ] of a matrix with entries all equal to 1 or -1? * ]: what is the largest ] of a matrix with entries all equal to 1 or –1?
* ]: put ] on a rigorous foundation. * ]: put ] on a rigorous foundation.
* ]: what are the possible configurations of the ] of ]? * ]: what are the possible configurations of the ] of ]?
* ] * ]
* ] that the intersection of all powers of the ] of a left-and-right ] is precisely 0. * ]: the intersection of all powers of the ] of a left-and-right ] is precisely 0.
* ] * ]
* ] that if a ring has no ] other than <math>\{0\}</math>, then it has no nil ] other than <math>\{0\}</math>. * ]: if a ring has no ] other than <math>\{0\}</math>, then it has no nil ] other than <math>\{0\}</math>.
* Existence of ]s and associated ] * Existence of ]s and associated ]
* ] that 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. * ]: 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.
* ] that 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. * ]: 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.
* ] that 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 ]. * ]: 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 ].
* ] that 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. * ]: 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.
* ] * ]
* ]: ]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. * ]: 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?
* ]: classification of pairs of ''n''×''n'' matrices under simultaneous conjugation and problems containing it such as a lot of classification problems * ]: classification of pairs of <math>n\times n</math> matrices under simultaneous conjugation and problems containing it such as a lot of classification problems
* ] that 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 ]. * ]: 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 on the existence of ]s in all dimensions * Zauner's conjecture: do ]s exist in all dimensions?

==== Notebook problems ====
* The Dniester Notebook ({{Lang-ru|Днестровская тетрадъ}}) lists several hundred unsolved problems in algebra, particularly ] and ].<ref>* {{citation |title=ДНЕСТРОВСКАЯ ТЕТРАДЪ |url=http://math.nsc.ru/LBRT/a1/files/dnestr93.pdf |year=1993 |trans-title=DNIESTER NOTEBOOK |publisher=The Russian Academy of Sciences |language=ru}}
* {{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 Erlagol Notebook ({{Lang-ru|Эрлаголъская тетрадъ}}) lists unsolved problems in algebra and model theory.<ref>{{citation |title=Эрлаголъская тетрадъ |url=http://uamt.conf.nstu.ru/erl_note.pdf |year=2018 |trans-title=Erlagol notebook |publisher=The Novosibirsk State University |language=ru}}</ref>


=== Analysis === === Analysis ===
{{Main|Mathematical analysis}} {{Main|Mathematical analysis}}
], which may or may not be a rational number.]] ], which may or may not be a rational number.]]
* 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>
====Conjectures and problems====
* The ]: the transcendence of at least one of four exponentials of combinations of irrationals<ref name=waldschmidt>{{citation|pages=14, 16|url=https://books.google.com/books?id=Wrj0CAAAQBAJ&pg=PA14|title=Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables|first=Michel|last=Waldschmidt|publisher=Springer|year=2013|isbn=9783662115695}}</ref>
* The ] on 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>
* The ] on the transcendence of at least one of four exponentials of combinations of irrationals<ref name=waldschmidt>{{citation|pages=14, 16|url=https://books.google.com/books?id=Wrj0CAAAQBAJ&pg=PA14|title=Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables|first=Michel|last=Waldschmidt|publisher=Springer|year=2013|isbn=9783662115695}}</ref>
* ] on the coefficients of ]s * ] on the coefficients of ]s
* ] – does every ] on a complex ] send some non-trivial ] subspace to itself? * ] – does every ] on a complex ] send some non-trivial ] subspace to itself?
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* ] on compact subsets of <math>\mathbb{C}</math> with analytic capacity <math>0</math> * ] on compact subsets of <math>\mathbb{C}</math> with analytic capacity <math>0</math>


* Are <math>\gamma</math> (the ]),<math>\pi + e, \pi - e, \pi e, \pi/e, \pi^e, \pi^{\sqrt{2}}, \pi^{\pi}, e^{\pi^2}, \ln\pi, 2^e, e^e</math>, ], or ] rational, ] irrational, or ]? What is the ] of each of these numbers?<ref>For some background on the numbers in this problem, see articles by ] at ] (all articles accessed 15 December 2014):
====Open questions====

* Are <math>\gamma</math> (the ]), ]&nbsp;+&nbsp;'']'', {{pi}}&nbsp;−&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 ( {{Webarchive|url=https://web.archive.org/web/20141206023912/http://mathworld.wolfram.com/Pi.html |date=2014-12-06 }}), e ( {{Webarchive|url=https://web.archive.org/web/20141121122615/http://mathworld.wolfram.com/e.html |date=2014-11-21 }}), Khinchin's Constant ( {{Webarchive|url=https://web.archive.org/web/20141105201509/http://mathworld.wolfram.com/KhinchinsConstant.html |date=2014-11-05 }}), irrational numbers ( {{Webarchive|url=https://web.archive.org/web/20150327024040/http://mathworld.wolfram.com/IrrationalNumber.html |date=2015-03-27 }}), transcendental numbers ( {{Webarchive|url=https://web.archive.org/web/20141113174913/http://mathworld.wolfram.com/TranscendentalNumber.html |date=2014-11-13 }}), and irrationality measures ( {{Webarchive|url=https://web.archive.org/web/20150421203736/http://mathworld.wolfram.com/IrrationalityMeasure.html |date=2015-04-21 }}) 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 {{Webarchive|url=https://web.archive.org/web/20141216004531/http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture5.pdf |date=2014-12-16 }}, 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 {{Webarchive|url=https://web.archive.org/web/20140117150133/http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf |date=2014-01-17 }}, accessed 15 December 2014.</ref>
* ({{Webarchive|url=https://web.archive.org/web/20141206023912/http://mathworld.wolfram.com/Pi.html|date=2014-12-06}})
* ({{Webarchive|url=https://web.archive.org/web/20141121122615/http://mathworld.wolfram.com/e.html|date=2014-11-21}})
* ({{Webarchive|url=https://web.archive.org/web/20141105201509/http://mathworld.wolfram.com/KhinchinsConstant.html|date=2014-11-05}})
* ({{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>{{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 |archive-url=https://web.archive.org/web/20140117150133/http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf |access-date=15 December 2014 |archive-date=17 December 2014}}</ref>
* What is the exact value of ], including ]? * What is the exact value of ], including ]?
* How are suspended infinite-infinitesimals paradoxes justified? * How are suspended infinite-infinitesimals paradoxes justified?


====Other====
* Regularity of solutions of ] * Regularity of solutions of ]
* Convergence of ] * Convergence of ]
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=== Combinatorics === === Combinatorics ===
{{Main|Combinatorics}} {{Main|Combinatorics}}
====Conjectures and problems====
* 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 * 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
| last1 = Brightwell | first1 = Graham R. | last1 = Brightwell | first1 = Graham R.
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}}</ref> }}</ref>


====Other====
* The values of the ]s <math>M(n)</math> for <math>n \ge 9</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 |url-status=dead }}</ref> * The values of the ]s <math>M(n)</math> for <math>n \ge 9</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 |url-status=dead }}</ref>
* Give a combinatorial interpretation of the ]s.<ref>{{citation * Give a combinatorial interpretation of the ]s.<ref>{{citation
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{{Main|Dynamical system}} {{Main|Dynamical system}}
]. It is not known whether the Mandelbrot set is ] or not.]] ]. It is not known whether the Mandelbrot set is ] or not.]]
====Conjectures and problems====
* ] and ] – relating symplectic geometry to Morse theory * ] and ] – relating symplectic geometry to Morse theory
* ] * ]
* ] 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> * ] 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> * ] 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>
* ] (3''n''&nbsp;+&nbsp;1 conjecture) * ] (''aka'' the <math>3n + 1</math> conjecture)
* ] conjecture that every component of the ] of an ] ] function is unbounded * ] conjecture: every component of the ] of an ] ] function is unbounded
* ] conjecture – is every invariant and ] measure for the <math>\times 2,\times 3</math> action on the circle either Lebesgue or atomic? * ] 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 * ] on the dimension of an ] in terms of its ]s
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* ] – does a regular compact ] ] of a ] on a ] carry at least one periodic orbit of the Hamiltonian flow? * ] – does a regular compact ] ] of a ] on a ] carry at least one periodic orbit of the Hamiltonian flow?


====Open questions====
* Does every positive integer generate a ] terminating at 1? * Does every positive integer generate a ] terminating at 1?
* ] – For what classes of ], describing dynamical systems, does the Lyapunov’s second method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion? * ] – For what classes of ], describing dynamical systems, does the 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>
* Is every ] in three or more dimensions locally reversible?<ref>{{citation
| last = Kari | first = Jarkko | author-link = Jarkko Kari
| contribution = Structure of reversible cellular automata
| doi = 10.1007/978-3-642-03745-0_5
| page = 6
| publisher = Springer
| series = ]
| title = Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7ÔÇô11, 2009, Proceedings
| volume = 5715
| year = 2009| bibcode = 2009LNCS.5715....6K| isbn = 978-3-642-03744-3 | doi-access = free
}}</ref>


=== Games and puzzles === === Games and puzzles ===
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==== Algebraic geometry ==== ==== Algebraic geometry ====
{{Main|Algebraic geometry}} {{Main|Algebraic geometry}}
* ]: if the ] of a ] with ] is ], then it is semiample.

=====Conjectures=====
* ] that if the ] of a ] with ] is ], then it is semiample.
* ] on the ] of certain ]. * ] on the ] of certain ].
* ]: any one of numerous named for ]. * ]: any one of numerous named for ].
* ] that any ] of a ] is an ]. * ]: any ] of a ] is an ].
* ] on the ] of a set of forms. * ] 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> * ] 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>
* 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> * 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>
* ] that if a ] over a ]-0 field has a constant nonzero ], then it has a ] (i.e. with polynomial components) inverse function. * ]: 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 the distribution of ]s of bounded ] in certain subsets of ]
* ] on an equivalence between ] and ]<ref>{{citation * ] on an equivalence between ] and ]<ref>{{citation
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|date=2004-06-05|bibcode=2003math.....12059M |date=2004-06-05|bibcode=2003math.....12059M
}}</ref> }}</ref>
* ] that if a ] has a ring of ]s generated by its contained ], then it must be ]. * ]: if a ] has a ring of ]s generated by its contained ], then it must be ].
* ] that the higher ] of any ] ] defined over a ] must vanish up to torsion. * ]: 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 splittings of ]s from ]s of complete ] over finitely-generated ] <math>k</math> to the ] of <math>k</math>.
* ] on algebraic cycles * ] on algebraic cycles
* ] on the connection between ]s on ] and ] on ]. * ] on the connection between ]s on ] and ] on ].
* ] that a certain ] encoding the ]s of a ] ] is fixed by an action of half of the ]. * ]: 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> * 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>


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


====Covering and packing==== ====Covering and packing====
=====Conjectures and problems=====
* ] on upper and lower bounds for the number of smaller-diameter subsets needed to cover a ] ''n''-dimensional set. * ] 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 ]: 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 ] that 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 ]: 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 ] 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>
* ] that the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets<ref>{{citation * ]: 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 | last = Hales | first = Thomas | author-link = Thomas Callister Hales
| arxiv = 1703.01352 | arxiv = 1703.01352
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==== Differential geometry ==== ==== Differential geometry ====
{{Main|Differential geometry}} {{Main|Differential geometry}}
* The ], a generalization of ]
=====Conjectures and problems=====
* ]: any convex, closed, and twice-differentiable surface in three-dimensional ] admits at least two ]s
* The ], a possible generalization of the original ]
* ] that 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 ]? * ]: Can the classical ] for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as ]?
* ] that the ] of a ] ] vanishes. * ] that the ] of a ] ] vanishes.
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| volume = 31 | volume = 31
| year = 1997}}</ref> | year = 1997}}</ref>
* ] that a ] ] ] has an infinite number of ] closed ] ]s. * ]: a ] ] ] has an infinite number of ] closed ] ]s.
* ] that the first ] for the ] on an embedded ] of <math>S^{n+1}</math> is <math>n</math>. * ] that the first ] for the ] on an embedded ] of <math>S^{n+1}</math> is <math>n</math>.


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{{Main|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.]] ] 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.]]
=====Conjectures and problems=====
* 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> * 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 * Solving the ] for arbitrary <math>n</math><ref>{{citation
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| year = 1989| s2cid = 8917264 }}.</ref> | year = 1989| s2cid = 8917264 }}.</ref>
* The ] on triangles in line arrangements<ref>{{MathWorld|urlname=KobonTriangle|title=Kobon Triangle}}</ref> * The ] on triangles in line arrangements<ref>{{MathWorld|urlname=KobonTriangle|title=Kobon Triangle}}</ref>
* The ] that at most <math>2d</math> points can be equidistant in <math>L^1</math> spaces<ref>{{citation * The ]: at most <math>2d</math> points can be equidistant in <math>L^1</math> spaces<ref>{{citation
| last = Guy | first = Richard K. | authorlink = Richard K. Guy | last = Guy | first = Richard K. | authorlink = Richard K. Guy
| issue = 3 | issue = 3
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*] on finding ]s for various planar shapes *] on finding ]s for various planar shapes


=====Open questions=====
* ] can be determined by a set of {{mvar|n}} points in the Euclidean plane?<ref>{{citation * ] can be determined by a set of {{mvar|n}} points in the Euclidean plane?<ref>{{citation
| last1 = Brass | first1 = Peter | last1 = Brass | first1 = Peter
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| year = 2005}}</ref> | year = 2005}}</ref>


=====Other=====
* Finding matching upper and lower bounds for ] and halving lines<ref>{{citation * Finding matching upper and lower bounds for ] and halving lines<ref>{{citation
| last = Dey | first = Tamal K. | author-link = Tamal Dey | last = Dey | first = Tamal K. | author-link = Tamal Dey
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====Euclidean geometry==== ====Euclidean geometry====
{{Main|Euclidean geometry}} {{Main|Euclidean geometry}}
=====Conjectures and problems=====
* 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> * 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> * ] – 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>
* 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|access-date=2019-02-12|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|url-status=live}}</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|access-date=2019-02-12|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|url-status=live}}</ref>
* ] that 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> * ]: 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>
* The {{not a typo|]}} – does there exist a two-dimensional shape that forms the ] for an ], but not for any periodic tiling?<ref>{{citation|last1=Socolar|first1=Joshua E. S.|last2=Taylor|first2=Joan M.|arxiv=1009.1419|doi=10.1007/s00283-011-9255-y|issue=1|journal=The Mathematical Intelligencer|mr=2902144|pages=18–28|title=Forcing nonperiodicity with a single tile|volume=34|year=2012|s2cid=10747746}}</ref> * The {{not a typo|]}} – does there exist a two-dimensional shape that forms the ] for an ], but not for any periodic tiling?<ref>{{citation|last1=Socolar|first1=Joshua E. S.|last2=Taylor|first2=Joan M.|arxiv=1009.1419|doi=10.1007/s00283-011-9255-y|issue=1|journal=The Mathematical Intelligencer|mr=2902144|pages=18–28|title=Forcing nonperiodicity with a single tile|volume=34|year=2012|s2cid=10747746}}</ref>
* ] that 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=9780821834848|doi-access=free}}</ref> * ]: 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=9780821834848|doi-access=free}}</ref>
* ], 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> * ], 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|department=Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000)|doi=10.5565/PUBLMAT_Esco02_07|issue=Vol. Extra|journal=Publicacions Matemàtiques|mr=1964819|pages=161–179|title=Recent progress on the Kakeya conjecture|year=2002|citeseerx=10.1.1.241.5335|s2cid=77088}}</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|department=Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000)|doi=10.5565/PUBLMAT_Esco02_07|issue=Vol. Extra|journal=Publicacions Matemàtiques|mr=1964819|pages=161–179|title=Recent progress on the Kakeya conjecture|year=2002|citeseerx=10.1.1.241.5335|s2cid=77088}}</ref>
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* 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> * 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>


=====Open questions=====
* ] — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?<ref>{{citation * ] — 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 | last = Howards | first = Hugh Nelson
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* ] – 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|access-date=2018-11-22|archive-url=https://web.archive.org/web/20181104211116/https://pure.uva.nl/ws/files/836396/73198_315330.pdf|archive-date=2018-11-04|url-status=live}}. See in particular Conjecture 23, p. 327.</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|access-date=2018-11-22|archive-url=https://web.archive.org/web/20181104211116/https://pure.uva.nl/ws/files/836396/73198_315330.pdf|archive-date=2018-11-04|url-status=live}}. See in particular Conjecture 23, p. 327.</ref>


=====Other=====
* ]s – find and classify the complete set of these shapes<ref>{{citation|url=http://www.openproblemgarden.org/op/convex_uniform_5_polytopes|work=Open Problem Garden|title=Convex uniform 5-polytopes|access-date=2016-10-04|date=May 24, 2012|author=ACW|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> * ]s – find and classify the complete set of these shapes<ref>{{citation|url=http://www.openproblemgarden.org/op/convex_uniform_5_polytopes|work=Open Problem Garden|title=Convex uniform 5-polytopes|access-date=2016-10-04|date=May 24, 2012|author=ACW|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>


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==== Graph coloring and labeling ==== ==== Graph coloring and labeling ====
] ]
=====Conjectures and problems=====
* ] on the diameter of the space of colorings of degenerate graphs<ref>{{citation * ] on the diameter of the space of colorings of degenerate graphs<ref>{{citation
| last1 = Bousquet | first1 = Nicolas | last1 = Bousquet | first1 = Nicolas
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| publisher = Springer-Verlag | publisher = Springer-Verlag
| year = 1991}}, Problem G10.</ref> | year = 1991}}, Problem G10.</ref>
* ] that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph<ref>{{citation * ]: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph<ref>{{citation
| last1 = Hägglund | last1 = Hägglund
| first1 = Jonas | first1 = Jonas
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| doi-access = free | doi-access = free
}}.</ref> }}.</ref>
* The ] that, 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 ]:, 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 ] of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree<ref>{{citation * The ] of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree<ref>{{citation
| last1 = Molloy | first1 = Michael | last1 = Molloy | first1 = Michael
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==== Graph drawing ==== ==== Graph drawing ====
* 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>
=====Conjectures and problems=====
* The ] that 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>
* ]<ref>{{citation |last1=Fulek |first1=Radoslav |last2=Pach |first2=János |authorlink2=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 * ]<ref>{{citation |last1=Fulek |first1=Radoslav |last2=Pach |first2=János |authorlink2=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
* ] that 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> * ]: 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> * ] 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 ] that 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> * 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> * ] – 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>


=====Other=====
* ]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> * ]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>


==== Paths and cycles in graphs ==== ==== Paths and cycles in graphs ====
* ]: every cubic bipartite three-connected planar graph has a Hamiltonian cycle<ref>{{citation
=====Conjectures and problems=====
* ] that every cubic bipartite three-connected planar graph has a Hamiltonian cycle<ref>{{citation
| last = Florek | first = Jan | last = Florek | first = Jan
| doi = 10.1016/j.disc.2010.01.018 | doi = 10.1016/j.disc.2010.01.018
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| year = 2014| s2cid = 1377980 | year = 2014| s2cid = 1377980
}}</ref> }}</ref>
* The ] that every bridgeless graph has a family of cycles that includes each edge twice<ref>{{citation * The ]: every bridgeless graph has a family of cycles that includes each edge twice<ref>{{citation
| last = Jaeger | first = F. | last = Jaeger | first = F.
| contribution = A survey of the cycle double cover conjecture | contribution = A survey of the cycle double cover conjecture
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| volume = 97 | volume = 97
| year = 1991}}</ref> | year = 1991}}</ref>
* ] that every ] on the <math>n</math>-dimensional doubly-] ] can be routed with edge-disjoint ]. * ]: every ] on the <math>n</math>-dimensional doubly-] ] can be routed with edge-disjoint ].


==== Word-representation of graphs ==== ==== 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|url=https://link.springer.com/book/10.1007/978-3-319-25859-1|title=Words and Graphs|series=Monographs in Theoretical Computer Science. An EATCS Series |year=2015 |doi=10.1007/978-3-319-25859-1 |isbn=978-3-319-25857-7 |s2cid=7727433 |via=link.springer.com}}</ref><ref name="K17">]</ref><ref name="KP18">{{Cite journal|url=https://doi.org/10.1134/S1990478918020084|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 web|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus|title=С.В.Китаев, А.В.Пяткин, "Графы, представимые в виде слов. Обзор результатов", Дискретн. анализ и исслед. опер., 25:2 (2018), 19–53; J. Appl. Industr. Math., 12:2 (2018), 278–296|website=www.mathnet.ru}}</ref> *Are there any graphs on ''n'' vertices whose ] requires more than floor(''n''/2) copies of each letter?<ref name="KL15">{{Cite book|url=https://link.springer.com/book/10.1007/978-3-319-25859-1|title=Words and Graphs|series=Monographs in Theoretical Computer Science. An EATCS Series |year=2015 |doi=10.1007/978-3-319-25859-1 |isbn=978-3-319-25857-7 |s2cid=7727433 |via=link.springer.com}}</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 |url=https://arxiv.org/abs/1705.05924v1 |conference=] |language=en |doi=10.1007/978-3-319-62809-7_2}}</ref><ref name="KP18">{{Cite journal|url=https://doi.org/10.1134/S1990478918020084|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 |last=Kitaev |first=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 (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 ]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>) *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">]</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 journal |last=Kitaev |first=Sergey |date=2014-03-06 |title=On graphs with representation number 3 |url=https://arxiv.org/abs/1403.1616v1 |language=en |doi=10.48550/arXiv.1403.1616}}</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 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"/> *Is the ] of a non-] graph always non-]?<ref name="KL15"/><ref name="K17"/><ref name="KP18"/><ref name="KP18-2"/>
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==== Miscellaneous graph theory ==== ==== Miscellaneous graph theory ====
=====Conjectures and problems=====
* ]: which groups are Babai invariant groups? * ]: which groups are Babai invariant groups?
* ] on upper bounds for sums of ] of ] of graphs in terms of their number of edges. * ] on upper bounds for sums of ] of ] of graphs in terms of their number of edges.
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* ]: how many edges can there be in a ] on a given number of vertices with no ] of a given size? * ]: how many edges can there be in a ] on a given number of vertices with no ] of a given size?


=====Open questions=====
* 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> * 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>
* What is the largest possible ] of an {{mvar|n}}-vertex ]?<ref>{{citation * What is the largest possible ] of an {{mvar|n}}-vertex ]?<ref>{{citation
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{{Main|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.]] ] <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

====Notebook problems====
* The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.<ref>{{citation
| last1 = Khukhro | first1 = Evgeny I.
| last2 = Mazurov | first2 = Victor D. |author-link2 = Victor Mazurov
| arxiv = 1401.0300v16
| title = Unsolved Problems in Group Theory. The Kourovka Notebook
| year = 2019}}</ref>
====Conjectures and problems====
* ] that every balanced ] of the ] can be transformed into a trivial presentation by a sequence of ]s on ] and conjugations of relators
* 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> * 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>
* ] that 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. * ]: 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? * The ]: is every finite group the Galois group of a Galois extension of the rationals?
* ] consider generalizations of groups * ] consider generalizations of groups


====Open questions====
* Are there an infinite number of ]s? * Are there an infinite number of ]s?
* Does ] exist? * Does ] exist?
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* Is every group ]? * Is every group ]?


==== Notebook problems ====
* The Kourovka Notebook is a collection of unsolved problems in group theory, 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>
=== Model theory and formal languages === === Model theory and formal languages ===
{{Main|Model theory|formal languages}} {{Main|Model theory|formal languages}}
====Conjectures and problems====
* 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? * ]: can all ]s be expressed using ] with limited nesting depths of ]s?
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* 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|s2cid=10425739 }}</ref> * 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|s2cid=10425739 }}</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>
* ] that the number of ] models of a ] ] in a countable ] is either finite, <math>\aleph_{0}</math>, or <math>2^{\aleph_{0}}</math>. * ]: the number of ] models of a ] ] in a countable ] is either finite, <math>\aleph_{0}</math>, or <math>2^{\aleph_{0}}</math>.


====Open questions====
* 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> * 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 ]? * Do the ]s have the ]?
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* 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> * 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>


====Other====
* 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>], ], "A dividing line in simple unstable theories." https://arxiv.org/abs/1208.2140 {{Webarchive|url=https://web.archive.org/web/20170802171447/https://arxiv.org/abs/1208.2140 |date=2017-08-02 }}</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>], ], "A dividing line in simple unstable theories." https://arxiv.org/abs/1208.2140 {{Webarchive|url=https://web.archive.org/web/20170802171447/https://arxiv.org/abs/1208.2140 |date=2017-08-02 }}</ref>


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==== General ==== ==== General ====
] 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 are odd.]] ] 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 are odd.]]
* ]: 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>.
=====Conjectures, problems and hypotheses=====
** ]: 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>.
* ]: a generalization of the abc conjecture to more than three integers.
** ] that for any <math>\epsilon > 0</math>, <math>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>.
** ] that 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>.
*] *]
* ]: classify ]s over ]s. * ]: 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. * ]: 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. * ]: extend the ] on ]s of <math>\mathbb{Q}</math> to any base number field.
*] that 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>?
**] that 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>?
***] that the nontrivial zeros of the ] lie on the critical line <math>1/2 + it</math> with real <math>t</math>. ***]: do the nontrivial zeros of the ] lie on the critical line <math>1/2 + it</math> with real <math>t</math>?
*] that every irreducible component of the ] of a set of special points in a ] is a special ]. *]: is every irreducible component of the ] of a set of special points in a ] a special ]?
*] *]
* ]: are there any integer solutions to <math>n! + 1 = m^{2}</math> other than <math>n = 4, 5, 7</math>? * ]: are there any integer solutions to <math>n! + 1 = m^{2}</math> other than <math>n = 4, 5, 7</math>?
* ] that all values of ] have ] greater than <math>1</math>. * ]: do all values of ] have ] greater than <math>1</math>?
* ] that 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 <math>f</math> must be the <math>d</math>-th power of a linear polynomial. * ]: 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?
* ]: no ]s are infinite but non-repeating. * ]: no ]s are infinite but non-repeating.
* ] (a corollary to ], per ]): determine precisely what rational numbers are ]s. * ] (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 ]? * Erdős–Moser problem: is <math>1^{1} + 2^{1} = 3^{1}</math> the only solution to the ]?
* ] that 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>. * ]: 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>.
* ]: is there a ] of points in the plane all at rational distances from one-another? * ]: is there a ] of points in the plane all at rational distances from one-another?
* ]: for all <math>\epsilon > 0</math>, the pair <math>(\epsilon, 1/2 + \epsilon)</math> is an ]. * ]: for all <math>\epsilon > 0</math>, the pair <math>(\epsilon, 1/2 + \epsilon)</math> is an ].
* The ]: how far can the number of integer points in a circle centered at the origin be from the area of the circle? * The ]: how far can the number of integer points in a circle centered at the origin be from the area of the circle?
* ] 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>. * ] 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>.
* ] that each element of a set of consecutive ]s can be assigned a distinct ] that divides it. * ]: each element of a set of consecutive ]s can be assigned a distinct ] that divides it.
* ] that 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>. * ]: 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>.
* ] that the nontrivial zeros of the ] correspond to ] of a ]. * ]: the nontrivial zeros of the ] correspond to ] of a ].
* Keating–Snaith conjecture concerning the asymptotics of an integral involving the ]<ref>{{citation * Keating–Snaith conjecture concerning the asymptotics of an integral involving the ]<ref>{{citation
|last=Conrey |first=Brian |author-link=Brian Conrey |last=Conrey |first=Brian |author-link=Brian Conrey
Line 922: Line 883:
|volume=53 |number=3 |pages=507–512 |year=2016|doi-access=free}}</ref> |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? *]: if <math>\phi(n)</math> divides <math>n - 1</math>, must <math>n</math> be prime?
* ] that a ] analogue of the ] of an ] does not vanish. * ]: 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> * ] that for all <math>\epsilon > 0</math>, <math>\zeta(1/2 + it) = o(t^{\epsilon})</math>
** The ] for zeroes of the Riemann zeta function ** The ] for zeroes of the Riemann zeta function
* ] that 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. * ]: 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>. * ] 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>.
* ] that the normalized pair ] between pairs of zeros of the ] is the same as the pair correlation function of ]. * ]: the normalized pair ] between pairs of zeros of the ] is the same as the pair correlation function of ].
* ] that the ] satisfies any arbitrary congruence infinitely often. * ]: the ] satisfies any arbitrary congruence infinitely often.
* ] that 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>. * ]: 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>.
* ] on bounding <math>\Delta_{k}(x) = D_{k}(x) - xP_{k}(log(x))</math> * ] 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> ** ]: 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. * ]: 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. * ]: also a number of related conjectures that are generalizations of the original conjecture.
* ] that 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>. * ]: 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? * Do ]s exist?
* ]: is there a finite upper bound on the multiplicities of the entries greater than 1 in ]?<ref>{{citation * ]: is there a finite upper bound on the multiplicities of the entries greater than 1 in ]?<ref>{{citation
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* ] on ] of points on ] over ]s. * ] on ] of points on ] over ]s.


=====Open questions=====
* Are there infinitely many ]s? * Are there infinitely many ]s?
*Do any ]s exist? *Do any ]s exist?
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| s2cid = 835158 | s2cid = 835158
}}</ref> }}</ref>
* 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|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> * 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>
* Is 10 a ]? * Is 10 a ]?
* Can a 3×3 ] be constructed from 9 distinct perfect square numbers?{{citation needed|date=April 2022}} * 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 |url=https://doi.org/10.1080/0025570X.1996.11996457 |journal=Mathematics Magazine |volume=69 |issue=4 |pages=289–293 |doi=10.1080/0025570X.1996.11996457 |issn=0025-570X}}</ref>
* 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> * 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>
* ] * ]


* Find the value of the ].
=====Other=====
* Find the value of the ]


==== Additive number theory ==== ==== Additive number theory ====
{{Main|Additive number theory }} {{Main|Additive number theory }}
=====Conjectures and problems=====
{{See also|Problems involving arithmetic progressions}} {{See also|Problems involving arithmetic progressions}}
* ] that 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. * ]: 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.
* ] that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long ]s. * ] that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long ]s.
* ] that 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. * ] that 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.
* ] that 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>. * ]: 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 consecutive applications of the unsigned ] operator to the sequence of ]s. * ] on consecutive applications of the unsigned ] operator to the sequence of ]s.
* ] that every even natural number greater than <math>2</math> is the sum of two ]s. * ]: every even natural number greater than <math>2</math> is the sum of two ]s.
* ] that 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>. * ]: 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>.
* ] that all odd integers greater than <math>5</math> can be represented as the sum of an odd ] and an even ]. * ]: 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> * ] 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>
* ] * ]
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* The values of ''g''(''k'') and ''G''(''k'') in ] * The values of ''g''(''k'') and ''G''(''k'') in ]


=====Open questions=====
* Do the ]s have a positive density? * Do the ]s have a positive density?


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


==== Algebraic number theory ==== ==== Algebraic number theory ====
{{Main|Algebraic number theory }} {{Main|Algebraic number theory }}
=====Conjectures and problems=====
* ]: are there infinitely many ] with ]? * ]: are there infinitely many ] with ]?
* ]: actually numerous conjectures, all proposed by ] and ]. * ]: actually numerous conjectures, all proposed by ] and ].
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* ] * ]
* ]: 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>? * ]: 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>?
* ] that primes <math>p</math> do not divide the ] of the maximal real ] of the <math>p</math>-th ]. * ]: 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> * 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>
* ] that the ] of the ] on ]s of ]s are at least <math>1/4</math>. * ]: the ] of the ] on ]s of ]s are at least <math>1/4</math>.
* ] (including ]) * ] (including ])


=====Other=====
* Characterize all algebraic number fields that have some ]. * Characterize all algebraic number fields that have some ].


====Computational number theory==== ====Computational number theory====
{{Main|Computational number theory}} {{Main|Computational number theory}}
* ]: Can integer factorization be done in polynomial time? * Can ] be done in ]?


==== Prime numbers ==== ==== Prime numbers ====
Line 1,048: Line 1,002:
* ] on the ]s that <math>p</math> is prime if and only if <math>pB_{p-1} \equiv -1 \pmod p</math> * ] on the ]s that <math>p</math> is prime if and only if <math>pB_{p-1} \equiv -1 \pmod p</math>
* ] that if an integer is neither a perfect square nor <math>-1</math>, then it is a ] modulo infinitely many ]s <math>p</math> * ] that if an integer is neither a perfect square nor <math>-1</math>, then it is a ] modulo infinitely many ]s <math>p</math>
* ] that 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>. * ]: 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>.
* ] that 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. * ]: 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.
* ] that some ] is composite and thus all Catalan–Mersenne numbers are composite after some point. * ]: some ] is composite and thus all Catalan–Mersenne numbers are composite after some point.
* ] that 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. * ]: 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.
* ] that every number greater than <math>2408</math> is the sum of two ] which both have ]. * ]: every number greater than <math>2408</math> is the sum of two ] which both have ].
* ] on the distribution of ]s in ]s. * ] on the distribution of ]s in ]s.
* ] that no three consecutive numbers are all ]. * ]: no three consecutive numbers are all ].
* ] that 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> * ]: 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. * 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? * 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 ]. * ] on the distribution of ] divisors of ].
* ] that all even ]s greater than <math>2</math> are the sum of two ]s. * ]: all even ]s greater than <math>2</math> are the sum of two ]s.
* ] * ]
* Problems associated to ] * Problems associated to ]
* ] that 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 true. * ]: 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 true.
* ] that for all positive even numbers <math>n</math>, there are infinitely many ]s of size <math>n</math>. * ]: 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>. * ] 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 ]? * ]: is 78,557 the lowest ]?
* ] that there are infinitely many ]s. * ]: there are infinitely many ]s.
* Does the ] hold for all natural numbers? * Does the ] hold for all natural numbers?


=====Open questions=====
* Are all ]s ]? * Are all ]s ]?
* Are all ]s ]? * Are all ]s ]?
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* Are there infinitely many ]s? * Are there infinitely many ]s?
* Are there infinitely many ]s? * Are there infinitely many ]s?
* 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>{{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> * 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>
* Does every prime number appear in the ]? * Does every prime number appear in the ]?
* Find the smallest ] * Find the smallest ]
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* 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'' 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 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 ''k'' ≥ 1, ''b'' ≥ 2, ''c'' ≠ 0, with {{nowrap|1=gcd(''k'', ''c'') = 1}} and {{nowrap|1=gcd(''b'', ''c'') = 1,}} are there infinitely many primes of the form (''k''×''b''<sup>''n''</sup>+''c'')/gcd(''k''+''c'',''b''−1) with integer ''n'' ≥ 1? * 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 ] 2<sup>2<sup>''n''</sup></sup>&nbsp;+&nbsp;1 composite for <math>n > 4</math>? * Is every ] <math>2^{2^n} + 1</math> composite for <math>n > 4</math>?
* Is 509,203 the lowest ]? * Is 509,203 the lowest ]?


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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 ]. 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 ].

====Conjectures, problems, and hypotheses====
* (]) Does the ] below a ] imply the ] everywhere? * (]) Does the ] below a ] imply the ] everywhere?
* Does the ] entail ] for every ] <math>\lambda</math>? * Does the ] entail ] for every ] <math>\lambda</math>?
* Does the ] imply the existence of an ]? * Does the ] imply the existence of an ]?
* 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 ]. * 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 ].
* ] ] that if there is a ] of ]s, then ] satisfies an analogue of ]. * ] ]: if there is a ] of ]s, then ] satisfies an analogue of ].


====Open questions====
* 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 there exist a ] on ℵ<sub>ω</sub>? * Does there exist a ] on ℵ<sub>ω</sub>?
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{{Main|Topology}} {{Main|Topology}}
] asks whether there is an efficient algorithm to identify when the shape presented in a ] is actually the ].]] ] asks whether there is an efficient algorithm to identify when the shape presented in a ] is actually the ].]]
* ]: the ] is an ].
====Conjectures and problems====
* ]: every <math>n</math>-dimensional ] ] is a ].
* ] that the ] is an ].
* ] that every <math>n</math>-dimensional ] ] is a ]. * ]: ] ]s are determined up to ] by their ]s.
* ] that ] ]s are determined up to ] by their ]s.
* ] on rational ]s of certain ]s. * ] on rational ]s of certain ]s.
* ] that if a ] ] has a ], ] on a ], then the group must be a ]. * ]: 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> * 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 ]. * ] on the ] of certain ]s in the ]es of a ], arising from the ].
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* ]: can ]s be recognized in ]? * ]: can ]s be recognized in ]?
* ] relating ]s of ] to the ] of their ]s. * ] relating ]s of ] to the ] of their ]s.
* ] that every ] ] of a two-dimensional ] ] is aspherical. * ]: every ] ] of a two-dimensional ] ] is aspherical.
* ]: given a finite ] two-dimensional ] <math>K</math>, is the space <math>K \times </math> ]? * ]: given a finite ] two-dimensional ] <math>K</math>, is the space <math>K \times </math> ]?


== Problems solved since 1995 == == Problems solved since 1995 ==
], here illustrated with a 2D manifold, was the key tool in ]'s ].]] {{Duplication|date=August 2022|section=y}}], here illustrated with a 2D manifold, was the key tool in ]'s ].]]


===Analysis=== ===Analysis===
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====20th century==== ====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> * ] (], 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 |journal = Annals of Mathematics |title = Ring theoretic properties of certain Hecke algebras |volume = 141 |issue = 3 |pages = 553–572 |oclc = 37032255 |url = http://www.math.harvard.edu/~rtaylor/hecke.ps |doi = 10.2307/2118560 |jstor = 2118560 |citeseerx = 10.1.1.128.531 |archive-url=https://web.archive.org/web/20050301000000*/http://www.math.harvard.edu/~rtaylor/hecke.ps |archive-date=1 March 2005 |url-status=dead}}</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 |url-status=dead |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=== ===Ramsey theory===
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===Theoretical computer science=== ===Theoretical computer science===
*] for Boolean functions (], 2019) <ref>{{cite web *] for Boolean functions (], 2019)<ref>{{cite web
| url = https://www.popularmechanics.com/science/math/g30346822/biggest-math-breakthroughs-2019/ | url = https://www.popularmechanics.com/science/math/g30346822/biggest-math-breakthroughs-2019/
|author =Linkletter, David |author =Linkletter, David
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===Topology=== ===Topology===
*Deciding whether the ] is a ] (], 2020)<ref>, ], volume 191, issue 2, pp. 581–591</ref><ref>, ] 19 May 2020</ref> *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 |pages=581–591 |doi=10.4007/annals.2020.191.2.5}}</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 * ] (], Daniel Groves, Jason Manning, 2012)<ref>{{Cite journal
| arxiv = 1204.2810v1 | arxiv = 1204.2810v1
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| date=2015 | date=2015
| doi=10.4007/annals.2015.182.3.5 | doi-access=free}}</ref> | doi=10.4007/annals.2015.182.3.5 | doi-access=free}}</ref>
* ] (], 2014)<ref>{{Cite magazine |date=March 28, 2017 |title=A Long-Sought Proof, Found and Almost Lost |url=https://www.quantamagazine.org/20170328-statistician-proves-gaussian-correlation-inequality/ |publisher=Natalie Wolchover |access-date=May 2, 2017 |magazine=] |archive-url=https://web.archive.org/web/20170424133433/https://www.quantamagazine.org/20170328-statistician-proves-gaussian-correlation-inequality/ |archive-date=April 24, 2017 |url-status=live }}</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=] |publisher= |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> * 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=School of Mathematics Institute for Advanced Study Einstein Drive Princeton, NJ 08540|publisher=annals.math.princeton.edu (])|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 * ] (], 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=School of Mathematics Institute for Advanced Study Einstein Drive Princeton, NJ 08540|publisher=annals.math.princeton.edu (])|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
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| volume = 2001 | volume = 2001
| year = 2001| issue = 530 | year = 2001| issue = 530
}}</ref><ref>{{cite web|url=https://webusers.imj-prg.fr/~bruno.kahn/preprints/kcag.pdf|title=page 359|access-date=2016-03-18|archive-url=https://web.archive.org/web/20160327035553/https://webusers.imj-prg.fr/~bruno.kahn/preprints/kcag.pdf|archive-date=2016-03-27|url-status=live}}</ref><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>) }}</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===== =====2000s=====
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* *
* *
* The collection of open problems in mathematics build on the principle of user editable ("wiki") site *
* *
* . MathPro Press. * . MathPro Press.

Revision as of 02:05, 17 August 2022

List article of unsolved mathematical problems 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. Although this list may never be comprehensive, 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
The Riemann zeta function, subject of the celebrated and influential unsolved problem known as the Riemann hypothesis

Millennium Prize Problems

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

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.

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.

Notebook problems

Analysis

Main article: Mathematical analysis
The area of the blue region converges to the Euler–Mascheroni constant, which may or may not be a rational number.

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?
  • Problems in Latin squares – Open questions concerning Latin squares
  • The lonely runner conjecture – if k + 1 {\displaystyle k+1} 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 + 1 ) {\displaystyle 1/(k+1)} from each other runner) at some time?
  • 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} ?
  • 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?
  • 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

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.
  • Finding matching upper and lower bounds for k-sets and halving lines
  • Tripod packing: how many tripods can have their apexes packed into a given cube?

Euclidean geometry

Main article: Euclidean geometry

Graph theory

Main article: Graph theory

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

Paths and cycles in graphs

Word-representation of graphs

Miscellaneous graph theory

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.

Notebook problems

  • The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.

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 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 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, 0 {\displaystyle \aleph _{0}} , or 2 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 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 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.)
  • (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering 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

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 are 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

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.
Conjectures, problems and hypotheses

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

This section duplicates the scope of other articles. Please discuss this issue and help introduce a summary style to the section by replacing the section with a link and a summary or by splitting the content into a new article. (August 2022)
Ricci flow, here illustrated with a 2D manifold, was the key tool in Grigori Perelman's solution of the Poincaré conjecture.

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

21st century

2010s
2000s

20th century

See also

References

  1. Thiele, Rüdiger (2005), "On Hilbert and his twenty-four problems", in Van Brummelen, Glen (ed.), Mathematics and the historian's craft. The Kenneth O. May Lectures, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 21, pp. 243–295, ISBN 978-0-387-25284-1
  2. Guy, Richard (1994), Unsolved Problems in Number Theory (2nd ed.), Springer, p. vii, ISBN 978-1-4899-3585-4, archived from the original on 2019-03-23, retrieved 2016-09-22.
  3. Shimura, G. (1989). "Yutaka Taniyama and his time". Bulletin of the London Mathematical Society. 21 (2): 186–196. doi:10.1112/blms/21.2.186.
  4. Friedl, Stefan (2014). "Thurston's vision and the virtual fibering theorem for 3-manifolds". Jahresbericht der Deutschen Mathematiker-Vereinigung. 116 (4): 223–241. doi:10.1365/s13291-014-0102-x. MR 3280572. S2CID 56322745.
  5. Thurston, William P. (1982). "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. New Series. 6 (3): 357–381. doi:10.1090/S0273-0979-1982-15003-0. MR 0648524.
  6. ^ "Millennium Problems". claymath.org. Archived from the original on 2017-06-06. Retrieved 2015-01-20.
  7. "Fields Medal awarded to Artur Avila". Centre national de la recherche scientifique. 2014-08-13. Archived from the original on 2018-07-10. Retrieved 2018-07-07.
  8. Bellos, Alex (2014-08-13). "Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained". The Guardian. Archived from the original on 2016-10-21. Retrieved 2018-07-07.
  9. Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. ISBN 978-9051994902.
  10. "DARPA invests in math". CNN. 2008-10-14. Archived from the original on 2009-03-04. Retrieved 2013-01-14.
  11. "Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO)". DARPA. 2007-09-10. Archived from the original on 2012-10-01. Retrieved 2013-06-25.
  12. "Poincaré Conjecture". Clay Mathematics Institute. Archived from the original on 2013-12-15.
  13. "Smooth 4-dimensional Poincare conjecture". Open Problem Garden. Archived from the original on 2018-01-25. Retrieved 2019-08-06.
  14. Dowling, T. A. (February 1973). "A class of geometric lattices based on finite groups". Journal of Combinatorial Theory. Series B. 14 (1): 61–86. doi:10.1016/S0095-8956(73)80007-3.
  15. * ДНЕСТРОВСКАЯ ТЕТРАДЪ [DNIESTER NOTEBOOK] (PDF) (in Russian), The Russian Academy of Sciences, 1993
  16. Эрлаголъская тетрадъ [Erlagol notebook] (PDF) (in Russian), The Novosibirsk State University, 2018
  17. ^ Waldschmidt, Michel (2013), Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables, Springer, pp. 14, 16, ISBN 9783662115695
  18. Kung, H. T.; Traub, Joseph Frederick (1974), "Optimal order of one-point and multipoint iteration", Journal of the ACM, 21 (4): 643–651, doi:10.1145/321850.321860, S2CID 74921
  19. Smyth, Chris (2008), "The Mahler measure of algebraic numbers: a survey", in McKee, James; Smyth, Chris (eds.), Number Theory and Polynomials, London Mathematical Society Lecture Note Series, vol. 352, Cambridge University Press, pp. 322–349, ISBN 978-0-521-71467-9
  20. Berenstein, Carlos A. (2001) , "Pompeiu problem", Encyclopedia of Mathematics, EMS Press
  21. For some background on the numbers in this problem, see articles by Eric W. Weisstein at Wolfram MathWorld (all articles accessed 15 December 2014):
  22. Waldschmidt, Michel (2008). An introduction to irrationality and transcendence methods (PDF). 2008 Arizona Winter School. Archived from the original (PDF) on 16 December 2014. Retrieved 15 December 2014.
  23. Albert, John, Some unsolved problems in number theory (PDF), archived from the original (PDF) on 17 December 2014, retrieved 15 December 2014 {{citation}}: |archive-date= / |archive-url= timestamp mismatch; 17 January 2014 suggested (help)
  24. Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order, 12 (4): 327–349, CiteSeerX 10.1.1.38.7841, doi:10.1007/BF01110378, MR 1368815, S2CID 14793475.
  25. Tao, Terence (2018). "Some remarks on the lonely runner conjecture". Contributions to Discrete Mathematics. 13 (2): 1–31. arXiv:1701.02048. doi:10.11575/cdm.v13i2.62728.
  26. Bruhn, Henning; Schaudt, Oliver (2015), "The journey of the union-closed sets conjecture" (PDF), Graphs and Combinatorics, 31 (6): 2043–2074, arXiv:1309.3297, doi:10.1007/s00373-014-1515-0, MR 3417215, S2CID 17531822, archived (PDF) from the original on 2017-08-08, retrieved 2017-07-18
  27. "Dedekind Numbers and Related Sequences" (PDF). Archived from the original (PDF) on 2015-03-15. Retrieved 2020-04-30.
  28. Murnaghan, F. D. (1938), "The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups", American Journal of Mathematics, 60 (1): 44–65, doi:10.2307/2371542, JSTOR 2371542, MR 1507301, PMC 1076971, PMID 16577800
  29. Liśkiewicz, Maciej; Ogihara, Mitsunori; Toda, Seinosuke (2003-07-28). "The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes". Theoretical Computer Science. 304 (1): 129–156. doi:10.1016/S0304-3975(03)00080-X.
  30. S. M. Ulam, Problems in Modern Mathematics. Science Editions John Wiley & Sons, Inc., New York, 1964, page 76.
  31. Kaloshin, Vadim; Sorrentino, Alfonso (2018). "On the local Birkhoff conjecture for convex billiards". Annals of Mathematics. 188 (1): 315–380. arXiv:1612.09194. doi:10.4007/annals.2018.188.1.6. S2CID 119171182.
  32. Sarnak, Peter (2011), "Recent progress on the quantum unique ergodicity conjecture", Bulletin of the American Mathematical Society, 48 (2): 211–228, doi:10.1090/S0273-0979-2011-01323-4, MR 2774090
  33. Paul Halmos, Ergodic theory. Chelsea, New York, 1956.
  34. Kari, Jarkko (2009). "Structure of reversible cellular automata". Structure of Reversible Cellular Automata. International Conference on Unconventional Computation. Lecture Notes in Computer Science. Vol. 5715. Springer. p. 6. Bibcode:2009LNCS.5715....6K. doi:10.1007/978-3-642-03745-0_5. ISBN 978-3-642-03744-3.
  35. ^ http://english.log-it-ex.com Archived 2017-11-10 at the Wayback Machine Ten open questions about Sudoku (2012-01-21).
  36. "Higher-Dimensional Tic-Tac-Toe". PBS Infinite Series. YouTube. 2017-09-21. Archived from the original on 2017-10-11. Retrieved 2018-07-29.
  37. Barlet, Daniel; Peternell, Thomas; Schneider, Michael (1990). "On two conjectures of Hartshorne's". Mathematische Annalen. 286 (1–3): 13–25. doi:10.1007/BF01453563. S2CID 122151259.
  38. Maulik, Davesh; Nekrasov, Nikita; Okounov, Andrei; Pandharipande, Rahul (2004-06-05), Gromov–Witten theory and Donaldson–Thomas theory, I, arXiv:math/0312059, Bibcode:2003math.....12059M
  39. Zariski, Oscar (1971). "Some open questions in the theory of singularities". Bulletin of the American Mathematical Society. 77 (4): 481–491. doi:10.1090/S0002-9904-1971-12729-5. MR 0277533.
  40. Bereg, Sergey; Dumitrescu, Adrian; Jiang, Minghui (2010), "On covering problems of Rado", Algorithmica, 57 (3): 538–561, doi:10.1007/s00453-009-9298-z, MR 2609053, S2CID 6511998
  41. Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", American Mathematical Monthly, 100 (10): 916–925, doi:10.2307/2324212, JSTOR 2324212, MR 1252928
  42. Conway, John H.; Neil J.A. Sloane (1999), Sphere Packings, Lattices and Groups (3rd ed.), New York: Springer-Verlag, pp. 21–22, ISBN 978-0-387-98585-5
  43. Hales, Thomas (2017), The Reinhardt conjecture as an optimal control problem, arXiv:1703.01352
  44. Brass, Peter; Moser, William; Pach, János (2005), Research Problems in Discrete Geometry, New York: Springer, p. 45, ISBN 978-0387-23815-9, MR 2163782
  45. Gardner, Martin (1995), New Mathematical Diversions (Revised Edition), Washington: Mathematical Association of America, p. 251
  46. Barros, Manuel (1997), "General Helices and a Theorem of Lancret", Proceedings of the American Mathematical Society, 125 (5): 1503–1509, doi:10.1090/S0002-9939-97-03692-7, JSTOR 2162098
  47. Katz, Mikhail G. (2007), Systolic geometry and topology, Mathematical Surveys and Monographs, vol. 137, American Mathematical Society, Providence, RI, p. 57, doi:10.1090/surv/137, ISBN 978-0-8218-4177-8, MR 2292367
  48. Rosenberg, Steven (1997), The Laplacian on a Riemannian Manifold: An introduction to analysis on manifolds, London Mathematical Society Student Texts, vol. 31, Cambridge: Cambridge University Press, pp. 62–63, doi:10.1017/CBO9780511623783, ISBN 978-0-521-46300-3, MR 1462892
  49. Boltjansky, V.; Gohberg, I. (1985), "11. Hadwiger's Conjecture", Results and Problems in Combinatorial Geometry, Cambridge University Press, pp. 44–46.
  50. Morris, Walter D.; Soltan, Valeriu (2000), "The Erdős-Szekeres problem on points in convex position—a survey", Bull. Amer. Math. Soc., 37 (4): 437–458, doi:10.1090/S0273-0979-00-00877-6, MR 1779413; Suk, Andrew (2016), "On the Erdős–Szekeres convex polygon problem", J. Amer. Math. Soc., 30 (4): 1047–1053, arXiv:1604.08657, doi:10.1090/jams/869, S2CID 15732134
  51. Kalai, Gil (1989), "The number of faces of centrally-symmetric polytopes", Graphs and Combinatorics, 5 (1): 389–391, doi:10.1007/BF01788696, MR 1554357, S2CID 8917264.
  52. Weisstein, Eric W. "Kobon Triangle". MathWorld.
  53. Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed", American Mathematical Monthly, 90 (3): 196–200, doi:10.2307/2975549, JSTOR 2975549, MR 1540158
  54. Matoušek, Jiří (2002), Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, p. 206, doi:10.1007/978-1-4613-0039-7, ISBN 978-0-387-95373-1, MR 1899299
  55. Brass, Peter; Moser, William; Pach, János (2005), "5.1 The Maximum Number of Unit Distances in the Plane", Research problems in discrete geometry, Springer, New York, pp. 183–190, ISBN 978-0-387-23815-9, MR 2163782
  56. Dey, Tamal K. (1998), "Improved bounds for planar k-sets and related problems", Discrete Comput. Geom., 19 (3): 373–382, doi:10.1007/PL00009354, MR 1608878; Tóth, Gábor (2001), "Point sets with many k-sets", Discrete Comput. Geom., 26 (2): 187–194, doi:10.1007/s004540010022, MR 1843435.
  57. Aronov, Boris; Dujmović, Vida; Morin, Pat; Ooms, Aurélien; Schultz Xavier da Silveira, Luís Fernando (2019), "More Turán-type theorems for triangles in convex point sets", Electronic Journal of Combinatorics, 26 (1): P1.8, arXiv:1706.10193, Bibcode:2017arXiv170610193A, doi:10.37236/7224, archived from the original on 2019-02-18, retrieved 2019-02-18
  58. Atiyah, Michael (2001), "Configurations of points", Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 359 (1784): 1375–1387, Bibcode:2001RSPTA.359.1375A, doi:10.1098/rsta.2001.0840, ISSN 1364-503X, MR 1853626, S2CID 55833332
  59. Finch, S. R.; Wetzel, J. E. (2004), "Lost in a forest", American Mathematical Monthly, 11 (8): 645–654, doi:10.2307/4145038, JSTOR 4145038, MR 2091541
  60. Solomon, Yaar; Weiss, Barak (2016), "Dense forests and Danzer sets", Annales Scientifiques de l'École Normale Supérieure, 49 (5): 1053–1074, arXiv:1406.3807, doi:10.24033/asens.2303, MR 3581810, S2CID 672315; Conway, John H., Five $1,000 Problems (Update 2017) (PDF), On-Line Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12
  61. Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer, 34 (1): 18–28, arXiv:1009.1419, doi:10.1007/s00283-011-9255-y, MR 2902144, S2CID 10747746
  62. Arutyunyants, G.; Iosevich, A. (2004), "Falconer conjecture, spherical averages and discrete analogs", in Pach, János (ed.), Towards a Theory of Geometric Graphs, Contemp. Math., vol. 342, Amer. Math. Soc., Providence, RI, pp. 15–24, doi:10.1090/conm/342/06127, ISBN 9780821834848, MR 2065249
  63. Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100
  64. Katz, Nets; Tao, Terence (2002), "Recent progress on the Kakeya conjecture", Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), Publicacions Matemàtiques (Vol. Extra): 161–179, CiteSeerX 10.1.1.241.5335, doi:10.5565/PUBLMAT_Esco02_07, MR 1964819, S2CID 77088 {{citation}}: |issue= has extra text (help)
  65. Weaire, Denis, ed. (1997), The Kelvin Problem, CRC Press, p. 1, ISBN 9780748406326
  66. Brass, Peter; Moser, William; Pach, János (2005), Research problems in discrete geometry, New York: Springer, p. 457, ISBN 9780387299297, MR 2163782
  67. Mahler, Kurt (1939). "Ein Minimalproblem für konvexe Polygone". Mathematica (Zutphen) B: 118–127.
  68. Norwood, Rick; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser", Discrete and Computational Geometry, 7 (2): 153–162, doi:10.1007/BF02187832, MR 1139077
  69. Wagner, Neal R. (1976), "The Sofa Problem" (PDF), The American Mathematical Monthly, 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022, archived (PDF) from the original on 2015-04-20, retrieved 2014-05-14
  70. Chai, Ying; Yuan, Liping; Zamfirescu, Tudor (June–July 2018), "Rupert Property of Archimedean Solids", The American Mathematical Monthly, 125 (6): 497–504, doi:10.1080/00029890.2018.1449505, S2CID 125508192
  71. Steininger, Jakob; Yurkevich, Sergey (December 27, 2021), An algorithmic approach to Rupert's problem, arXiv:2112.13754
  72. Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338
  73. Ghomi, Mohammad (2018-01-01). "D "urer's Unfolding Problem for Convex Polyhedra". Notices of the American Mathematical Society. 65 (1): 25–27. doi:10.1090/noti1609. ISSN 0002-9920.
  74. Whyte, L. L. (1952), "Unique arrangements of points on a sphere", The American Mathematical Monthly, 59 (9): 606–611, doi:10.2307/2306764, JSTOR 2306764, MR 0050303
  75. Howards, Hugh Nelson (2013), "Forming the Borromean rings out of arbitrary polygonal unknots", Journal of Knot Theory and Its Ramifications, 22 (14): 1350083, 15, arXiv:1406.3370, doi:10.1142/S0218216513500831, MR 3190121, S2CID 119674622
  76. Brandts, Jan; Korotov, Sergey; Křížek, Michal; Šolc, Jakub (2009), "On nonobtuse simplicial partitions" (PDF), SIAM Review, 51 (2): 317–335, Bibcode:2009SIAMR..51..317B, doi:10.1137/060669073, MR 2505583, archived (PDF) from the original on 2018-11-04, retrieved 2018-11-22. See in particular Conjecture 23, p. 327.
  77. ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, archived from the original on October 5, 2016, retrieved 2016-10-04.
  78. Bousquet, Nicolas; Bartier, Valentin (2019), "Linear Transformations Between Colorings in Chordal Graphs", in Bender, Michael A.; Svensson, Ola; Herman, Grzegorz (eds.), 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, LIPIcs, vol. 144, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 24:1–24:15, doi:10.4230/LIPIcs.ESA.2019.24, ISBN 9783959771245, S2CID 195791634{{citation}}: CS1 maint: unflagged free DOI (link)
  79. Chung, Fan; Graham, Ron (1998), Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, pp. 97–99.
  80. Chudnovsky, Maria; Seymour, Paul (2014), "Extending the Gyárfás-Sumner conjecture", Journal of Combinatorial Theory, Series B, 105: 11–16, doi:10.1016/j.jctb.2013.11.002, MR 3171779
  81. Toft, Bjarne (1996), "A survey of Hadwiger's conjecture", Congressus Numerantium, 115: 249–283, MR 1411244.
  82. Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991), Unsolved Problems in Geometry, Springer-Verlag, Problem G10.
  83. Hägglund, Jonas; Steffen, Eckhard (2014), "Petersen-colorings and some families of snarks", Ars Mathematica Contemporanea, 7 (1): 161–173, doi:10.26493/1855-3974.288.11a, MR 3047618, archived from the original on 2016-10-03, retrieved 2016-09-30.
  84. Jensen, Tommy R.; Toft, Bjarne (1995), "12.20 List-Edge-Chromatic Numbers", Graph Coloring Problems, New York: Wiley-Interscience, pp. 201–202, ISBN 978-0-471-02865-9.
  85. Molloy, Michael; Reed, Bruce (1998), "A bound on the total chromatic number", Combinatorica, 18 (2): 241–280, CiteSeerX 10.1.1.24.6514, doi:10.1007/PL00009820, MR 1656544, S2CID 9600550.
  86. Barát, János; Tóth, Géza (2010), "Towards the Albertson Conjecture", Electronic Journal of Combinatorics, 17 (1): R73, arXiv:0909.0413, Bibcode:2009arXiv0909.0413B, doi:10.37236/345.
  87. Fulek, Radoslav; Pach, János (2011), "A computational approach to Conway's thrackle conjecture", Computational Geometry, 44 (6–7): 345–355, arXiv:1002.3904, doi:10.1016/j.comgeo.2011.02.001, MR 2785903.
  88. Hartsfield, Nora; Ringel, Gerhard (2013), Pearls in Graph Theory: A Comprehensive Introduction, Dover Books on Mathematics, Courier Dover Publications, p. 247, ISBN 978-0-486-31552-2, MR 2047103.
  89. Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, CiteSeerX 10.1.1.605.4932, doi:10.1007/s00373-010-0934-9, MR 2669457, S2CID 121645, archived (PDF) from the original on 2016-03-04, retrieved 2016-10-04.
  90. Nöllenburg, Martin; Prutkin, Roman; Rutter, Ignaz (2016), "On self-approaching and increasing-chord drawings of 3-connected planar graphs", Journal of Computational Geometry, 7 (1): 47–69, arXiv:1409.0315, doi:10.20382/jocg.v7i1a3, MR 3463906, S2CID 1500695
  91. Pach, János; Sharir, Micha (2009), "5.1 Crossings—the Brick Factory Problem", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, vol. 152, American Mathematical Society, pp. 126–127.
  92. Demaine, E.; O'Rourke, J. (2002–2012), "Problem 45: Smallest Universal Set of Points for Planar Graphs", The Open Problems Project, archived from the original on 2012-08-14, retrieved 2013-03-19.
  93. Florek, Jan (2010), "On Barnette's conjecture", Discrete Mathematics, 310 (10–11): 1531–1535, doi:10.1016/j.disc.2010.01.018, MR 2601261.
  94. Broersma, Hajo; Patel, Viresh; Pyatkin, Artem (2014), "On toughness and Hamiltonicity of $2K_2$-free graphs", Journal of Graph Theory, 75 (3): 244–255, doi:10.1002/jgt.21734, MR 3153119, S2CID 1377980
  95. Jaeger, F. (1985), "A survey of the cycle double cover conjecture", Annals of Discrete Mathematics 27 – Cycles in Graphs, North-Holland Mathematics Studies, vol. 27, pp. 1–12, doi:10.1016/S0304-0208(08)72993-1, ISBN 9780444878038.
  96. Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7, doi:10.37236/3252.
  97. Akiyama, Jin; Exoo, Geoffrey; Harary, Frank (1981), "Covering and packing in graphs. IV. Linear arboricity", Networks, 11 (1): 69–72, doi:10.1002/net.3230110108, MR 0608921.
  98. L. Babai, Automorphism groups, isomorphism, reconstruction Archived 2007-06-13 at the Wayback Machine, in Handbook of Combinatorics, Vol. 2, Elsevier, 1996, 1447–1540.
  99. Lenz, Hanfried; Ringel, Gerhard (1991), "A brief review on Egmont Köhler's mathematical work", Discrete Mathematics, 97 (1–3): 3–16, doi:10.1016/0012-365X(91)90416-Y, MR 1140782
  100. ^ Words and Graphs. Monographs in Theoretical Computer Science. An EATCS Series. 2015. doi:10.1007/978-3-319-25859-1. ISBN 978-3-319-25857-7. S2CID 7727433 – via link.springer.com.
  101. ^ Kitaev, Sergey (2017-05-16). A Comprehensive Introduction to the Theory of Word-Representable Graphs. International Conference on Developments in Language Theory. doi:10.1007/978-3-319-62809-7_2.
  102. ^ Kitaev, S. V.; Pyatkin, A. V. (April 1, 2018). "Word-Representable Graphs: a Survey". Journal of Applied and Industrial Mathematics. 12 (2): 278–296. doi:10.1134/S1990478918020084. S2CID 125814097 – via Springer Link.
  103. ^ Kitaev, Sergey V.; Pyatkin, Artem V. (2018). "Графы, представимые в виде слов. Обзор результатов" [Word-representable graphs: A survey]. Дискретн. анализ и исслед. опер. (in Russian). 25 (2): 19–53. doi:10.17377/daio.2018.25.588.
  104. Marc Elliot Glen (2016). "Colourability and word-representability of near-triangulations". arXiv:1605.01688 .
  105. Kitaev, Sergey (2014-03-06). "On graphs with representation number 3". doi:10.48550/arXiv.1403.1616. {{cite journal}}: Cite journal requires |journal= (help)
  106. Glen, Marc; Kitaev, Sergey; Pyatkin, Artem (2018). "On the representation number of a crown graph". Discrete Applied Mathematics. 244: 89–93. arXiv:1609.00674. doi:10.1016/j.dam.2018.03.013. S2CID 46925617.
  107. Conway, John H., Five $1,000 Problems (Update 2017) (PDF), Online Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12
  108. Chudnovsky, Maria (2014), "The Erdös–Hajnal conjecture—a survey" (PDF), Journal of Graph Theory, 75 (2): 178–190, arXiv:1606.08827, doi:10.1002/jgt.21730, MR 3150572, S2CID 985458, Zbl 1280.05086, archived (PDF) from the original on 2016-03-04, retrieved 2016-09-22.
  109. Gupta, Anupam; Newman, Ilan; Rabinovich, Yuri; Sinclair, Alistair (2004), "Cuts, trees and 1 {\displaystyle \ell _{1}} -embeddings of graphs", Combinatorica, 24 (2): 233–269, CiteSeerX 10.1.1.698.8978, doi:10.1007/s00493-004-0015-x, MR 2071334, S2CID 46133408
  110. Pleanmani, Nopparat (2019), "Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph", Discrete Mathematics, Algorithms and Applications, 11 (6): 1950068, 7, doi:10.1142/s179383091950068x, MR 4044549, S2CID 204207428
  111. Spinrad, Jeremy P. (2003), "2. Implicit graph representation", Efficient Graph Representations, pp. 17–30, ISBN 978-0-8218-2815-1.
  112. "Jorgensen's Conjecture", Open Problem Garden, archived from the original on 2016-11-14, retrieved 2016-11-13.
  113. Baird, William; Bonato, Anthony (2012), "Meyniel's conjecture on the cop number: a survey", Journal of Combinatorics, 3 (2): 225–238, arXiv:1308.3385, doi:10.4310/JOC.2012.v3.n2.a6, MR 2980752, S2CID 18942362
  114. Schwenk, Allen (2012), "Some History on the Reconstruction Conjecture" (PDF), Joint Mathematics Meetings, archived (PDF) from the original on 2015-04-09, retrieved 2018-11-26
  115. Ramachandran, S. (1981), "On a new digraph reconstruction conjecture", Journal of Combinatorial Theory, Series B, 31 (2): 143–149, doi:10.1016/S0095-8956(81)80019-6, MR 0630977
  116. Seymour's 2nd Neighborhood Conjecture Archived 2019-01-11 at the Wayback Machine, Open Problems in Graph Theory and Combinatorics, Douglas B. West.
  117. Blokhuis, A.; Brouwer, A. E. (1988), "Geodetic graphs of diameter two", Geometriae Dedicata, 25 (1–3): 527–533, doi:10.1007/BF00191941, MR 0925851, S2CID 189890651
  118. Kühn, Daniela; Mycroft, Richard; Osthus, Deryk (2011), "A proof of Sumner's universal tournament conjecture for large tournaments", Proceedings of the London Mathematical Society, Third Series, 102 (4): 731–766, arXiv:1010.4430, doi:10.1112/plms/pdq035, MR 2793448, S2CID 119169562, Zbl 1218.05034.
  119. 4-flow conjecture Archived 2018-11-26 at the Wayback Machine and 5-flow conjecture Archived 2018-11-26 at the Wayback Machine, Open Problem Garden
  120. Brešar, Boštjan; Dorbec, Paul; Goddard, Wayne; Hartnell, Bert L.; Henning, Michael A.; Klavžar, Sandi; Rall, Douglas F. (2012), "Vizing's conjecture: a survey and recent results", Journal of Graph Theory, 69 (1): 46–76, CiteSeerX 10.1.1.159.7029, doi:10.1002/jgt.20565, MR 2864622, S2CID 9120720.
  121. Ducey, Joshua E. (2017), "On the critical group of the missing Moore graph", Discrete Mathematics, 340 (5): 1104–1109, arXiv:1509.00327, doi:10.1016/j.disc.2016.10.001, MR 3612450, S2CID 28297244
  122. Fomin, Fedor V.; Høie, Kjartan (2006), "Pathwidth of cubic graphs and exact algorithms", Information Processing Letters, 97 (5): 191–196, doi:10.1016/j.ipl.2005.10.012, MR 2195217
  123. Aschbacher, Michael (1990), "On Conjectures of Guralnick and Thompson", Journal of Algebra, 135 (2): 277–343, doi:10.1016/0021-8693(90)90292-V
  124. Khukhro, Evgeny I.; Mazurov, Victor D. (2019), Unsolved Problems in Group Theory. The Kourovka Notebook, arXiv:1401.0300v16
  125. Hrushovski, Ehud (1989). "Kueker's conjecture for stable theories". Journal of Symbolic Logic. 54 (1): 207–220. doi:10.2307/2275025. JSTOR 2275025. S2CID 41940041.
  126. ^ Shelah S, Classification Theory, North-Holland, 1990
  127. Shelah, Saharon (2009). Classification theory for abstract elementary classes. College Publications. ISBN 978-1-904987-71-0.
  128. Peretz, Assaf (2006). "Geometry of forking in simple theories". Journal of Symbolic Logic. 71 (1): 347–359. arXiv:math/0412356. doi:10.2178/jsl/1140641179. S2CID 9380215.
  129. Cherlin, G.; Shelah, S. (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory, Series B. 97 (3): 293–333. arXiv:math/0512218. doi:10.1016/j.jctb.2006.05.008. S2CID 10425739.
  130. Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
  131. Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae. 159 (1): 1–50. arXiv:math/9802134. Bibcode:1998math......2134S. doi:10.4064/fm-159-1-1-50. S2CID 8846429.
  132. Baldwin, John T. (July 24, 2009). Categoricity (PDF). American Mathematical Society. ISBN 978-0-8218-4893-7. Archived (PDF) from the original on July 29, 2010. Retrieved February 20, 2014.
  133. Shelah, Saharon (2009). "Introduction to classification theory for abstract elementary classes". arXiv:0903.3428. Bibcode:2009arXiv0903.3428S. {{cite journal}}: Cite journal requires |journal= (help)
  134. Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
  135. Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
  136. Keisler, HJ (1967). "Ultraproducts which are not saturated". J. Symb. Log. 32 (1): 23–46. doi:10.2307/2271240. JSTOR 2271240. S2CID 250345806.
  137. Malliaris M, Shelah S, "A dividing line in simple unstable theories." https://arxiv.org/abs/1208.2140 Archived 2017-08-02 at the Wayback Machine
  138. Conrey, Brian (2016), "Lectures on the Riemann zeta function (book review)", Bulletin of the American Mathematical Society, 53 (3): 507–512, doi:10.1090/bull/1525
  139. Singmaster, D. (1971), "Research Problems: How often does an integer occur as a binomial coefficient?", American Mathematical Monthly, 78 (4): 385–386, doi:10.2307/2316907, JSTOR 2316907, MR 1536288.
  140. Aigner, Martin (2013), Markov's theorem and 100 years of the uniqueness conjecture, Cham: Springer, doi:10.1007/978-3-319-00888-2, ISBN 978-3-319-00887-5, MR 3098784
  141. Guo, Song; Sun, Zhi-Wei (2005), "On odd covering systems with distinct moduli", Advances in Applied Mathematics, 35 (2): 182–187, arXiv:math/0412217, doi:10.1016/j.aam.2005.01.004, MR 2152886, S2CID 835158
  142. "Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key". Archived from the original on 2016-03-27. Retrieved 2016-03-18.
  143. Robertson, John P. (1996-10-01). "Magic Squares of Squares". Mathematics Magazine. 69 (4): 289–293. doi:10.1080/0025570X.1996.11996457. ISSN 0025-570X.
  144. Huisman, Sander G. (2016). "Newer sums of three cubes". arXiv:1604.07746 .
  145. Dobson, J. B. (1 April 2017), "On Lerch's formula for the Fermat quotient", p. 23, arXiv:1103.3907v6
  146. Ribenboim, P. (2006). Die Welt der Primzahlen. Springer-Lehrbuch (in German) (2nd ed.). Springer. pp. 242–243. doi:10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
  147. Mazur, Barry (1992), "The topology of rational points", Experimental Mathematics, 1 (1): 35–45, doi:10.1080/10586458.1992.10504244, S2CID 17372107, archived from the original on 2019-04-07, retrieved 2019-04-07
  148. Kuperberg, Greg (1994), "Quadrisecants of knots and links", Journal of Knot Theory and Its Ramifications, 3: 41–50, arXiv:math/9712205, doi:10.1142/S021821659400006X, MR 1265452, S2CID 6103528
  149. Casazza, Peter G.; Fickus, Matthew; Tremain, Janet C.; Weber, Eric (2006). "The Kadison-Singer problem in mathematics and engineering: A detailed account". In Han, Deguang; Jorgensen, Palle E. T.; Larson, David Royal (eds.). Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida. Contemporary Mathematics. Vol. 414. American Mathematical Society. pp. 299–355. doi:10.1090/conm/414/07820. ISBN 978-0-8218-3923-2. Retrieved 24 April 2015.
  150. Mackenzie, Dana. "Kadison–Singer Problem Solved" (PDF). SIAM News. No. January/February 2014. Society for Industrial and Applied Mathematics. Archived (PDF) from the original on 23 October 2014. Retrieved 24 April 2015.
  151. ^ Agol, Ian (2004). "Tameness of hyperbolic 3-manifolds". arXiv:math/0405568.
  152. Kurdyka, Krzysztof; Mostowski, Tadeusz; Parusiński, Adam (2000). "Proof of the gradient conjecture of R. Thom". Annals of Mathematics. 152 (3): 763–792. arXiv:math/9906212. doi:10.2307/2661354. JSTOR 2661354. S2CID 119137528.
  153. Moreira, Joel; Richter, Florian K.; Robertson, Donald (2019). "A proof of a sumset conjecture of Erdős". Annals of Mathematics. 189 (2): 605–652. arXiv:1803.00498. doi:10.4007/annals.2019.189.2.4. S2CID 119158401.
  154. Stanley, Richard P. (1994), "A survey of Eulerian posets", in Bisztriczky, T.; McMullen, P.; Schneider, R.; Weiss, A. Ivić (eds.), Polytopes: abstract, convex and computational (Scarborough, ON, 1993), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 440, Dordrecht: Kluwer Academic Publishers, pp. 301–333, MR 1322068. See in particular p. 316.
  155. Kalai, Gil (2018-12-25). "Amazing: Karim Adiprasito proved the g-conjecture for spheres!". Archived from the original on 2019-02-16. Retrieved 2019-02-15.
  156. Santos, Franciscos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics. 176 (1): 383–412. arXiv:1006.2814. doi:10.4007/annals.2012.176.1.7. S2CID 15325169.
  157. Ziegler, Günter M. (2012). "Who solved the Hirsch conjecture?". Documenta Mathematica. Extra Volume "Optimization Stories": 75–85.
  158. Chung, Fan; Greene, Curtis; Hutchinson, Joan (April 2015). "Herbert S. Wilf (1931–2012)". Notices of the AMS. 62 (4): 358. doi:10.1090/noti1247. ISSN 1088-9477. OCLC 34550461. The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004.
  159. Savchev, Svetoslav (2005). "Kemnitz' conjecture revisited". Discrete Mathematics. 297 (1–3): 196–201. doi:10.1016/j.disc.2005.02.018.
  160. Green, Ben (2004). "The Cameron–Erdős conjecture". The Bulletin of the London Mathematical Society. 36 (6): 769–778. arXiv:math.NT/0304058. doi:10.1112/S0024609304003650. MR 2083752. S2CID 119615076.
  161. "News from 2007". American Mathematical Society. AMS. 31 December 2007. Archived from the original on 17 November 2015. Retrieved 2015-11-13. The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..."
  162. Xue, Jinxin (2014). "Noncollision Singularities in a Planar Four-body Problem" (Document). {{cite document}}: Cite document requires |publisher= (help); Unknown parameter |arxiv= ignored (help)
  163. Xue, Jinxin (2020). "Non-collision singularities in a planar 4-body problem". Acta Mathematica. 224 (2): 253–388. doi:10.4310/ACTA.2020.v224.n2.a2. S2CID 226420221.
  164. Bowditch, Brian H. (2006). "The angel game in the plane" (PDF). School of Mathematics, University of Southampton: warwick.ac.uk Warwick University. Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18.
  165. Kloster, Oddvar. "A Solution to the Angel Problem" (PDF). SINTEF ICT, Postboks 124 Blindern, 0314 Oslo, Norway. Archived from the original (PDF) on 2016-01-07. Retrieved 2016-03-18.{{cite web}}: CS1 maint: location (link)
  166. Mathe, Andras (2007). "The Angel of power 2 wins" (PDF). Combinatorics, Probability and Computing. 16 (3): 363–374. doi:10.1017/S0963548306008303. S2CID 16892955. Archived (PDF) from the original on 2016-10-13. Retrieved 2016-03-18.
  167. Gacs, Peter. "THE ANGEL WINS" (PDF). Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18.
  168. Song, Antoine. "Existence of infinitely many minimal hypersurfaces in closed manifolds" (PDF). www.ams.org. Retrieved 19 June 2021. ..I will present a solution of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves..
  169. "Antoine Song | Clay Mathematics Institute". ...Building on work of Codá Marques and Neves, in 2018 Song proved Yau's conjecture in complete generality
  170. Wolchover, Natalie (July 11, 2017), "Pentagon Tiling Proof Solves Century-Old Math Problem", Quanta Magazine, archived from the original on August 6, 2017, retrieved July 18, 2017
  171. Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics. 179 (2): 683–782. arXiv:1202.6036. doi:10.4007/annals.2014.179.2.6. S2CID 50742102.
  172. Guth, Larry; Katz, Nets Hawk (2015). "On the Erdos distinct distance problem in the plane". Annals of Mathematics. 181 (1): 155–190. arXiv:1011.4105. doi:10.4007/annals.2015.181.1.2.
  173. Henle, Frederick V.; Henle, James M. "Squaring the Plane" (PDF). www.maa.org Mathematics Association of America. Archived (PDF) from the original on 2016-03-24. Retrieved 2016-03-18.
  174. Brock, Jeffrey F.; Canary, Richard D.; Minsky, Yair N. (2012). "The classification of Kleinian surface groups, II: The Ending Lamination Conjecture". Annals of Mathematics. 176 (1): 1–149. arXiv:math/0412006. doi:10.4007/annals.2012.176.1.1.
  175. Connelly, Robert; Demaine, Erik D.; Rote, Günter (2003), "Straightening polygonal arcs and convexifying polygonal cycles" (PDF), Discrete & Computational Geometry, 30 (2): 205–239, doi:10.1007/s00454-003-0006-7, MR 1931840, S2CID 40382145
  176. Shestakov, Ivan P.; Umirbaev, Ualbai U. (2004). "The tame and the wild automorphisms of polynomial rings in three variables". Journal of the American Mathematical Society. 17 (1): 197–227. doi:10.1090/S0894-0347-03-00440-5. MR 2015334.
  177. Hutchings, Michael; Morgan, Frank; Ritoré, Manuel; Ros, Antonio (2002). "Proof of the double bubble conjecture". Annals of Mathematics. Second Series. 155 (2): 459–489. arXiv:math/0406017. doi:10.2307/3062123. hdl:10481/32449. JSTOR 3062123. MR 1906593.
  178. Hales, Thomas C. (2001). "The Honeycomb Conjecture". Discrete & Computational Geometry. 25: 1–22. arXiv:math/9906042. doi:10.1007/s004540010071.
  179. Ullmo, E (1998). "Positivité et Discrétion des Points Algébriques des Courbes". Annals of Mathematics. 147 (1): 167–179. arXiv:alg-geom/9606017. doi:10.2307/120987. JSTOR 120987. S2CID 119717506. Zbl 0934.14013.
  180. Zhang, S.-W. (1998). "Equidistribution of small points on abelian varieties". Annals of Mathematics. 147 (1): 159–165. doi:10.2307/120986. JSTOR 120986.
  181. Hales, Thomas; Adams, Mark; Bauer, Gertrud; Dang, Dat Tat; Harrison, John; Hoang, Le Truong; Kaliszyk, Cezary; Magron, Victor; McLaughlin, Sean; Nguyen, Tat Thang; Nguyen, Quang Truong; Nipkow, Tobias; Obua, Steven; Pleso, Joseph; Rute, Jason; Solovyev, Alexey; Ta, Thi Hoai An; Tran, Nam Trung; Trieu, Thi Diep; Urban, Josef; Ky, Vu; Zumkeller, Roland (2017). "A formal proof of the Kepler conjecture". Forum of Mathematics, Pi. 5: e2. arXiv:1501.02155. doi:10.1017/fmp.2017.1.
  182. Hales, Thomas C.; McLaughlin, Sean (2010). "The dodecahedral conjecture". Journal of the American Mathematical Society. 23 (2): 299–344. arXiv:math/9811079. Bibcode:2010JAMS...23..299H. doi:10.1090/S0894-0347-09-00647-X.
  183. Dujmović, Vida; Eppstein, David; Hickingbotham, Robert; Morin, Pat; Wood, David R. (August 2021). "Stack-number is not bounded by queue-number". Combinatorica. 42 (2): 151–164. arXiv:2011.04195. doi:10.1007/s00493-021-4585-7. S2CID 226281691.
  184. Huang, C.; Kotzig, A.; Rosa, A. (1982). "Further results on tree labellings". Utilitas Mathematica. 21: 31–48. MR 0668845..
  185. Hartnett, Kevin (19 February 2020). "Rainbow Proof Shows Graphs Have Uniform Parts". Quanta Magazine. Retrieved 2020-02-29.
  186. Shitov, Yaroslav (1 September 2019). "Counterexamples to Hedetniemi's conjecture". Annals of Mathematics. 190 (2): 663–667. arXiv:1905.02167. doi:10.4007/annals.2019.190.2.6. JSTOR 10.4007/annals.2019.190.2.6. MR 3997132. S2CID 146120733. Zbl 1451.05087. Retrieved 19 July 2021.
  187. Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872.2014.967398. S2CID 117651702.
  188. Chalopin, Jérémie; Gonçalves, Daniel (2009). "Every planar graph is the intersection graph of segments in the plane: extended abstract". In Mitzenmacher, Michael (ed.). Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009. ACM. pp. 631–638. doi:10.1145/1536414.1536500.
  189. Aharoni, Ron; Berger, Eli (2009). "Menger's theorem for infinite graphs". Inventiones Mathematicae. 176 (1): 1–62. arXiv:math/0509397. Bibcode:2009InMat.176....1A. doi:10.1007/s00222-008-0157-3.
  190. Seigel-Itzkovich, Judy (2008-02-08). "Russian immigrant solves math puzzle". The Jerusalem Post. Retrieved 2015-11-12.
  191. "Graph Theory". Archived from the original on 2016-03-08. Retrieved 2016-03-18.
  192. Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2002). "The strong perfect graph theorem". Annals of Mathematics. 164: 51–229. arXiv:math/0212070. Bibcode:2002math.....12070C. doi:10.4007/annals.2006.164.51. S2CID 119151552.
  193. Joel Friedman, "Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: With an Appendix by Warren Dicks" Mem. Amer. Math. Soc., 233 (2015), no. 1100.
  194. Mineyev, Igor (2012). "Submultiplicativity and the Hanna Neumann conjecture". Annals of Mathematics. Second Series. 175 (1): 393–414. doi:10.4007/annals.2012.175.1.11. MR 2874647.
  195. Namazi, Hossein; Souto, Juan (2012). "Non-realizability and ending laminations: Proof of the density conjecture". Acta Mathematica. 209 (2): 323–395. doi:10.1007/s11511-012-0088-0.
  196. Bourgain, Jean; Ciprian, Demeter; Larry, Guth (2015). "Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three". Annals of Mathematics. 184 (2): 633–682. arXiv:1512.01565. Bibcode:2015arXiv151201565B. doi:10.4007/annals.2016.184.2.7. hdl:1721.1/115568. S2CID 43929329.
  197. Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 .
  198. Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 .
  199. Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 .
  200. Zhang, Yitang (2014-05-01). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. ISSN 0003-486X.
  201. "Bounded gaps between primes - Polymath Wiki". asone.ai. Retrieved 2021-08-27.
  202. Maynard, James (2015-01-01). "Small gaps between primes". Annals of Mathematics: 383–413. arXiv:1311.4600. doi:10.4007/annals.2015.181.1.7. ISSN 0003-486X. S2CID 55175056.
  203. Cilleruelo, Javier (2010). "Generalized Sidon sets". Advances in Mathematics. 225 (5): 2786–2807. doi:10.1016/j.aim.2010.05.010. hdl:10261/31032. S2CID 7385280.
  204. Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)", Inventiones Mathematicae, 178 (3): 485–504, Bibcode:2009InMat.178..485K, CiteSeerX 10.1.1.518.4611, doi:10.1007/s00222-009-0205-7, S2CID 14846347
  205. Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)", Inventiones Mathematicae, 178 (3): 505–586, Bibcode:2009InMat.178..505K, CiteSeerX 10.1.1.228.8022, doi:10.1007/s00222-009-0206-6, S2CID 189820189
  206. "2011 Cole Prize in Number Theory" (PDF). Notices of the AMS. 58 (4): 610–611. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2015-11-06. Retrieved 2015-11-12.
  207. "Bombieri and Tao Receive King Faisal Prize" (PDF). Notices of the AMS. 57 (5): 642–643. May 2010. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18. Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem.
  208. Metsänkylä, Tauno (5 September 2003). "Catalan's conjecture: another old diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. 41 (1): 43–57. doi:10.1090/s0273-0979-03-00993-5. ISSN 0273-0979. Archived (PDF) from the original on 4 March 2016. Retrieved 13 November 2015. The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu.
  209. Croot, Ernest S., III (2000). Unit Fractions. Ph.D. thesis. University of Georgia, Athens.{{cite book}}: CS1 maint: multiple names: authors list (link) Croot, Ernest S., III (2003). "On a coloring conjecture about unit fractions". Annals of Mathematics. 157 (2): 545–556. arXiv:math.NT/0311421. Bibcode:2003math.....11421C. doi:10.4007/annals.2003.157.545. S2CID 13514070.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  210. Lafforgue, Laurent (1998), "Chtoucas de Drinfeld et applications" [Drinfelʹd shtukas and applications], Documenta Mathematica (in French), II: 563–570, ISSN 1431-0635, MR 1648105, archived from the original on 2018-04-27, retrieved 2016-03-18
  211. Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255. Archived (PDF) from the original on 2011-05-10. Retrieved 2016-03-06.
  212. Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255. Archived from the original on 16 September 2000.
  213. Lee, Choongbum (2017). "Ramsey numbers of degenerate graphs". Annals of Mathematics. 185 (3): 791–829. arXiv:1505.04773. doi:10.4007/annals.2017.185.3.2. S2CID 7974973.
  214. Lamb, Evelyn (26 May 2016). "Two-hundred-terabyte maths proof is largest ever". Nature. 534 (7605): 17–18. Bibcode:2016Natur.534...17L. doi:10.1038/nature.2016.19990. PMID 27251254.
  215. Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016). "Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer". In Creignou, N.; Le Berre, D. (eds.). Theory and Applications of Satisfiability Testing – SAT 2016. Lecture Notes in Computer Science. Vol. 9710. Springer, . pp. 228–245. arXiv:1605.00723. doi:10.1007/978-3-319-40970-2_15. ISBN 978-3-319-40969-6. MR 3534782. S2CID 7912943.
  216. Linkletter, David (27 December 2019). "The 10 Biggest Math Breakthroughs of 2019". www.popularmechanics.com. Hearst Digital Media. Retrieved 20 June 2021.
  217. Piccirillo, Lisa (2020). "The Conway knot is not slice". Annals of Mathematics. 191: 581–591. doi:10.4007/annals.2020.191.2.5.
  218. Klarreich, Erica (2020-05-19). "Graduate Student Solves Decades-Old Conway Knot Problem". Quanta Magazine. Retrieved 2022-08-17.
  219. Agol, Ian (2013). "The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves, and Jason Manning)" (PDF). Documenta Mathematica. 18: 1045–1087. arXiv:1204.2810v1.
  220. Brendle, Simon (2013). "Embedded minimal tori in S 3 {\displaystyle S^{3}} and the Lawson conjecture". Acta Mathematica. 211 (2): 177–190. arXiv:1203.6597. doi:10.1007/s11511-013-0101-2.
  221. Kahn, Jeremy; Markovic, Vladimir (2015). "The good pants homology and the Ehrenpreis conjecture". Annals of Mathematics. 182 (1): 1–72. arXiv:1101.1330. doi:10.4007/annals.2015.182.1.1.
  222. Austin, Tim (December 2013). "Rational group ring elements with kernels having irrational dimension". Proceedings of the London Mathematical Society. 107 (6): 1424–1448. arXiv:0909.2360. Bibcode:2009arXiv0909.2360A. doi:10.1112/plms/pdt029. S2CID 115160094.
  223. Lurie, Jacob (2009). "On the classification of topological field theories". Current Developments in Mathematics. 2008: 129–280. arXiv:0905.0465. Bibcode:2009arXiv0905.0465L. doi:10.4310/cdm.2008.v2008.n1.a3. S2CID 115162503.
  224. ^ "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (PDF) (Press release). Clay Mathematics Institute. March 18, 2010. Archived from the original on March 22, 2010. Retrieved November 13, 2015. The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.
  225. Morgan, John; Tian, Gang (2008). "Completion of the Proof of the Geometrization Conjecture". arXiv:0809.4040 .
  226. Norio Iwase (1 November 1998). "Ganea's Conjecture on Lusternik-Schnirelmann Category". ResearchGate.
  227. Tao, Terence (2015). "The Erdős discrepancy problem". arXiv:1509.05363v5 .
  228. Duncan, John F. R.; Griffin, Michael J.; Ono, Ken (1 December 2015). "Proof of the umbral moonshine conjecture". Research in the Mathematical Sciences. 2 (1): 26. arXiv:1503.01472. Bibcode:2015arXiv150301472D. doi:10.1186/s40687-015-0044-7. S2CID 43589605.{{cite journal}}: CS1 maint: unflagged free DOI (link)
  229. Cheeger, Jeff; Naber, Aaron (2015). "Regularity of Einstein Manifolds and the Codimension 4 Conjecture". Annals of Mathematics. 182 (3): 1093–1165. arXiv:1406.6534. doi:10.4007/annals.2015.182.3.5.
  230. Wolchover, Natalie (March 28, 2017). "A Long-Sought Proof, Found and Almost Lost". Quanta Magazine. Archived from the original on April 24, 2017. Retrieved May 2, 2017.
  231. Newman, Alantha; Nikolov, Aleksandar (2011). "A counterexample to Beck's conjecture on the discrepancy of three permutations". arXiv:1104.2922 .
  232. Voevodsky, Vladimir (1 July 2011). "On motivic cohomology with Z/l-coefficients" (PDF). School of Mathematics Institute for Advanced Study Einstein Drive Princeton, NJ 08540: annals.math.princeton.edu (Princeton University). pp. 401–438. Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.{{cite web}}: CS1 maint: location (link)
  233. Geisser, Thomas; Levine, Marc (2001). "The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky". Journal für die Reine und Angewandte Mathematik. 2001 (530): 55–103. doi:10.1515/crll.2001.006. MR 1807268.
  234. Kahn, Bruno. "Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry" (PDF). webusers.imj-prg.fr. Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.
  235. "motivic cohomology – Milnor–Bloch–Kato conjecture implies the Beilinson-Lichtenbaum conjecture – MathOverflow". Retrieved 2016-03-18.
  236. Mattman, Thomas W.; Solis, Pablo (2009). "A proof of the Kauffman-Harary Conjecture". Algebraic & Geometric Topology. 9 (4): 2027–2039. arXiv:0906.1612. Bibcode:2009arXiv0906.1612M. doi:10.2140/agt.2009.9.2027. S2CID 8447495.
  237. Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics. 175 (3): 1127–1190. arXiv:0910.5501. doi:10.4007/annals.2012.175.3.4.
  238. Lu, Zhiqin (September 2011) . "Normal Scalar Curvature Conjecture and its applications". Journal of Functional Analysis. 261 (5): 1284–1308. arXiv:0711.3510. doi:10.1016/j.jfa.2011.05.002.
  239. Dencker, Nils (2006), "The resolution of the Nirenberg–Treves conjecture" (PDF), Annals of Mathematics, 163 (2): 405–444, doi:10.4007/annals.2006.163.405, S2CID 16630732, archived (PDF) from the original on 2018-07-20, retrieved 2019-04-07
  240. "Research Awards". Clay Mathematics Institute. Archived from the original on 2019-04-07. Retrieved 2019-04-07.
  241. Lewis, A. S.; Parrilo, P. A.; Ramana, M. V. (2005). "The Lax conjecture is true". Proceedings of the American Mathematical Society. 133 (9): 2495–2499. doi:10.1090/S0002-9939-05-07752-X. MR 2146191. S2CID 17436983.
  242. "Fields Medal – Ngô Bảo Châu". International Congress of Mathematicians 2010. ICM. 19 August 2010. Archived from the original on 24 September 2015. Retrieved 2015-11-12. 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.
  243. Voevodsky, Vladimir (2003). "Reduced power operations in motivic cohomology" (PDF). Publications Mathématiques de l'IHÉS. 98: 1–57. arXiv:math/0107109. CiteSeerX 10.1.1.170.4427. doi:10.1007/s10240-003-0009-z. S2CID 8172797. Archived from the original on 2017-07-28. Retrieved 2016-03-18.
  244. Baruch, Ehud Moshe (2003). "A proof of Kirillov's conjecture". Annals of Mathematics. Second Series. 158 (1): 207–252. doi:10.4007/annals.2003.158.207. MR 1999922.
  245. Haas, Bertrand. "A Simple Counterexample to Kouchnirenko's Conjecture" (PDF). Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.
  246. Haiman, Mark (2001). "Hilbert schemes, polygraphs and the Macdonald positivity conjecture". Journal of the American Mathematical Society. 14 (4): 941–1006. doi:10.1090/S0894-0347-01-00373-3. MR 1839919. S2CID 9253880.
  247. Auscher, Pascal; Hofmann, Steve; Lacey, Michael; McIntosh, Alan; Tchamitchian, Ph. (2002). "The solution of the Kato square root problem for second order elliptic operators on R n {\displaystyle \mathbb {R} ^{n}} ". Annals of Mathematics. Second Series. 156 (2): 633–654. doi:10.2307/3597201. JSTOR 3597201. MR 1933726.
  248. Barbieri-Viale, Luca; Rosenschon, Andreas; Saito, Morihiko (2003). "Deligne's Conjecture on 1-Motives". Annals of Mathematics. 158 (2): 593–633. arXiv:math/0102150. doi:10.4007/annals.2003.158.593.
  249. Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3-adic exercises", Journal of the American Mathematical Society, 14 (4): 843–939, doi:10.1090/S0894-0347-01-00370-8, ISSN 0894-0347, MR 1839918
  250. Luca, Florian (2000). "On a conjecture of Erdős and Stewart" (PDF). Mathematics of Computation. 70 (234): 893–897. Bibcode:2001MaCom..70..893L. doi:10.1090/s0025-5718-00-01178-9. Archived (PDF) from the original on 2016-04-02. Retrieved 2016-03-18.
  251. Atiyah, Michael (2000). "The geometry of classical particles". In Yau, Shing-Tung (ed.). Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer. Surveys in Differential Geometry. Vol. 7. Somerville, Massachusetts: International Press. pp. 1–15. doi:10.4310/SDG.2002.v7.n1.a1. MR 1919420.
  252. Merel, Loïc (1996). ""Bornes pour la torsion des courbes elliptiques sur les corps de nombres" ". Inventiones Mathematicae. 124 (1): 437–449. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. MR 1369424. S2CID 3590991.
  253. Chen, Zhibo (1996). "Harary's conjectures on integral sum graphs". Discrete Mathematics. 160 (1–3): 241–244. doi:10.1016/0012-365X(95)00163-Q.

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