List 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.

File:Riemann-Zeta-Func.png, subject of the Riemann hypothesis{{Cite web |title=Math Problems Guide: From Simple to Hardest Math Problems Tips & Examples. |url=https://blendedlearningmath.com/math-word-problems-to-challenge-university-students/ |access-date=2024-11-28 |website=blendedlearningmath |language=en-US}}]]

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

• Birch and Swinnerton-Dyer conjecture
• Hodge conjecture
• Navier–Stokes existence and smoothness
• P versus NP
• Riemann hypothesis
• Yang–Mills existence and mass gap

The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman in 2003.{{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}} 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.{{cite web |last=rybu |date=November 7, 2009 |title=Smooth 4-dimensional Poincare conjecture |url=http://www.openproblemgarden.org/?q=op/smooth_4_dimensional_poincare_conjecture |url-status=live |archive-url=https://web.archive.org/web/20180125203721/http://www.openproblemgarden.org/?q=op%2Fsmooth_4_dimensional_poincare_conjecture |archive-date=2018-01-25 |access-date=2019-08-06 |website=Open Problem Garden}}

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

{{Main|Algebra}}

File:Regular tetrahedron inscribed in a sphere.svg 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.]]

• Birch–Tate conjecture on the relation between the order of the center of the Steinberg group of the ring of integers of a number field to the field's Dedekind zeta function.
• Bombieri–Lang conjectures on densities of rational points of algebraic surfaces and algebraic varieties defined on number fields and their field extensions.
• Connes embedding problem in Von Neumann algebra theory
• Crouzeix's conjecture: the matrix norm of a complex function $f$ applied to a complex matrix $A$ is at most twice the supremum of $|f(z)|$ over the field of values of $A$.
• Determinantal conjecture on the determinant of the sum of two normal matrices.
• Eilenberg–Ganea conjecture: a group with cohomological dimension 2 also has a 2-dimensional Eilenberg–MacLane space $K(G,\; 1)$.
• Farrell–Jones conjecture on whether certain assembly maps are isomorphisms.
• Bost conjecture: a specific case of the Farrell–Jones conjecture
• Finite lattice representation problem: is every finite lattice isomorphic to the congruence lattice of some finite algebra?{{cite journal |last1=Dowling |first1=T. A. |title=A class of geometric lattices based on finite groups|journal=Journal of Combinatorial Theory |series=Series B |date=February 1973 |volume=14 |issue=1 |pages=61–86 |doi=10.1016/S0095-8956(73)80007-3 | doi-access=free }}
• Goncharov conjecture on the cohomology of certain motivic complexes.
• Green's conjecture: the Clifford index of a non-hyperelliptic curve is determined by the extent to which it, as a canonical curve, has linear syzygies.
• Grothendieck–Katz p-curvature conjecture: a conjectured local–global principle for linear ordinary differential equations.
• Hadamard conjecture: for every positive integer $k$, a Hadamard matrix of order $4k$ exists.
• Williamson conjecture: the problem of finding Williamson matrices, which can be used to construct Hadamard matrices.
• Hadamard's maximal determinant problem: what is the largest determinant of a matrix with entries all equal to 1 or −1?
• Hilbert's fifteenth problem: put Schubert calculus on a rigorous foundation.
• Hilbert's sixteenth problem: what are the possible configurations of the connected components of M-curves?
• Homological conjectures in commutative algebra
• Jacobson's conjecture: the intersection of all powers of the Jacobson radical of a left-and-right Noetherian ring is precisely 0.
• Kaplansky's conjectures
• Köthe conjecture: if a ring has no nil ideal other than $\backslash \{0\backslash \}$, then it has no nil one-sided ideal other than $\backslash \{0\backslash \}$.
• Monomial conjecture on Noetherian local rings
• Existence of perfect cuboids and associated cuboid conjectures
• Pierce–Birkhoff conjecture: every piecewise-polynomial $f:\backslash mathbb\{R\}^\{n\}\backslash rightarrow\backslash mathbb\{R\}$ is the maximum of a finite set of minimums of finite collections of polynomials.
• Rota's basis conjecture: for matroids of rank $n$ with $n$ disjoint bases $B\_\{i\}$, it is possible to create an $n\; \backslash times\; n$ matrix whose rows are $B\_\{i\}$ and whose columns are also bases.
• Serre's conjecture II: if $G$ is a simply connected semisimple algebraic group over a perfect field of cohomological dimension at most $2$, then the Galois cohomology set $H^\{1\}(F,\; G)$ is zero.
• Serre's positivity conjecture that if $R$ is a commutative regular local ring, and $P,\; Q$ are prime ideals of $R$, then $\backslash dim\; (R/P)\; +\; \backslash dim\; (R/Q)\; =\; \backslash dim\; (R)$ implies $\backslash chi(R/P,\; R/Q)\; >\; 0$.
• Uniform boundedness conjecture for rational points: do algebraic curves of genus $g\; \backslash geq\; 2$ over number fields $K$ have at most some bounded number $N(K,\; g)$ of $K$-rational points?
• Wild problems: problems involving classification of pairs of $n\backslash times\; n$ matrices under simultaneous conjugation.
• Zariski–Lipman conjecture: for a complex algebraic variety $V$ with coordinate ring $R$, if the derivations of $R$ are a free module over $R$, then $V$ is smooth.
• Zauner's conjecture: do SIC-POVMs exist in all dimensions?
• Zilber–Pink conjecture that if $X$ is a mixed Shimura variety or semiabelian variety defined over $\backslash mathbb\{C\}$, and $V\; \backslash subseteq\; X$ is a subvariety, then $V$ contains only finitely many atypical subvarieties.

{{Main|Group theory }}

File:FreeBurnsideGroupExp3Gens2.png $B(2,3)$ is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups $B(m,n)$ are finite remains open.]]

• Andrews–Curtis conjecture: every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on relators and conjugations of relators
• Bounded Burnside problem: for which positive integers m, n is the free Burnside group {{nowrap|B(m,n)}} finite? In particular, is {{nowrap|B(2, 5)}} finite?
• Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems{{citation |last=Aschbacher |first=Michael |author-link=Michael Aschbacher |title=On Conjectures of Guralnick and Thompson |journal=Journal of Algebra |volume=135 |issue=2 |pages=277–343 |year=1990 |doi=10.1016/0021-8693(90)90292-V}}
• Herzog–Schönheim conjecture: if a finite system of left cosets of subgroups of a group $G$ form a partition of $G$, then the finite indices of said subgroups cannot be distinct.
• The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
• Isomorphism problem of Coxeter groups
• Are there an infinite number of Leinster groups?
• Does generalized moonshine exist?
• Is every finitely presented periodic group finite?
• Is every group surjunctive?
• Is every discrete, countable group sofic?
• Problems in loop theory and quasigroup theory consider generalizations of groups

• Arthur's conjectures
• Dade's conjecture relating the numbers of characters of blocks of a finite group to the numbers of characters of blocks of local subgroups.
• Demazure conjecture on representations of algebraic groups over the integers.
• Kazhdan–Lusztig conjectures relating the values of the Kazhdan–Lusztig polynomials at 1 with representations of complex semisimple Lie groups and Lie algebras.
• McKay conjecture: in a group $G$, the number of irreducible complex characters of degree not divisible by a prime number $p$ is equal to the number of irreducible complex characters of the normalizer of any Sylow $p$-subgroup within $G$.

{{Main|Mathematical analysis}}

• The Brennan conjecture: estimating the integral of powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of $\backslash mathbb\{C\}$
• Fuglede's conjecture on whether nonconvex sets in $\backslash mathbb\{R\}$ and $\backslash mathbb\{R\}^\{2\}$ are spectral if and only if they tile by translation.
• Goodman's conjecture on the coefficients of multivalued functions
• Invariant subspace problem – does every bounded operator on a complex Banach space send some non-trivial closed subspace to itself?
• Kung–Traub conjecture on the optimal order of a multipoint iteration without memory{{citation |last1=Kung |first1=H. T. |last2=Traub |first2=Joseph Frederick |author-link1=H. T. Kung |author-link2=Joseph F. Traub |title=Optimal order of one-point and multipoint iteration |journal=Journal of the ACM |year=1974 |volume=21 |number=4 |pages=643–651|doi=10.1145/321850.321860 |s2cid=74921 }}
• Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials{{citation | first=Chris | last=Smyth | chapter=The Mahler measure of algebraic numbers: a survey | pages=322–349 | editor1-first=James | editor1-last=McKee | editor2-last=Smyth | editor2-first=Chris | title=Number Theory and Polynomials | series=London Mathematical Society Lecture Note Series | volume=352 | publisher=Cambridge University Press | year=2008 | isbn=978-0-521-71467-9 }}
• The mean value problem: given a complex polynomial $f$ of degree $d\; \backslash ge\; 2$ and a complex number $z$, is there a critical point $c$ of $f$ such that $|f(z)-f(c)|\; \backslash le\; |f\text{'}(z)||z-c|$?
• The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy{{SpringerEOM|title=Pompeiu problem|id=Pompeiu_problem&oldid=14506|author-last1=Berenstein|author-first1=Carlos A.}}
• Sendov's conjecture: if a complex polynomial with degree at least $2$ has all roots in the closed unit disk, then each root is within distance $1$ from some critical point.
• Vitushkin's conjecture on compact subsets of $\backslash mathbb\{C\}$ with analytic capacity $0$
• What is the exact value of Landau's constants, including Bloch's constant?

• Regularity of solutions of Euler equations
• Convergence of [https://mathworld.wolfram.com/FlintHillsSeries.html Flint Hills series]
• Regularity of solutions of Vlasov–Maxwell equations

{{Main|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?{{citation

| last1 = Brightwell | first1 = Graham R.

| last2 = Felsner | first2 = Stefan

| last3 = Trotter | first3 = William T.

| doi = 10.1007/BF01110378

| mr = 1368815

| issue = 4

| journal = Order

| pages = 327–349

| title = Balancing pairs and the cross product conjecture

| volume = 12

| year = 1995| citeseerx = 10.1.1.38.7841

| s2cid = 14793475

}}.

• The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
• Problems in Latin squares – open questions concerning Latin squares
• The lonely runner conjecture – if $k$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance $1/k$ from each other runner) at some time?{{cite journal

| last=Tao | first=Terence | author-link=Terence Tao

| title=Some remarks on the lonely runner conjecture

| journal=Contributions to Discrete Mathematics

| volume=13

| issue=2

| pages=1–31

| date=2018

| arxiv=1701.02048

| doi=10.11575/cdm.v13i2.62728 | doi-access=free}}

• Map folding – various problems in map folding and stamp folding.
• No-three-in-line problem – how many points can be placed in the $n\; \backslash times\; n$ grid so that no three of them lie on a line?
• Rudin's conjecture on the number of squares in finite arithmetic progressions{{cite journal|journal=LMS Journal of Computation and Mathematics|volume=17|issue=1|year=2014|pages=58–76|title=On a conjecture of Rudin on squares in arithmetic progressions|author=González-Jiménez, Enrique|author2=Xarles, Xavier|doi=10.1112/S1461157013000259|arxiv=1301.5122|s2cid=11615385 }}
• The sunflower conjecture – can the number of $k$ size sets required for the existence of a sunflower of $r$ sets be bounded by an exponential function in $k$ for every fixed $r>2$?
• Frankl's union-closed sets conjecture – for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets{{citation

| last1 = Bruhn

| first1 = Henning

| last2 = Schaudt

| first2 = Oliver

| doi = 10.1007/s00373-014-1515-0

| issue = 6

| journal = Graphs and Combinatorics

| mr = 3417215

| pages = 2043–2074

| title = The journey of the union-closed sets conjecture

| url = http://www.zaik.uni-koeln.de/~schaudt/UCSurvey.pdf

| volume = 31

| year = 2015

| arxiv = 1309.3297

| s2cid = 17531822

| access-date = 2017-07-18

| archive-url = https://web.archive.org/web/20170808104232/http://www.zaik.uni-koeln.de/~schaudt/UCSurvey.pdf

| archive-date = 2017-08-08

| url-status = live

}}

• Give a combinatorial interpretation of the Kronecker coefficients{{citation

| last = Murnaghan | first = F. D.

| doi = 10.2307/2371542

| issue = 1

| journal = American Journal of Mathematics

| mr = 1507301

| pages = 44–65

| title = The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups

| volume = 60

| year = 1938| pmc = 1076971

| pmid=16577800

| jstor = 2371542

}}

• The values of the Dedekind numbers $M(n)$ for $n\; \backslash ge\; 10${{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 }}
• The values of the Ramsey numbers, particularly $R(5,\; 5)$
• The values of the Van der Waerden numbers
• Finding a function to model n-step self-avoiding walks{{Cite journal|last1=Liśkiewicz|first1=Maciej|last2=Ogihara|first2=Mitsunori|last3=Toda|first3=Seinosuke|date=2003-07-28|title=The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes|journal=Theoretical Computer Science|volume=304|issue=1|pages=129–156|doi=10.1016/S0304-3975(03)00080-X|s2cid=33806100 }}

{{Main|Dynamical system}}

File:Mandel zoom 07 satellite.jpg. It is not known whether the Mandelbrot set is locally connected or not.]]

• Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theory.
• Berry–Tabor conjecture in quantum chaos
• Banach's problem – is there an ergodic system with simple Lebesgue spectrum?S. M. Ulam, Problems in Modern Mathematics. Science Editions John Wiley & Sons, Inc., New York, 1964, page 76.
• Birkhoff conjecture – if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse?{{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=Annals of Mathematics|arxiv=1612.09194 |s2cid=119171182 }}
• Collatz conjecture (also known as the $3n\; +\; 1$ conjecture)
• Eden's conjecture that the supremum of the local Lyapunov dimensions on the global attractor is achieved on a stationary point or an unstable periodic orbit embedded into the attractor.
• Eremenko's conjecture: every component of the escaping set of an entire transcendental function is unbounded.
• Fatou conjecture that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters.
• Furstenberg conjecture – is every invariant and ergodic measure for the $\backslash times\; 2,\backslash times\; 3$ action on the circle either Lebesgue or atomic?
• Kaplan–Yorke conjecture on the dimension of an attractor in terms of its Lyapunov exponents
• Margulis conjecture – measure classification for diagonalizable actions in higher-rank groups.
• Hilbert–Arnold problem – is there a uniform bound on limit cycles in generic finite-parameter families of vector fields on a sphere?
• MLC conjecture – is the Mandelbrot set locally connected?
• Many problems concerning an outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.
• Quantum unique ergodicity conjecture on the distribution of large-frequency eigenfunctions of the Laplacian on a negatively-curved manifold{{citation |last=Sarnak |first=Peter |author-link=Peter Sarnak |title=Recent progress on the quantum unique ergodicity conjecture |journal=Bulletin of the American Mathematical Society |volume=48 |issue=2 |year=2011 |pages=211–228 |doi=10.1090/S0273-0979-2011-01323-4 |mr=2774090|doi-access=free }}
• Rokhlin's multiple mixing problem – are all strongly mixing systems also strongly 3-mixing?Paul Halmos, Ergodic theory. Chelsea, New York, 1956.
• Weinstein conjecture – does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow?

• Does every positive integer generate a juggler sequence terminating at 1?
• Lyapunov function: Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion?
• Is every reversible cellular automaton in three or more dimensions locally reversible?{{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=Lecture Notes in Computer Science |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}}

{{Main|Game theory}}

{{Main|Combinatorial game theory}}

• Sudoku:
• How many puzzles have exactly one solution?
• How many puzzles with exactly one solution are minimal?
• What is the maximum number of givens for a minimal puzzle?{{Cite web |title=Open Q – Solving and rating of hard Sudoku |url=http://english.log-it-ex.com/2.html |archive-url=https://web.archive.org/web/20171110030932/http://english.log-it-ex.com/2.html |archive-date=10 November 2017 |website=english.log-it-ex.com}}
• Tic-tac-toe variants:
• Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also Hales–Jewett theorem and n^d game){{cite web |url=https://www.youtube.com/watch?v=FwJZa-helig |title=Higher-Dimensional Tic-Tac-Toe |website=PBS Infinite Series |publisher=YouTube |date=2017-09-21 |access-date=2018-07-29 |archive-url=https://web.archive.org/web/20171011000653/https://www.youtube.com/watch?v=FwJZa-helig |archive-date=2017-10-11 |url-status=live }}
• Chess:
• What is the outcome of a perfectly played game of chess? (See also first-move advantage in chess)
• Go:
• What is the perfect value of Komi?
• Are the nim-sequences of all finite octal games eventually periodic?
• Is the nim-sequence of Grundy's game eventually periodic?

• Rendezvous problem

{{Main|Geometry}}

{{Main|Algebraic geometry}}

• Abundance conjecture: if the canonical bundle of a projective variety with Kawamata log terminal singularities is nef, then it is semiample.
• Bass conjecture on the finite generation of certain algebraic K-groups.
• Bass–Quillen conjecture relating vector bundles over a regular Noetherian ring and over the polynomial ring $A[t\_\{1\},\; \backslash ldots,\; t\_\{n\}]$.
• Deligne conjecture: any one of numerous named for Pierre Deligne.
• Deligne's conjecture on Hochschild cohomology about the operadic structure on Hochschild cochain complex.
• Dixmier conjecture: any endomorphism of a Weyl algebra is an automorphism.
• Fröberg conjecture on the Hilbert functions of a set of forms.
• Fujita conjecture regarding the line bundle $K\_\{M\}\; \backslash otimes\; L^\{\backslash otimes\; m\}$ constructed from a positive holomorphic line bundle $L$ on a compact complex manifold $M$ and the canonical line bundle $K\_\{M\}$ of $M$
• General elephant problem: do general elephants have at most Du Val singularities?
• Hartshorne's conjectures{{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=Mathematische Annalen |year=1990 |volume=286 |issue=1–3 |pages=13–25|s2cid=122151259 }}
• In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent?{{citation

|last = Dupont

|first = Johan L.

|doi = 10.1142/9789812810335

|isbn = 978-981-02-4507-8

|mr = 1832859

|page = 6

|publisher = World Scientific Publishing Co., Inc., River Edge, NJ

|series = Nankai Tracts in Mathematics

|title = Scissors congruences, group homology and characteristic classes

|url = http://home.math.au.dk/dupont/scissors.ps

|volume = 1

|year = 2001

|url-status = dead

|archive-url = https://web.archive.org/web/20160429152252/http://home.math.au.dk/dupont/scissors.ps

|archive-date = 2016-04-29

}}.

• Jacobian conjecture: if a polynomial mapping over a characteristic-0 field has a constant nonzero Jacobian determinant, then it has a regular (i.e. with polynomial components) inverse function.
• Manin conjecture on the distribution of rational points of bounded height in certain subsets of Fano varieties
• Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory{{citation

|last1=Maulik |first1=Davesh

|last2=Nekrasov |first2=Nikita |author-link2=Nikita Nekrasov

|last3=Okounov |first3=Andrei |author-link3=Andrei Okounov

|last4=Pandharipande |first4=Rahul |author-link4=Rahul Pandharipande

|title=Gromov–Witten theory and Donaldson–Thomas theory, I

|arxiv=math/0312059

|date=2004-06-05|bibcode=2003math.....12059M

}}

• Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities.
• Nagata–Biran conjecture that if $X$ is a smooth algebraic surface and $L$ is an ample line bundle on $X$ of degree $d$, then for sufficiently large $r$, the Seshadri constant satisfies $\backslash varepsilon(p\_1,\backslash ldots,p\_r;X,L)\; =\; d/\backslash sqrt\{r\}$.
• Nakai conjecture: if a complex algebraic variety has a ring of differential operators generated by its contained derivations, then it must be smooth.
• Parshin's conjecture: the higher algebraic K-groups of any smooth projective variety defined over a finite field must vanish up to torsion.
• Section conjecture on splittings of group homomorphisms from fundamental groups of complete smooth curves over finitely-generated fields $k$ to the Galois group of $k$.
• Standard conjectures on algebraic cycles
• Tate conjecture on the connection between algebraic cycles on algebraic varieties and Galois representations on étale cohomology groups.
• Virasoro conjecture: a certain generating function encoding the Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra.
• Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points{{cite journal|last=Zariski |first=Oscar |author-link=Oscar Zariski |title=Some open questions in the theory of singularities |journal=Bulletin of the American Mathematical Society |volume=77 |issue=4 |year=1971 |pages=481–491 |doi=10.1090/S0002-9904-1971-12729-5 |mr=0277533|doi-access=free }}

• Are infinite sequences of flips possible in dimensions greater than 3?
• Resolution of singularities in characteristic $p$

• 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?{{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}}
• The Erdős–Oler conjecture: when $n$ is a triangular number, packing $n-1$ circles in an equilateral triangle requires a triangle of the same size as packing $n$ circles.{{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}}
• The disk covering problem abount finding the smallest real number $r(n)$ such that $n$ disks of radius $r(n)$ can be arranged in such a way as to cover the unit disk.
• The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24{{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=[https://books.google.com/books?id=upYwZ6cQumoC&pg=PA21 21–22]}}
• Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets{{citation

| last = Hales | first = Thomas | author-link = Thomas Callister Hales

| arxiv = 1703.01352

| title = The Reinhardt conjecture as an optimal control problem

| year = 2017}}

• 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?{{citation|last1=Brass|first1=Peter|last2=Moser|first2=William|last3=Pach|first3=János|author3-link=János Pach|isbn=978-0387-23815-9|mr=2163782|page=45|publisher=Springer|location=New York|title=Research Problems in Discrete Geometry|url=https://books.google.com/books?id=WehCspo0Qa0C&pg=PA45|year=2005}}
• Ulam's packing conjecture about the identity of the worst-packing convex solid{{citation |last=Gardner |first=Martin |date=1995 |title=New Mathematical Diversions (Revised Edition) |location=Washington |publisher=Mathematical Association of America |page=251 }}
• The Tammes problem for numbers of nodes greater than 14 (except 24).{{cite journal |last1=Musin |first1=Oleg R. |last2=Tarasov |first2=Alexey S. |title=The Tammes Problem for N = 14 |journal=Experimental Mathematics |date=2015 |volume=24 |issue=4 |pages=460–468 |doi=10.1080/10586458.2015.1022842|s2cid=39429109 }}

{{Main|Differential geometry}}

• The spherical Bernstein's problem, a generalization of Bernstein's problem
• Carathéodory conjecture: any convex, closed, and twice-differentiable surface in three-dimensional Euclidean space admits at least two umbilical points.
• Cartan–Hadamard conjecture: can the classical isoperimetric inequality for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as Cartan–Hadamard manifolds?
• Chern's conjecture (affine geometry) that the Euler characteristic of a compact affine manifold vanishes.
• Chern's conjecture for hypersurfaces in spheres, a number of closely related conjectures.
• Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.{{citation

| last = Barros | first = Manuel

| jstor = 2162098

| journal = Proceedings of the American Mathematical Society

| pages = 1503–1509

| title = General Helices and a Theorem of Lancret

| volume = 125

| issue = 5

| year = 1997| doi = 10.1090/S0002-9939-97-03692-7

| doi-access = free

}}

• The filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length{{citation

| last = Katz | first = Mikhail G.

| doi = 10.1090/surv/137

| isbn = 978-0-8218-4177-8

| mr = 2292367

| page = 57

| publisher = American Mathematical Society, Providence, RI

| series = Mathematical Surveys and Monographs

| title = Systolic geometry and topology

| url = https://books.google.com/books?id=R5_zBwAAQBAJ&pg=PA57

| volume = 137

| year = 2007}}

• The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds{{citation

| last = Rosenberg | first = Steven

| doi = 10.1017/CBO9780511623783

| isbn = 978-0-521-46300-3

| location = Cambridge

| mr = 1462892

| pages = 62–63

| publisher = Cambridge University Press

| series = London Mathematical Society Student Texts

| title = The Laplacian on a Riemannian Manifold: An introduction to analysis on manifolds

| url = https://books.google.com/books?id=gzJ6Vn0y7XQC&pg=PA62

| volume = 31

| year = 1997}}

• Osserman conjecture: that every Osserman manifold is either flat or locally isometric to a rank-one symmetric space{{citation | last = Nikolayevsky | first = Y. | journal = Differential Geometry and Its Applications | title = Two theorems on Osserman manifolds | volume = 18 | pages = 239–253 | year = 2003 | issue = 3 | doi = 10.1016/S0926-2245(02)00160-2}}
• Yau's conjecture on the first eigenvalue that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of $S^\{n+1\}$ is $n$.

{{Main|Discrete geometry }}

File:Kissing-3d.png 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.]]

• The big-line-big-clique conjecture on the existence of either many collinear points or many mutually visible points in large planar point sets{{citation

| last1 = Ghosh | first1 = Subir Kumar

| last2 = Goswami | first2 = Partha P.

| arxiv = 1012.5187

| doi = 10.1145/2543581.2543589

| issue = 2

| journal = ACM Computing Surveys

| pages = 22:1–22:29

| title = Unsolved problems in visibility graphs of points, segments, and polygons

| volume = 46

| year = 2013| s2cid = 8747335

}}

• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2ⁿ smaller copies{{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}}.
• Solving the happy ending problem for arbitrary $n${{citation

| last1 = Morris | first1 = Walter D.

| last2 = Soltan | first2 = Valeriu

| doi = 10.1090/S0273-0979-00-00877-6

| issue = 4

| journal = Bull. Amer. Math. Soc.

| mr = 1779413

| pages = 437–458

| title = The Erdős-Szekeres problem on points in convex position—a survey

| volume = 37

| year = 2000| doi-access = free

}}; {{citation

| last = Suk | first = Andrew

| arxiv = 1604.08657

| doi = 10.1090/jams/869

| journal = J. Amer. Math. Soc.

| title = On the Erdős–Szekeres convex polygon problem

| year = 2016

| volume=30

| issue = 4

| pages=1047–1053| s2cid = 15732134

}}

• Improving lower and upper bounds for the Heilbronn triangle problem.
• Kalai's 3^d conjecture on the least possible number of faces of centrally symmetric polytopes.{{citation

| last = Kalai | first = Gil | author-link = Gil Kalai

| doi = 10.1007/BF01788696

| issue = 1

| journal = Graphs and Combinatorics

| mr = 1554357

| pages = 389–391

| title = The number of faces of centrally-symmetric polytopes

| volume = 5

| year = 1989| s2cid = 8917264 }}.

• The Kobon triangle problem on triangles in line arrangements{{cite journal

| last1 = Moreno | first1 = José Pedro

| last2 = Prieto-Martínez | first2 = Luis Felipe

| hdl = 10486/705416

| issue = 1

| journal = La Gaceta de la Real Sociedad Matemática Española

| language = es

| mr = 4225268

| pages = 111–130

| title = El problema de los triángulos de Kobon

| trans-title = The Kobon triangles problem

| volume = 24

| year = 2021}}

• The Kusner conjecture: at most $2d$ points can be equidistant in $L^1$ spaces{{citation

| last = Guy | first = Richard K. | author-link = Richard K. Guy

| issue = 3

| journal = American Mathematical Monthly

| mr = 1540158

| pages = 196–200

| title = An olla-podrida of open problems, often oddly posed

| jstor = 2975549

| volume = 90

| year = 1983

| doi = 10.2307/2975549 }}

• The McMullen problem on projectively transforming sets of points into convex position{{citation

| last = Matoušek | first = Jiří | author-link = Jiří Matoušek (mathematician)

| doi = 10.1007/978-1-4613-0039-7

| isbn = 978-0-387-95373-1

| mr = 1899299

| page = 206

| publisher = Springer-Verlag, New York

| series = Graduate Texts in Mathematics

| title = Lectures on discrete geometry

| volume = 212

| year = 2002}}

• Opaque forest problem on finding opaque sets for various planar shapes
• How many unit distances can be determined by a set of {{mvar|n}} points in the Euclidean plane?{{citation

| last1 = Brass | first1 = Peter

| last2 = Moser | first2 = William

| last3 = Pach | first3 = János

| contribution = 5.1 The Maximum Number of Unit Distances in the Plane

| isbn = 978-0-387-23815-9

| mr = 2163782

| pages = 183–190

| publisher = Springer, New York

| title = Research problems in discrete geometry

| year = 2005}}

• Finding matching upper and lower bounds for k-sets and halving lines{{citation

| last = Dey | first = Tamal K. | author-link = Tamal Dey

| doi = 10.1007/PL00009354

| journal = Discrete & Computational Geometry

| mr = 1608878

| pages = 373–382

| title = Improved bounds for planar k-sets and related problems

| volume = 19

| issue = 3

| year = 1998| doi-access = free

}}; {{citation

| last = Tóth | first = Gábor

| doi = 10.1007/s004540010022

| issue = 2

| journal = Discrete & Computational Geometry

| mr = 1843435

| pages = 187–194

| title = Point sets with many k-sets

| volume = 26

| year = 2001| doi-access = free

}}.

• Tripod packing:{{citation|last1=Aronov|first1=Boris|author1-link=Boris Aronov|last2=Dujmović|first2=Vida|author2-link=Vida Dujmović|last3=Morin|first3=Pat|author3-link= Pat Morin |last4=Ooms|first4=Aurélien|last5=Schultz Xavier da Silveira |first5=Luís Fernando|issue=1|journal=Electronic Journal of Combinatorics|page=P1.8|title=More Turán-type theorems for triangles in convex point sets |url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p8 |volume=26 |year=2019 |bibcode=2017arXiv170610193A |arxiv=1706.10193 |access-date=2019-02-18 |archive-url=https://web.archive.org/web/20190218082023/https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i1p8|archive-date=2019-02-18|url-status=live|doi-access=free|doi=10.37236/7224}} how many tripods can have their apexes packed into a given cube?

{{Main|Euclidean geometry}}

• The Atiyah conjecture on configurations on the invertibility of a certain $n$-by-$n$ matrix depending on $n$ points in $\backslash mathbb\{R\}^\{3\}${{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 }}
• Bellman's lost-in-a-forest problem – 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{{citation |last1=Finch |first1=S. R. |last2=Wetzel |first2=J. E. |title=Lost in a forest |volume=11 |issue=8 |year=2004 |journal=American Mathematical Monthly |pages=645–654 |mr=2091541 |doi=10.2307/4145038 |jstor=4145038}}
• Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?{{citation

| last = Howards | first = Hugh Nelson

| arxiv = 1406.3370

| doi = 10.1142/S0218216513500831

| issue = 14

| journal = Journal of Knot Theory and Its Ramifications

| mr = 3190121

| pages = 1350083, 15

| title = Forming the Borromean rings out of arbitrary polygonal unknots

| volume = 22

| year = 2013| s2cid = 119674622

}}

• Connelly’s blooming conjecture: Does every net of a convex polyhedron have a blooming?{{citation

| last1 = Miller | first1 = Ezra

| last2 = Pak | first2 = Igor | author2-link = Igor Pak

| doi = 10.1007/s00454-008-9052-3

| issue = 1–3

| journal = Discrete & Computational Geometry

| mr = 2383765

| pages = 339–388

| title = Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings

| volume = 39

| year = 2008| doi-access = free

}}. Announced in 2003.

• Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?{{citation |last1=Solomon |first1=Yaar |last2=Weiss |first2=Barak |arxiv=1406.3807 |doi=10.24033/asens.2303 |issue=5 |journal=Annales Scientifiques de l'École Normale Supérieure |mr=3581810 |pages=1053–1074 |title=Dense forests and Danzer sets |volume=49 |year=2016 |s2cid=672315}}; {{citation |last=Conway |first=John H. |author-link=John Horton Conway |publisher=On-Line Encyclopedia of Integer Sequences |title=Five $1,000 Problems (Update 2017) |url=https://oeis.org/A248380/a248380.pdf |archive-url=https://web.archive.org/web/20190213123825/https://oeis.org/A248380/a248380.pdf |archive-date=2019-02-13 |access-date=2019-02-12 |url-status=live}}
• Dissection into orthoschemes – is it possible for simplices of every dimension?{{citation |last1=Brandts |first1=Jan |last2=Korotov |first2=Sergey |last3=Křížek |first3=Michal |last4=Šolc |first4=Jakub |doi=10.1137/060669073 |issue=2 |journal=SIAM Review |mr=2505583 |pages=317–335 |title=On nonobtuse simplicial partitions |volume=51 |year=2009 |url=https://pure.uva.nl/ws/files/836396/73198_315330.pdf |bibcode=2009SIAMR..51..317B |s2cid=216078793 |access-date=2018-11-22 |archive-date=2018-11-04 |archive-url=https://web.archive.org/web/20181104211116/https://pure.uva.nl/ws/files/836396/73198_315330.pdf |url-status=live}}. See in particular Conjecture 23, p. 327.
• Ehrhart's volume conjecture: a convex body $K$ in $n$ dimensions containing a single lattice point in its interior as its center of mass cannot have volume greater than $(n+1)^\{n\}/n!$
• Falconer's conjecture: sets of Hausdorff dimension greater than $d/2$ in $\backslash mathbb\{R\}^d$ must have a distance set of nonzero Lebesgue measure{{citation |last1=Arutyunyants |first1=G. |last2=Iosevich |first2=A. |editor-last=Pach |editor-first=János |editor-link=János Pach |contribution=Falconer conjecture, spherical averages and discrete analogs |doi=10.1090/conm/342/06127 |mr=2065249 |pages=15–24 |publisher=Amer. Math. Soc., Providence, RI|series=Contemp. Math. |title=Towards a Theory of Geometric Graphs |volume=342 |year=2004 |isbn=978-0-8218-3484-8 |doi-access=free}}
• The values of the Hermite constants for dimensions other than 1–8 and 24
• What is the lowest number of faces possible for a holyhedron?
• Inscribed square problem, also known as Toeplitz' conjecture and the square peg problem – does every Jordan curve have an inscribed square?{{citation|last=Matschke|first=Benjamin|date=2014|title=A survey on the square peg problem|journal=Notices of the American Mathematical Society|volume=61|issue=4|pages=346–352|doi=10.1090/noti1100|doi-access=free}}
• The Kakeya conjecture – do $n$-dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to $n$?{{citation |last1=Katz |first1=Nets |author1-link=Nets Katz|last2=Tao|first2=Terence|author2-link=Terence Tao|title=Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000) |doi=10.5565/PUBLMAT_Esco02_07|series=Publicacions Matemàtiques|mr=1964819 |pages=161–179 |contribution=Recent progress on the Kakeya conjecture |year=2002 |citeseerx=10.1.1.241.5335 |s2cid=77088}}
• The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem{{citation |title=The Kelvin Problem |editor-first=Denis |editor-last=Weaire |editor-link=Denis Weaire |publisher=CRC Press |year=1997 |isbn=978-0-7484-0632-6 |page=1 |url=https://books.google.com/books?id=otokU4KQnXIC&pg=PA1}}
• Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one{{citation |last1=Brass |first1=Peter |last2=Moser |first2=William |last3=Pach |first3=János |location=New York |mr=2163782 |page=457 |publisher=Springer |title=Research problems in discrete geometry|url=https://books.google.com/books?id=cT7TB20y3A8C&pg=PA457 |year=2005 |isbn=978-0-387-29929-7}}
• Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar.{{Cite journal|last1=Mahler|first1=Kurt|title=Ein Minimalproblem für konvexe Polygone |journal=Mathematica (Zutphen) B|pages=118–127|year=1939}}
• Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?{{citation |last1=Norwood |first1=Rick |last2=Poole |first2=George |last3=Laidacker |first3=Michael |doi=10.1007/BF02187832 |issue=2 |journal=Discrete & Computational Geometry |mr=1139077 |pages=153–162 |title=The worm problem of Leo Moser |volume=7 |year=1992 |doi-access=free}}
• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?{{citation |last=Wagner |first=Neal R. |date=1976 |title=The Sofa Problem |journal=The American Mathematical Monthly |doi=10.2307/2977022 |jstor=2977022 |volume=83 |issue=3 |pages=188–189 |url=http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |access-date=2014-05-14 |archive-url=https://web.archive.org/web/20150420160001/http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |archive-date=2015-04-20 |url-status=live}}
• In parallelohedron:
• Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combinatorial structure as a parallelohedron?{{cite journal|last1=Senechal|first1=Marjorie|author1-link=Marjorie Senechal|last2=Galiulin|first2=R. V.|hdl=2099/1195|issue=10|journal=Structural Topology|language=en,fr|mr=768703|pages=5–22|title=An introduction to the theory of figures: the geometry of E. S. Fedorov|year=1984}}
• Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram?{{cite journal|last1=Grünbaum|first1=Branko|author1-link=Branko Grünbaum|last2=Shephard|first2=G. C.|author2-link=Geoffrey Colin Shephard|doi=10.1090/S0273-0979-1980-14827-2|issue=3|journal=Bulletin of the American Mathematical Society|mr=585178|pages=951–973|series=New Series|title=Tilings with congruent tiles|volume=3|year=1980|doi-access=free}}
• Does every convex polyhedron have Rupert's property?{{citation |first1=Ying |last1=Chai |first2=Liping |last2=Yuan |first3=Tudor |last3=Zamfirescu |title=Rupert Property of Archimedean Solids |journal=The American Mathematical Monthly |volume=125 |issue=6 |pages=497–504 |date=June–July 2018 |doi=10.1080/00029890.2018.1449505| s2cid=125508192}}{{citation|title=An algorithmic approach to Rupert's problem |first1=Jakob |last1=Steininger |first2=Sergey |last2=Yurkevich| date=December 27, 2021 |arxiv=2112.13754}}
• Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net, or simple edge-unfolding?{{citation |last1=Demaine |first1=Erik D. |author1-link=Erik Demaine |last2=O'Rourke |first2=Joseph |author2-link=Joseph O'Rourke (professor) |date=2007 |title=Geometric Folding Algorithms: Linkages, Origami, Polyhedra |title-link=Geometric Folding Algorithms |publisher=Cambridge University Press |contribution=Chapter 22. Edge Unfolding of Polyhedra |pages=306–338}}{{Cite journal |last=Ghomi |first=Mohammad |date=2018-01-01 |title=Dürer's Unfolding Problem for Convex Polyhedra |journal=Notices of the American Mathematical Society |volume=65 |issue=1 |pages=25–27 |doi=10.1090/noti1609 |issn=0002-9920 |doi-access=free}}
• Is there a non-convex polyhedron without self-intersections with more than seven faces, all of which share an edge with each other?
• The Thomson problem – what is the minimum energy configuration of $n$ mutually-repelling particles on a unit sphere?{{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}}
• Convex uniform 5-polytopes – find and classify the complete set of these shapes{{citation |author=ACW |date=May 24, 2012 |title=Convex uniform 5-polytopes |url=http://www.openproblemgarden.org/op/convex_uniform_5_polytopes |work=Open Problem Garden |access-date=2016-10-04 |archive-url=https://web.archive.org/web/20161005164840/http://www.openproblemgarden.org/op/convex_uniform_5_polytopes |archive-date=October 5, 2016 |url-status=live}}.

{{Main|Graph theory}}

• Babai's problem: which groups are Babai invariant groups?
• Brouwer's conjecture on upper bounds for sums of eigenvalues of Laplacians of graphs in terms of their number of edges

• Does there exist a graph $G$ such that the dominating number $\backslash gamma(G)$ equals the eternal dominating number $\backslash gamma$_∞$(G)$ of $G$ and $\backslash gamma(G)$ is less than the clique covering number of $G$? {{cite journal

|last1=Klostermeyer |first1=W.

|last2=Mynhardt |first2=C.

|year=2015

|title=Protecting a graph with mobile guards

|journal=Applicable Analysis and Discrete Mathematics

|volume=10 |pages=21

|arxiv=1407.5228

|doi=10.2298/aadm151109021k

}}.

• Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs{{cite journal

| last = Pleanmani | first = Nopparat

| doi = 10.1142/s179383091950068x

| issue = 6

| journal = Discrete Mathematics, Algorithms and Applications

| mr = 4044549

| pages = 1950068, 7

| title = Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph

| volume = 11

| year = 2019| s2cid = 204207428

}}

• Meyniel's conjecture that cop number is $O(\backslash sqrt\; n)${{cite journal

| last1 = Baird | first1 = William

| last2 = Bonato | first2 = Anthony

| arxiv = 1308.3385

| doi = 10.4310/JOC.2012.v3.n2.a6

| issue = 2

| journal = Journal of Combinatorics

| mr = 2980752

| pages = 225–238

| title = Meyniel's conjecture on the cop number: a survey

| volume = 3

| year = 2012| s2cid = 18942362

}}

• Suppose Alice has a winning strategy for the vertex coloring game on a graph $G$ with $k$ colors. Does she have one for $k+1$ colors?{{cite journal

| last = Zhu

| first = Xuding

| date = 1999

| title = The Game Coloring Number of Planar Graphs

| journal = Journal of Combinatorial Theory, Series B

| volume = 75

| issue = 2

| pages =245–258

| doi=10.1006/jctb.1998.1878

| doi-access= free

}}

File:Erdős–Faber–Lovász conjecture.svg

• The 1-factorization conjecture that if $n$ is odd or even and $k\; \backslash geq\; n,\; n\; -\; 1$ respectively, then a $k$-regular graph with $2n$ vertices is 1-factorable.
• The perfect 1-factorization conjecture that every complete graph on an even number of vertices admits a perfect 1-factorization.
• Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs{{citation

| last1 = Bousquet | first1 = Nicolas

| last2 = Bartier | first2 = Valentin

| editor1-last = Bender | editor1-first = Michael A.

| editor2-last = Svensson | editor2-first = Ola

| editor3-last = Herman | editor3-first = Grzegorz

| contribution = Linear Transformations Between Colorings in Chordal Graphs

| doi = 10.4230/LIPIcs.ESA.2019.24

| pages = 24:1–24:15

| publisher = Schloss Dagstuhl – Leibniz-Zentrum für Informatik

| series = LIPIcs

| title = 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany

| volume = 144

| year = 2019| doi-access = free

| isbn = 978-3-95977-124-5

| s2cid = 195791634

}}

• The Earth–Moon problem: what is the maximum chromatic number of biplanar graphs?{{citation

| last = Gethner | first = Ellen | author-link = Ellen Gethner

| editor1-last = Gera | editor1-first = Ralucca | editor1-link = Ralucca Gera

| editor2-last = Haynes | editor2-first = Teresa W. | editor2-link = Teresa W. Haynes

| editor3-last = Hedetniemi | editor3-first = Stephen T.

| contribution = To the Moon and beyond

| doi = 10.1007/978-3-319-97686-0_11

| mr = 3930641

| pages = 115–133

| publisher = Springer International Publishing

| series = Problem Books in Mathematics

| title = Graph Theory: Favorite Conjectures and Open Problems, II

| year = 2018| isbn = 978-3-319-97684-6 }}

• The Erdős–Faber–Lovász conjecture on coloring unions of cliques{{citation

| last1 = Chung | first1 = Fan | author-link1 = Fan Chung

| last2 = Graham | first2 = Ron | author-link2 = Ronald Graham

| title = Erdős on Graphs: His Legacy of Unsolved Problems

| year = 1998

| publisher = A K Peters

| pages = 97–99}}.

• The graceful tree conjecture that every tree admits a graceful labeling
• Rosa's conjecture that all triangular cacti are graceful or nearly-graceful
• The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree{{citation

| last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky

| last2 = Seymour | first2 = Paul | author2-link = Paul Seymour (mathematician)

| doi = 10.1016/j.jctb.2013.11.002

| journal = Journal of Combinatorial Theory

| mr = 3171779

| pages = 11–16

| series = Series B

| title = Extending the Gyárfás-Sumner conjecture

| volume = 105

| year = 2014| doi-access = free

}}

• The Hadwiger conjecture relating coloring to clique minors{{citation

| last = Toft | first = Bjarne

| journal = Congressus Numerantium

| mr = 1411244

| pages = 249–283

| title = A survey of Hadwiger's conjecture

| volume = 115

| year = 1996}}.

• The Hadwiger–Nelson problem on the chromatic number of unit distance graphs{{citation

| last1 = Croft | first1 = Hallard T.

| last2 = Falconer | first2 = Kenneth J.

| last3 = Guy | first3 = Richard K. | author-link3 = Richard K. Guy

| title = Unsolved Problems in Geometry

| publisher = Springer-Verlag

| year = 1991}}, Problem G10.

• Jaeger's Petersen-coloring conjecture: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph{{citation

| last1 = Hägglund

| first1 = Jonas

| last2 = Steffen

| first2 = Eckhard

| issue = 1

| journal = Ars Mathematica Contemporanea

| mr = 3047618

| pages = 161–173

| title = Petersen-colorings and some families of snarks

| url = http://amc-journal.eu/index.php/amc/article/viewFile/288/247

| volume = 7

| year = 2014

| doi = 10.26493/1855-3974.288.11a

| access-date = 2016-09-30

| archive-url = https://web.archive.org/web/20161003070647/http://amc-journal.eu/index.php/amc/article/viewFile/288/247

| archive-date = 2016-10-03

| url-status = live

| doi-access = free

}}.

• The list coloring conjecture: for every graph, the list chromatic index equals the chromatic index{{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}}.
• The overfull conjecture that a graph with maximum degree $\backslash Delta(G)\; \backslash geq\; n/3$ is class 2 if and only if it has an overfull subgraph $S$ satisfying $\backslash Delta(S)\; =\; \backslash Delta(G)$.
• The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree{{citation

| last1 = Molloy | first1 = Michael

| last2 = Reed | first2 = Bruce | author1-link = Bruce Reed (mathematician)

| doi = 10.1007/PL00009820

| issue = 2

| journal = Combinatorica

| mr = 1656544

| pages = 241–280

| title = A bound on the total chromatic number

| volume = 18

| year = 1998| citeseerx = 10.1.1.24.6514

| s2cid = 9600550

}}.

• The Albertson conjecture: the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number{{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}}.
• Conway's thrackle conjecture{{citation |last1=Fulek |first1=Radoslav |last2=Pach |first2=János |author-link2=János Pach |title=A computational approach to Conway's thrackle conjecture|journal=Computational Geometry |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 }}. that thrackles cannot have more edges than vertices
• The GNRS conjecture on whether minor-closed graph families have $\backslash ell\_1$ embeddings with bounded distortion{{citation

| last1 = Gupta | first1 = Anupam

| last2 = Newman | first2 = Ilan

| last3 = Rabinovich | first3 = Yuri

| last4 = Sinclair | first4 = Alistair | author4-link = Alistair Sinclair

| doi = 10.1007/s00493-004-0015-x

| issue = 2

| journal = Combinatorica

| mr = 2071334

| pages = 233–269

| title = Cuts, trees and $\backslash ell\_1$-embeddings of graphs

| volume = 24

| year = 2004| citeseerx = 10.1.1.698.8978

| s2cid = 46133408

}}

• Harborth's conjecture: every planar graph can be drawn with integer edge lengths{{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=[https://books.google.com/books?id=VMjDAgAAQBAJ&pg=PA247 p. 247]|mr=2047103}}.
• Negami's conjecture on projective-plane embeddings of graphs with planar covers{{citation | last = Hliněný | first = Petr | doi = 10.1007/s00373-010-0934-9 | issue = 4 | journal = Graphs and Combinatorics | 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 }}.
• The strong Papadimitriou–Ratajczak conjecture: every polyhedral graph has a convex greedy embedding{{citation | last1 = Nöllenburg | first1 = Martin | last2 = Prutkin | first2 = Roman | last3 = Rutter | first3 = Ignaz | doi = 10.20382/jocg.v7i1a3 | issue = 1 | journal = Journal of Computational Geometry | 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 }}
• Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?{{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 = American Mathematical Society | series = Mathematical Surveys and Monographs | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures | volume = 152 | year = 2009}}.

• Universal point sets of subquadratic size for planar graphs{{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 }}.

• Conway's 99-graph problem: does there exist a strongly regular graph with parameters (99,14,1,2)?{{citation

| last = Conway

| first = John H.

| author-link = John Horton Conway

| access-date = 2019-02-12

| publisher = Online Encyclopedia of Integer Sequences

| 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

}}

• Degree diameter problem: given two positive integers $d,\; k$, what is the largest graph of diameter $k$ such that all vertices have degrees at most $d$?
• Jørgensen's conjecture that every 6-vertex-connected K₆-minor-free graph is an apex graph{{citation |last1=mdevos |title=Jorgensen's Conjecture |date=December 7, 2019 |url=http://www.openproblemgarden.org/op/jorgensens_conjecture |work=Open Problem Garden |archive-url=https://web.archive.org/web/20161114232136/http://www.openproblemgarden.org/op/jorgensens_conjecture |access-date=2016-11-13 |archive-date=2016-11-14 |last2=Wood |first2=David |url-status=live}}.
• Does a Moore graph with girth 5 and degree 57 exist?{{citation

| last=Ducey

| first=Joshua E.

| doi=10.1016/j.disc.2016.10.001

| issue=5

| journal=Discrete Mathematics

| mr=3612450

| pages=1104–1109

| title=On the critical group of the missing Moore graph

| volume=340

| year=2017

| arxiv=1509.00327

| s2cid=28297244}}

• Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?{{citation

| last1 = Blokhuis | first1 = A.

| last2 = Brouwer | first2 = A. E. | author-link = Andries Brouwer

| doi = 10.1007/BF00191941

| issue = 1–3

| journal = Geometriae Dedicata

| mr = 925851

| pages = 527–533

| title = Geodetic graphs of diameter two

| volume = 25

| year = 1988| s2cid = 189890651

}}

• Barnette's conjecture: every cubic bipartite three-connected planar graph has a Hamiltonian cycle{{citation

| last = Florek | first = Jan

| doi = 10.1016/j.disc.2010.01.018

| issue = 10–11

| journal = Discrete Mathematics

| mr = 2601261

| pages = 1531–1535

| title = On Barnette's conjecture

| volume = 310

| year = 2010}}.

• Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane that the Steiner ratio is $\backslash sqrt\{3\}/2$
• Chvátal's toughness conjecture, that there is a number {{mvar|t}} such that every {{mvar|t}}-tough graph is Hamiltonian{{citation

| last1 = Broersma | first1 = Hajo

| last2 = Patel | first2 = Viresh

| last3 = Pyatkin | first3 = Artem

| doi = 10.1002/jgt.21734

| issue = 3

| journal = Journal of Graph Theory

| mr = 3153119

| pages = 244–255

| title = On toughness and Hamiltonicity of $2K_2$-free graphs

| volume = 75

| year = 2014| s2cid = 1377980

| url = https://ris.utwente.nl/ws/files/6416631/jgt21734.pdf

}}

• The cycle double cover conjecture: every bridgeless graph has a family of cycles that includes each edge twice{{citation

| last = Jaeger | first = F.

| contribution = A survey of the cycle double cover conjecture

| doi = 10.1016/S0304-0208(08)72993-1

| pages = 1–12

| series = North-Holland Mathematics Studies

| title = Annals of Discrete Mathematics 27 – Cycles in Graphs

| volume = 27

| year = 1985| isbn = 978-0-444-87803-8

}}.

• The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs{{citation|title=Erdös-Gyárfás conjecture for cubic planar graphs |first1=Christopher Carl |last1=Heckman |first2=Roi |last2=Krakovski |volume=20 |issue=2 |year=2013 |at=P7 |journal=Electronic Journal of Combinatorics |doi-access=free |doi=10.37236/3252}}.
• The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph{{citation

| last = Chudnovsky

| first = Maria

| author-link = Maria Chudnovsky

| arxiv = 1606.08827

| doi = 10.1002/jgt.21730

| issue = 2

| journal = Journal of Graph Theory

| mr = 3150572

| zbl = 1280.05086

| pages = 178–190

| title = The Erdös–Hajnal conjecture—a survey

| url = http://www.columbia.edu/~mc2775/EHsurvey.pdf

| volume = 75

| year = 2014

| s2cid = 985458

| access-date = 2016-09-22

| archive-url = https://web.archive.org/web/20160304102611/http://www.columbia.edu/~mc2775/EHsurvey.pdf

| archive-date = 2016-03-04

| url-status = live

}}.

• The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree{{citation

| last1 = Akiyama | first1 = Jin | author1-link = Jin Akiyama

| last2 = Exoo | first2 = Geoffrey

| last3 = Harary | first3 = Frank

| doi = 10.1002/net.3230110108

| issue = 1

| journal = Networks

| mr = 608921

| pages = 69–72

| title = Covering and packing in graphs. IV. Linear arboricity

| volume = 11

| year = 1981}}.

• The Lovász conjecture on Hamiltonian paths in symmetric graphs{{Cite book |last=Babai |first=László |url=http://newtraell.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps |title=Handbook of Combinatorics |date=June 9, 1994 |chapter=Automorphism groups, isomorphism, reconstruction |format=PostScript |author-link=László Babai |archive-url=https://web.archive.org/web/20070613201449/http://www.cs.uchicago.edu/research/publications/techreports/TR-94-10 |archive-date=13 June 2007}}
• The Oberwolfach problem on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.{{citation

| last1 = Lenz | first1 = Hanfried

| last2 = Ringel | first2 = Gerhard

| doi = 10.1016/0012-365X(91)90416-Y

| issue = 1–3

| journal = Discrete Mathematics

| mr = 1140782

| pages = 3–16

| title = A brief review on Egmont Köhler's mathematical work

| volume = 97

| year = 1991}}

• What is the largest possible pathwidth of an {{mvar|n}}-vertex cubic graph?{{citation

| last1 = Fomin | first1 = Fedor V.

| last2 = Høie | first2 = Kjartan

| doi = 10.1016/j.ipl.2005.10.012

| issue = 5

| journal = Information Processing Letters

| mr = 2195217

| pages = 191–196

| title = Pathwidth of cubic graphs and exact algorithms

| volume = 97

| year = 2006}}

• The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.{{cite conference |last=Schwenk |first=Allen |year=2012 |title=Some History on the Reconstruction Conjecture |url=http://faculty.nps.edu/rgera/conjectures/jmm2012/Schwenk,%20%20Some%20History%20on%20the%20RC.pdf |conference=Joint Mathematics Meetings |archive-url=https://web.archive.org/web/20150409233306/http://faculty.nps.edu/rgera/Conjectures/jmm2012/Schwenk,%20%20Some%20History%20on%20the%20RC.pdf |archive-date=2015-04-09 |access-date=2018-11-26}}{{citation

| last = Ramachandran | first = S.

| doi = 10.1016/S0095-8956(81)80019-6

| issue = 2

| journal = Journal of Combinatorial Theory

| mr = 630977

| pages = 143–149

| series = Series B

| title = On a new digraph reconstruction conjecture

| volume = 31

| year = 1981| doi-access = free

}}

• The snake-in-the-box problem: what is the longest possible induced path in an $n$-dimensional hypercube graph?
• Sumner's conjecture: does every $(2n-2)$-vertex tournament contain as a subgraph every $n$-vertex oriented tree?{{citation

| last1 = Kühn | first1 = Daniela | author1-link = Daniela Kühn

| last2 = Mycroft | first2 = Richard

| last3 = Osthus | first3 = Deryk

| arxiv = 1010.4430

| doi = 10.1112/plms/pdq035

| issue = 4

| journal = Proceedings of the London Mathematical Society | series = Third Series

| mr = 2793448 | zbl=1218.05034

| pages = 731–766

| title = A proof of Sumner's universal tournament conjecture for large tournaments

| volume = 102

| year = 2011| s2cid = 119169562 }}.

• Szymanski's conjecture: every permutation on the $n$-dimensional doubly-directed hypercube graph can be routed with edge-disjoint paths.
• Tuza's conjecture: if the maximum number of disjoint triangles is $\backslash nu$, can all triangles be hit by a set of at most $2\backslash nu$ edges?{{cite journal

| last = Tuza | first = Zsolt

| doi = 10.1007/BF01787705

| issue = 4

| journal = Graphs and Combinatorics

| mr = 1092587

| pages = 373–380

| title = A conjecture on triangles of graphs

| volume = 6

| year = 1990| s2cid = 38821128

}}

• Vizing's conjecture on the domination number of cartesian products of graphs{{citation

| last1 = Brešar | first1 = Boštjan

| last2 = Dorbec | first2 = Paul

| last3 = Goddard | first3 = Wayne

| last4 = Hartnell | first4 = Bert L.

| last5 = Henning | first5 = Michael A.

| last6 = Klavžar | first6 = Sandi

| last7 = Rall | first7 = Douglas F.

| doi = 10.1002/jgt.20565

| issue = 1

| journal = Journal of Graph Theory

| mr = 2864622

| pages = 46–76

| title = Vizing's conjecture: a survey and recent results

| volume = 69

| year = 2012| citeseerx = 10.1.1.159.7029

| s2cid = 9120720

}}.

• Zarankiewicz problem: how many edges can there be in a bipartite graph on a given number of vertices with no complete bipartite subgraphs of a given size?

• Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter?{{Cite book |last1=Kitaev |first1=Sergey | author1-link = Sergey Kitaev|url=https://link.springer.com/book/10.1007/978-3-319-25859-1 |title=Words and Graphs |last2=Lozin |first2=Vadim |year=2015 |isbn=978-3-319-25857-7 |series=Monographs in Theoretical Computer Science. An EATCS Series |doi=10.1007/978-3-319-25859-1 |via=link.springer.com |s2cid=7727433}}{{Cite conference |last=Kitaev |first=Sergey |date=2017-05-16 |title=A Comprehensive Introduction to the Theory of Word-Representable Graphs |conference=International Conference on Developments in Language Theory |language=en |doi=10.1007/978-3-319-62809-7_2|arxiv=1705.05924v1 }}{{Cite journal|title=Word-Representable Graphs: a Survey|first1=S. V.|last1=Kitaev|first2=A. V.|last2=Pyatkin|date=April 1, 2018|journal=Journal of Applied and Industrial Mathematics|volume=12|issue=2|pages=278–296|via=Springer Link|doi=10.1134/S1990478918020084|s2cid=125814097 }}{{Cite journal |last1=Kitaev |first1=Sergey V. |last2=Pyatkin |first2=Artem V. |date=2018 |title=Графы, представимые в виде слов. Обзор результатов |trans-title=Word-representable graphs: A survey |url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus |journal=Дискретн. анализ и исслед. опер. |language=ru |volume=25 |issue=2 |pages=19–53 |doi=10.17377/daio.2018.25.588}}
• Characterise (non-)word-representable planar graphs
• Characterise word-representable graphs in terms of (induced) forbidden subgraphs.
• Characterise word-representable near-triangulations containing the complete graph K₄ (such a characterisation is known for K₄-free planar graphs{{cite arXiv |eprint=1605.01688|author1=Marc Elliot Glen|title=Colourability and word-representability of near-triangulations|class=math.CO|year=2016}})
• Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter{{Cite arXiv|last=Kitaev |first=Sergey |date=2014-03-06 |title=On graphs with representation number 3 |class=math.CO |eprint=1403.1616v1 }}
• Is it true that out of all bipartite graphs, crown graphs require longest word-representants?{{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 }}
• Is the line graph of a non-word-representable graph always non-word-representable?
• Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?

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

{{Main|Model theory|formal languages}}

• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in $\backslash aleph\_0$ 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{{cite journal |last1=Hrushovski |first1=Ehud |year=1989 |title=Kueker's conjecture for stable theories |journal=Journal of Symbolic Logic |volume=54 |issue=1| pages=207–220 |doi=10.2307/2275025| jstor=2275025 |s2cid=41940041}}
• The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for $\backslash aleph\_1$-saturated models of a countable theory.{{cite book |vauthors=Shelah S |title=Classification Theory |publisher=North-Holland |year=1990}}
• Shelah's categoricity conjecture for $L\_\{\backslash omega\_1,\backslash omega\}$: 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 $\backslash lambda$ there exists a cardinal $\backslash mu(\backslash lambda)$ such that if an AEC K with LS(K)$\{\}\; \backslash le\; \backslash lambda$ is categorical in a cardinal above $\backslash mu(\backslash lambda)$ then it is categorical in all cardinals above $\backslash mu(\backslash lambda)$.{{Cite book

| title = Classification theory for abstract elementary classes

| last = Shelah

| first = Saharon

| publisher = College Publications

| year = 2009

| isbn = 978-1-904987-71-0

}}

• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• The stable forking conjecture for simple theories{{cite journal | last1 = Peretz | first1 = Assaf | year = 2006 | title = Geometry of forking in simple theories | journal = Journal of Symbolic Logic| volume = 71 | issue = 1| pages = 347–359 | doi=10.2178/jsl/1140641179| arxiv = math/0412356| s2cid = 9380215 }}
• 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?{{cite journal |last1=Cherlin |first1=Gregory |last2=Shelah |first2=Saharon | author-link2=Saharon Shelah|date=May 2007 |title=Universal graphs with a forbidden subtree |journal=Journal of Combinatorial Theory | series=Series B |arxiv=math/0512218 |doi=10.1016/j.jctb.2006.05.008 | doi-access=free |volume=97 |issue=3 |pages=293–333|s2cid=10425739 }}
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
• Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is either finite, $\backslash aleph\_0$, or $2^\{\backslash aleph\_0\}$.

• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality $\backslash aleph\_\{\backslash omega\_1\}$ does it have a model of cardinality continuum?{{cite journal |last=Shelah |first=Saharon |author-link=Saharon Shelah |date=1999 |title=Borel sets with large squares |journal=Fundamenta Mathematicae |arxiv=math/9802134 |volume=159 |issue=1 |pages=1–50|bibcode=1998math......2134S |doi=10.4064/fm-159-1-1-50 |s2cid=8846429 }}
• 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 $\backslash aleph\_n$, is it categorical in every cardinal?{{cite book |last=Baldwin |first=John T. |date=July 24, 2009 |title=Categoricity |publisher=American Mathematical Society |isbn=978-0-8218-4893-7 |url=http://www.math.uic.edu/~jbaldwin/pub/AEClec.pdf |access-date=February 20, 2014 |archive-url=https://web.archive.org/web/20100729073738/http://www.math.uic.edu/%7Ejbaldwin/pub/AEClec.pdf |archive-date=July 29, 2010 |url-status=live }}{{cite arXiv |last=Shelah |first=Saharon |title=Introduction to classification theory for abstract elementary classes |year=2009 |class=math.LO |eprint=0903.3428 }}
• Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
• Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
• Is the theory of the field of Laurent series over $\backslash mathbb\{Z\}\_p$ decidable? of the field of polynomials over $\backslash mathbb\{C\}$?
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.

• Determine the structure of Keisler's order.{{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 }}{{Cite arXiv |eprint=1208.2140 |class=math.LO |first1=Maryanthe |last1=Malliaris |first2=Saharon |last2=Shelah |author-link=Maryanthe Malliaris |author-link2=Saharon Shelah |title=A Dividing Line Within Simple Unstable Theories |date=10 August 2012}} {{Cite arXiv |title=A Dividing Line within Simple Unstable Theories |eprint=1208.2140 |last1=Malliaris |first1=M. |last2=Shelah |first2=S. |date=2012 |class=math.LO }}

{{Main|Probability theory}}

• Ibragimov–Iosifescu conjecture for φ-mixing sequences

{{Main articles|Category:Unsolved problems in number theory}}

{{See also|Number theory }}

File:Perfect number Cuisenaire rods 6 exact.svg because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them is odd.]]

• Beilinson's conjectures
• Brocard's problem: are there any integer solutions to $n!\; +\; 1\; =\; m^\{2\}$ other than $n\; =\; 4,\; 5,\; 7$?
• Büchi's problem on sufficiently large sequences of square numbers with constant second difference.
• Carmichael's totient function conjecture: do all values of Euler's totient function have multiplicity greater than $1$?
• Casas-Alvero conjecture: if a polynomial of degree $d$ defined over a field $K$ of characteristic $0$ has a factor in common with its first through $d\; -\; 1$-th derivative, then must $f$ be the $d$-th power of a linear polynomial?
• Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating.
• Erdős–Ulam problem: is there a dense set of points in the plane all at rational distances from one-another?
• Exponent pair conjecture: for all $\backslash varepsilon\; >\; 0$, is the pair $(\backslash varepsilon,\; 1/2\; +\; \backslash varepsilon)$ an exponent pair?
• The Gauss circle problem: how far can the number of integer points in a circle centered at the origin be from the area of the circle?
• Grand Riemann hypothesis: do the nontrivial zeros of all automorphic L-functions lie on the critical line $1/2\; +\; it$ with real $t$?
• Generalized Riemann hypothesis: do the nontrivial zeros of all Dirichlet L-functions lie on the critical line $1/2\; +\; it$ with real $t$?
• Riemann hypothesis: do the nontrivial zeros of the Riemann zeta function lie on the critical line $1/2\; +\; it$ with real $t$?
• Grimm's conjecture: each element of a set of consecutive composite numbers can be assigned a distinct prime number that divides it.
• Hall's conjecture: for any $\backslash varepsilon\; >\; 0$, there is some constant $c(\backslash varepsilon)$ such that either $y^2\; =\; x^3$ or $|y^2\; -\; x^3|\; >\; c(\backslash varepsilon)x^\{1/2\; -\; \backslash varepsilon\}$.
• Hardy–Littlewood zeta function conjectures
• Hilbert–Pólya conjecture: the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator.
• Hilbert's eleventh problem: classify quadratic forms over algebraic number fields.
• Hilbert's ninth problem: find the most general reciprocity law for the norm residues of $k$-th order in a general algebraic number field, where $k$ is a power of a prime.
• Hilbert's twelfth problem: extend the Kronecker–Weber theorem on Abelian extensions of $\backslash mathbb\{Q\}$ to any base number field.
• Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function{{citation

|last=Conrey |first=Brian |author-link=Brian Conrey

|doi=10.1090/bull/1525

|title=Lectures on the Riemann zeta function (book review)

|journal=Bulletin of the American Mathematical Society

|volume=53 |number=3 |pages=507–512 |year=2016|doi-access=free}}

• Lehmer's totient problem: if $\backslash phi(n)$ divides $n\; -\; 1$, must $n$ be prime?
• Leopoldt's conjecture: a p-adic analogue of the regulator of an algebraic number field does not vanish.
• Lindelöf hypothesis that for all $\backslash varepsilon\; >\; 0$, $\backslash zeta(1/2\; +\; it)\; =\; o(t^\backslash varepsilon)$
• The density hypothesis for zeroes of the Riemann zeta function
• Littlewood conjecture: for any two real numbers $\backslash alpha,\; \backslash beta$, $\backslash liminf\_\{n\; \backslash rightarrow\; \backslash infty\}\; n\backslash ,\backslash Vert\; n\backslash alpha\backslash Vert\backslash ,\backslash Vert\; n\backslash beta\backslash Vert\; =\; 0$, where $\backslash Vert\; x\backslash Vert$ is the distance from $x$ to the nearest integer.
• Mahler's 3/2 problem that no real number $x$ has the property that the fractional parts of $x(3/2)^n$ are less than $1/2$ for all positive integers $n$.
• Montgomery's pair correlation conjecture: the normalized pair correlation function between pairs of zeros of the Riemann zeta function is the same as the pair correlation function of random Hermitian matrices.
• n conjecture: a generalization of the abc conjecture to more than three integers.
• abc conjecture: for any $\backslash varepsilon\; >\; 0$, is true for only finitely many positive $a,\; b,\; c$ such that $a\; +\; b\; =\; c$.
• Szpiro's conjecture: for any $\backslash varepsilon\; >\; 0$, there is some constant $C(\backslash varepsilon)$ such that, for any elliptic curve $E$ defined over $\backslash mathbb\{Q\}$ with minimal discriminant $\backslash Delta$ and conductor $f$, we have $|\backslash Delta|\; \backslash leq\; C(\backslash varepsilon)\; \backslash cdot\; f^\{6+\backslash varepsilon\}$.
• Newman's conjecture: the partition function satisfies any arbitrary congruence infinitely often.
• Piltz divisor problem on bounding $\backslash Delta\_k(x)\; =\; D\_k(x)\; -\; xP\_k(\backslash log(x))$
• Dirichlet's divisor problem: the specific case of the Piltz divisor problem for $k\; =\; 1$
• Ramanujan–Petersson conjecture: a number of related conjectures that are generalizations of the original conjecture.
• Sato–Tate conjecture: also a number of related conjectures that are generalizations of the original conjecture.
• Scholz conjecture: the length of the shortest addition chain producing $2^n\; -\; 1$ is at most $n\; -\; 1$ plus the length of the shortest addition chain producing $n$.
• Do Siegel zeros exist?
• Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?{{citation |last=Singmaster |first=David |title=Research Problems: How often does an integer occur as a binomial coefficient? |journal=American Mathematical Monthly |volume=78 |issue=4 |pages=385–386 |year=1971 |doi=10.2307/2316907 |jstor=2316907 |mr=1536288 |author-link=David Singmaster}}.
• Vojta's conjecture on heights of points on algebraic varieties over algebraic number fields.

• Are there infinitely many perfect numbers?
• Do any odd perfect numbers exist?
• Do quasiperfect numbers exist?
• Do any non-power of 2 almost perfect numbers exist?
• Are there 65, 66, or 67 idoneal numbers?
• Are there any pairs of amicable numbers which have opposite parity?
• Are there any pairs of betrothed numbers which have same parity?
• Are there any pairs of relatively prime amicable numbers?
• Are there infinitely many amicable numbers?
• Are there infinitely many betrothed numbers?
• Are there infinitely many Giuga numbers?
• Does every rational number with an odd denominator have an odd greedy expansion?
• Do any Lychrel numbers exist?
• Do any odd noncototients exist?
• Do any odd weird numbers exist?
• Do any (2, 5)-perfect numbers exist?
• Do any Taxicab(5, 2, n) exist for n > 1?
• Is there a covering system with odd distinct moduli?{{citation

| last1 = Guo | first1 = Song

| last2 = Sun | first2 = Zhi-Wei

| doi = 10.1016/j.aam.2005.01.004

| issue = 2

| journal = Advances in Applied Mathematics

| mr = 2152886

| pages = 182–187

| title = On odd covering systems with distinct moduli

| volume = 35

| year = 2005| arxiv = math/0412217

| s2cid = 835158

}}

• Is $\backslash pi$ a normal number (i.e., is each digit 0–9 equally frequent)?{{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}}
• Are all irrational algebraic numbers normal?
• Is 10 a solitary number?
• Can a 3×3 magic square be constructed from 9 distinct perfect square numbers?{{Cite journal |last=Robertson |first=John P. |date=1996-10-01 |title=Magic Squares of Squares |journal=Mathematics Magazine |volume=69 |issue=4 |pages=289–293 |doi=10.1080/0025570X.1996.11996457 |issn=0025-570X}}
• Find the value of the De Bruijn–Newman constant.

{{Main|Additive number theory }}

{{See also|Problems involving arithmetic progressions}}

• Erdős conjecture on arithmetic progressions that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long arithmetic progressions.
• Erdős–Turán conjecture on additive bases: if $B$ is an additive basis of order $2$, then the number of ways that positive integers $n$ can be expressed as the sum of two numbers in $B$ must tend to infinity as $n$ tends to infinity.
• Gilbreath's conjecture on consecutive applications of the unsigned forward difference operator to the sequence of prime numbers.
• Goldbach's conjecture: every even natural number greater than $2$ is the sum of two prime numbers.
• Lander, Parkin, and Selfridge conjecture: if the sum of $m$ $k$-th powers of positive integers is equal to a different sum of $n$ $k$-th powers of positive integers, then $m\; +\; n\; \backslash geq\; k$.
• Lemoine's conjecture: all odd integers greater than $5$ can be represented as the sum of an odd prime number and an even semiprime.
• Minimum overlap problem of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set $\backslash \{1,\; \backslash ldots,\; 2n\backslash \}$
• Pollock's conjectures
• Does every nonnegative integer appear in Recamán's sequence?
• Skolem problem: can an algorithm determine if a constant-recursive sequence contains a zero?
• The values of g(k) and G(k) in Waring's problem

• Do the Ulam numbers have a positive density?

• Determine growth rate of r_k(N) (see Szemerédi's theorem)

{{Main|Algebraic number theory }}

• Class number problem: are there infinitely many real quadratic number fields with unique factorization?
• Fontaine–Mazur conjecture: actually numerous conjectures, all proposed by Jean-Marc Fontaine and Barry Mazur.
• Gan–Gross–Prasad conjecture: a restriction problem in representation theory of real or p-adic Lie groups.
• Greenberg's conjectures
• Hermite's problem: is it possible, for any natural number $n$, to assign a sequence of natural numbers to each real number such that the sequence for $x$ is eventually periodic if and only if $x$ is algebraic of degree $n$?
• Kummer–Vandiver conjecture: primes $p$ do not divide the class number of the maximal real subfield of the $p$-th cyclotomic field.
• Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant $X$ is within a constant multiple of $\backslash sqrt\{X\}/\backslash ln\{X\}$
• Selberg's 1/4 conjecture: the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least $1/4$.
• Stark conjectures (including Brumer–Stark conjecture)

• Characterize all algebraic number fields that have some power basis.

{{Main|Computational number theory}}

• Can integer factorization be done in polynomial time?

{{Further|Diophantine approximation|Transcendental number theory}}

File:gamma-area.svg, which may or may not be a rational number.]]

• Schanuel's conjecture on the transcendence degree of certain field extensions of the rational numbers.{{citation |last=Waldschmidt |first=Michel |title=Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables |pages=14, 16 |year=2013 |url=https://books.google.com/books?id=Wrj0CAAAQBAJ&pg=PA14 |publisher=Springer |isbn=978-3-662-11569-5}} In particular: Are $\backslash pi$ and $e$ algebraically independent? Which nontrivial combinations of transcendental numbers (such as $e\; +\; \backslash pi,\; e\backslash pi,\; \backslash pi^e,\; \backslash pi^\{\backslash pi\},\; e^e$) are themselves transcendental?{{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}}{{Citation |last=Albert |first=John |title=Some unsolved problems in number theory |url=http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf |access-date=15 December 2014 |archive-url=https://web.archive.org/web/20140117150133/http://www2.math.ou.edu/~jalbert/courses/openprob2.pdf |archive-date=17 January 2014}}
• The four exponentials conjecture: the transcendence of at least one of four exponentials of combinations of irrationals
• Are Euler's constant $\backslash gamma$ and Catalan's constant $G$ irrational? Are they transcendental? Is Apéry's constant $\backslash zeta(3)$ transcendental?For some background on the numbers in this problem, see articles by Eric W. Weisstein at Wolfram MathWorld (all articles accessed 22 August 2024):

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

{{Further|Diophantine equation}}

• Beal's conjecture: for all integral solutions to $A^x\; +\; B^y\; =\; C^z$ where $x,\; y,\; z\; >\; 2$, all three numbers $A,\; B,\; C$ must share some prime factor.
• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem): determine precisely what rational numbers are congruent numbers.
• Erdős–Moser problem: is $1^1\; +\; 2^1\; =\; 3^1$ the only solution to the Erdős–Moser equation?
• Erdős–Straus conjecture: for every $n\; \backslash geq\; 2$, there are positive integers $x,\; y,\; z$ such that $4/n\; =\; 1/x\; +\; 1/y\; +\; 1/z$.
• Fermat–Catalan conjecture: there are finitely many distinct solutions $(a^m,\; b^n,\; c^k)$ to the equation $a^m\; +\; b^n\; =\; c^k$ with $a,\; b,\; c$ being positive coprime integers and $m,\; n,\; k$ being positive integers satisfying .
• Goormaghtigh conjecture on solutions to $(x^m\; -\; 1)/(x\; -\; 1)\; =\; (y^n\; -\; 1)/(y\; -\; 1)$ where $x\; >\; y\; >\; 1$ and $m,\; n\; >\; 2$.
• The uniqueness conjecture for Markov numbers{{citation

| last = Aigner | first = Martin

| doi = 10.1007/978-3-319-00888-2

| isbn = 978-3-319-00887-5

| location = Cham

| mr = 3098784

| publisher = Springer

| title = Markov's theorem and 100 years of the uniqueness conjecture

| year = 2013}} that every Markov number is the largest number in exactly one normalized solution to the Markov Diophantine equation.

• Pillai's conjecture: for any $A,\; B,\; C$, the equation $Ax^m\; -\; By^n\; =\; C$ has finitely many solutions when $m,\; n$ are not both $2$.
• Which integers can be written as the sum of three perfect cubes?{{Cite arXiv |eprint = 1604.07746|last1 = Huisman |first1 = Sander G.|title = Newer sums of three cubes|class = math.NT|year = 2016}}
• Can every integer be written as a sum of four perfect cubes?

{{Main|Prime numbers}}

{{Prime number conjectures}}

File:Goldbach partitions of the even integers from 4 to 50 rev4b.svg 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.]]

• Agoh–Giuga conjecture on the Bernoulli numbers that $p$ is prime if and only if $pB\_\{p-1\}\; \backslash equiv\; -1\; \backslash pmod\; p$
• Agrawal's conjecture that given coprime positive integers $n$ and $r$, if $(X\; -\; 1)^n\; \backslash equiv\; X^n\; -\; 1\; \backslash pmod\{n,\; X^r\; -\; 1\}$, then either $n$ is prime or $n^\{2\}\; \backslash equiv\; 1\; \backslash pmod\{r\}$
• Artin's conjecture on primitive roots that if an integer is neither a perfect square nor $-1$, then it is a primitive root modulo infinitely many prime numbers $p$
• Brocard's conjecture: there are always at least $4$ prime numbers between consecutive squares of prime numbers, aside from $2^\{2\}$ and $3^\{2\}$.
• Bunyakovsky conjecture: if an integer-coefficient polynomial $f$ has a positive leading coefficient, is irreducible over the integers, and has no common factors over all $f(x)$ where $x$ is a positive integer, then $f(x)$ is prime infinitely often.
• Catalan's Mersenne conjecture: some Catalan–Mersenne number is composite and thus all Catalan–Mersenne numbers are composite after some point.
• Dickson's conjecture: for a finite set of linear forms $a\_1\; +\; b\_1\; n,\; \backslash ldots,\; a\_k\; +\; b\_k\; n$ with each $b\_i\; \backslash geq\; 1$, there are infinitely many $n$ for which all forms are prime, unless there is some congruence condition preventing it.
• Dubner's conjecture: every even number greater than $4208$ is the sum of two primes which both have a twin.
• Elliott–Halberstam conjecture on the distribution of prime numbers in arithmetic progressions.
• Erdős–Mollin–Walsh conjecture: no three consecutive numbers are all powerful.
• Feit–Thompson conjecture: for all distinct prime numbers $p$ and $q$, $(p^q\; -\; 1)/(p\; -\; 1)$ does not divide $(q^p\; -\; 1)/(q\; -\; 1)$
• Fortune's conjecture that no Fortunate number is composite.
• The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
• Gillies' conjecture on the distribution of prime divisors of Mersenne numbers.
• Landau's problems
• Goldbach conjecture: all even natural numbers greater than $2$ are the sum of two prime numbers.
• Legendre's conjecture: for every positive integer $n$, there is a prime between $n^\{2\}$ and $(n+1)^\{2\}$.
• Twin prime conjecture: there are infinitely many twin primes.
• Are there infinitely many primes of the form $n^\{2\}\; +\; 1$?
• Problems associated to Linnik's theorem
• New Mersenne conjecture: for any odd natural number $p$, if any two of the three conditions $p\; =\; 2^k\; \backslash pm\; 1$ or $p\; =\; 4^k\; \backslash pm\; 3$, $2^p\; -\; 1$ is prime, and $(2^\{p\}\; +\; 1)/3$ is prime are true, then the third condition is also true.
• Polignac's conjecture: for all positive even numbers $n$, there are infinitely many prime gaps of size $n$.
• Schinzel's hypothesis H that for every finite collection $\backslash \{f\_1,\; \backslash ldots,\; f\_k\backslash \}$ of nonconstant irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers $n$ for which $f\_1(n),\; \backslash ldots,\; f\_k(n)$ are all primes, or there is some fixed divisor $m\; >\; 1$ which, for all $n$, divides some $f\_i(n)$.
• Selfridge's conjecture: is 78,557 the lowest Sierpiński number?
• Does the converse of Wolstenholme's theorem hold for all natural numbers?
• Are all Euclid numbers square-free?
• Are all Fermat numbers square-free?
• Are all Mersenne numbers of prime index square-free?
• Are there any composite c satisfying 2^{c − 1} ≡ 1 (mod c²)?
• Are there any Wall–Sun–Sun primes?
• Are there any Wieferich primes in base 47?
• Are there infinitely many balanced primes?
• Are there infinitely many Carol primes?
• Are there infinitely many cluster primes?
• Are there infinitely many cousin primes?
• Are there infinitely many Cullen primes?
• Are there infinitely many Euclid primes?
• Are there infinitely many Fibonacci primes?
• Are there infinitely many Kummer primes?
• Are there infinitely many Kynea primes?
• Are there infinitely many Lucas primes?
• Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
• Are there infinitely many Newman–Shanks–Williams primes?
• Are there infinitely many palindromic primes to every base?
• Are there infinitely many Pell primes?
• Are there infinitely many Pierpont primes?
• Are there infinitely many prime quadruplets?
• Are there infinitely many prime triplets?
• Siegel's conjecture: are there infinitely many regular primes, and if so is their natural density as a subset of all primes $e^\{-1/2\}$?
• Are there infinitely many sexy primes?
• Are there infinitely many safe and Sophie Germain primes?
• Are there infinitely many Wagstaff primes?
• Are there infinitely many Wieferich primes?
• Are there infinitely many Wilson primes?
• Are there infinitely many Wolstenholme primes?
• Are there infinitely many Woodall primes?
• Can a prime p satisfy $2^\{p-1\}\backslash equiv\; 1\backslash pmod\{p^2\}$ and $3^\{p-1\}\backslash equiv\; 1\backslash pmod\{p^2\}$ simultaneously?{{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 }}
• Does every prime number appear in the Euclid–Mullin sequence?
• What is the smallest Skewes's number?
• For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)
• For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p²)?{{cite book |last=Ribenboim |first=P. |author-link=Paulo Ribenboim |date=2006 |title=Die Welt der Primzahlen |edition=2nd |language=de |publisher=Springer |doi=10.1007/978-3-642-18079-8 |isbn=978-3-642-18078-1 |pages=242–243 |url=https://books.google.com/books?id=XMyzh-2SClUC&q=die+folgenden+probleme+sind+ungel%C3%B6st&pg=PA242|series=Springer-Lehrbuch }}
• For any given integer b which is not a perfect power and not of the form −4k⁴ for integer k, are there infinitely many repunit primes to base b?
• For any given integers $k\backslash geq\; 1,\; b\backslash geq\; 2,\; c\backslash neq\; 0$, with {{nowrap|1=gcd(k, c) = 1}} and {{nowrap|1=gcd(b, c) = 1,}} are there infinitely many primes of the form $(k\backslash times\; b^n+c)/\backslash gcd(k+c,b-1)$ with integer n ≥ 1?
• Is every Fermat number $2^\{2^n\}\; +\; 1$ composite for $n\; >\; 4$?
• Is 509,203 the lowest Riesel number?

{{Main|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.

• (Woodin) Does the generalized continuum hypothesis below a strongly compact cardinal imply the generalized continuum hypothesis everywhere?
• Does the generalized continuum hypothesis entail $\{\backslash diamondsuit(E^\{\backslash lambda^+\}\_\{\backslash operatorname\{cf\}(\backslash lambda)\}\})$ for every singular cardinal $\backslash lambda$?
• Does the generalized continuum hypothesis imply the existence of an ℵ₂-Suslin tree?
• If ℵ_ω is a strong limit cardinal, is (see Singular cardinals hypothesis)? The best bound, ℵ_ω4, was obtained by Shelah using his PCF theory.
• The problem of finding the ultimate core model, one that contains all large cardinals.
• Woodin's Ω-conjecture: if there is a proper class of Woodin cardinals, then Ω-logic satisfies an analogue of Gödel's completeness theorem.
• Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
• Does there exist a Jónsson algebra on ℵ_ω?
• Is OCA (the open coloring axiom) consistent with $2^\{\backslash aleph\_\{0\}\}>\backslash aleph\_\{2\}$?
• Reinhardt cardinals: Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?

{{Main|Topology}}

File:Ochiai_unknot.svg asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot.]]

• Baum–Connes conjecture: the assembly map is an isomorphism.
• Berge conjecture that the only knots in the 3-sphere which admit lens space surgeries are Berge knots.
• Bing–Borsuk conjecture: every $n$-dimensional homogeneous absolute neighborhood retract is a topological manifold.
• Borel conjecture: aspherical closed manifolds are determined up to homeomorphism by their fundamental groups.
• Halperin conjecture on rational Serre spectral sequences of certain fibrations.
• Hilbert–Smith conjecture: if a locally compact topological group has a continuous, faithful group action on a topological manifold, then the group must be a Lie group.
• Mazur's conjectures{{citation |last=Mazur |first=Barry |author-link=Barry Mazur |title=The topology of rational points |journal=Experimental Mathematics |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 }}
• Novikov conjecture on the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group.
• Quadrisecants of wild knots: it has been conjectured that wild knots always have infinitely many quadrisecants.{{citation

| last = Kuperberg | first = Greg | author-link = Greg Kuperberg

| arxiv = math/9712205

| doi = 10.1142/S021821659400006X

| journal = Journal of Knot Theory and Its Ramifications

| mr = 1265452

| pages = 41–50

| title = Quadrisecants of knots and links

| volume = 3

| year = 1994| s2cid = 6103528 }}

• Telescope conjecture: the last of Ravenel's conjectures in stable homotopy theory to be resolved.{{efn|A disproof has been announced, with a preprint made available on arXiv.{{cite arXiv |last1=Burklund |first1=Robert |last2=Hahn |first2=Jeremy |last3=Levy |first3=Ishan |last4=Schlank |first4=Tomer |title=K-theoretic counterexamples to Ravenel's telescope conjecture |date=2023 |class=math.AT |eprint=2310.17459 }}}}
• Unknotting problem: can unknots be recognized in polynomial time?
• Volume conjecture relating quantum invariants of knots to the hyperbolic geometry of their knot complements.
• Whitehead conjecture: every connected subcomplex of a two-dimensional aspherical CW complex is aspherical.
• Zeeman conjecture: given a finite contractible two-dimensional CW complex $K$, is the space $K\; \backslash times\; [0,\; 1]$ collapsible?

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

• Mazur's conjecture B (Vessilin Dimitrov, Ziyang Gao, and Philipp Habegger, 2020){{cite journal

|first1=Vessilin

|last1=Dimitrov

|first2=Ziyang

|last2=Gao

|first3=Philipp

|last3=Habegger

|title=Uniformity in Mordell–Lang for curves

|journal = Annals of Mathematics

|volume = 194

|year=2021

|pages=237–298

|doi=10.4007/annals.2021.194.1.4

|arxiv=2001.10276

|s2cid=210932420

|url=https://hal.sorbonne-universite.fr/hal-03374335/file/Dimitrov%20et%20al.%20-%202021%20-%20Uniformity%20in%20Mordell%E2%80%93Lang%20for%20curves.pdf}}

• Suita conjecture (Qi'an Guan and Xiangyu Zhou, 2015) {{cite journal

| jstor=24523356

| last1=Guan

| first1=Qi'an

| last2=Zhou

| first2=Xiangyu

| author2-link=Xiangyu Zhou

| title=A solution of an $L^2$ extension problem with optimal estimate and applications

| journal=Annals of Mathematics

| year=2015

| volume=181

| issue=3

| pages=1139–1208

| doi=10.4007/annals.2015.181.3.6

| s2cid=56205818

| arxiv=1310.7169}}

• Torsion conjecture (Loïc Merel, 1996){{cite journal

| last1 = Merel

| first1 = Loïc

| year = 1996

| title = "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]

| journal = Inventiones Mathematicae

| volume = 124

| issue = 1

| pages = 437–449

| doi = 10.1007/s002220050059

| mr = 1369424

| bibcode = 1996InMat.124..437M

| s2cid = 3590991 }}

• Carlitz–Wan conjecture (Hendrik Lenstra, 1995){{citation

| last1=Cohen

| first1=Stephen D.

| last2=Fried

| first2=Michael D.

| author2-link=Michael D. Fried

| doi=10.1006/ffta.1995.1027

| issue=3

| journal=Finite Fields and Their Applications

| mr=1341953

| pages=372–375

| title=Lenstra's proof of the Carlitz–Wan conjecture on exceptional polynomials: an elementary version

| volume=1

| year=1995

| doi-access=free}}

• Serre's nonnegativity conjecture (Ofer Gabber, 1995)

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

| arxiv=math/9906212

| last1=Kurdyka | first1=Krzysztof

| last2=Mostowski | first2=Tadeusz

| last3=Parusiński | first3=Adam

| title = Proof of the gradient conjecture of R. Thom

| journal=Annals of Mathematics

| pages=763–792

| volume=152

| date=2000

| issue=3

| doi=10.2307/2661354| jstor=2661354 | s2cid=119137528 }}

• Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018){{Cite journal |last1=Moreira |first1=Joel |last2=Richter |first2=Florian K. |last3=Robertson |first3=Donald |title=A proof of a sumset conjecture of Erdős |journal=Annals of Mathematics |doi=10.4007/annals.2019.189.2.4 |volume=189 |number=2 |pages=605–652 |language=en-US|year=2019 |arxiv=1803.00498 |s2cid=119158401 }}
• McMullen's g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018){{citation|last=Stanley|first=Richard P. |editor1-last=Bisztriczky|editor1-first=T.|editor2-last=McMullen|editor2-first=P.|editor3-last=Schneider|editor3-first=R.|editor4-last=Weiss|editor4-first=A. Ivić|contribution=A survey of Eulerian posets|location=Dordrecht|mr=1322068|pages=301–333 |publisher=Kluwer Academic Publishers|series=NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences |title=Polytopes: abstract, convex and computational (Scarborough, ON, 1993)|volume=440|year=1994}}. See in particular [https://books.google.com/books?id=gHjrCAAAQBAJ&pg=PA316 p. 316].{{cite web |last1=Kalai |first1=Gil |title=Amazing: Karim Adiprasito proved the g-conjecture for spheres! |url=https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/ |access-date=2019-02-15 |archive-url=https://web.archive.org/web/20190216031650/https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/ |archive-date=2019-02-16 |url-status=live |date=2018-12-25 }}
• Hirsch conjecture (Francisco Santos Leal, 2010){{cite journal |last=Santos |first=Franciscos |date=2012 |title=A counterexample to the Hirsch conjecture |journal=Annals of Mathematics |volume=176 |issue=1 |pages=383–412 |doi=10.4007/annals.2012.176.1.7 |arxiv=1006.2814 |s2cid=15325169 }}{{cite journal |last=Ziegler |first=Günter M. |date=2012 |title=Who solved the Hirsch conjecture? |journal=Documenta Mathematica |series=Documenta Mathematica Series |volume=6 |issue=Extra Volume "Optimization Stories" |pages=75–85 |doi=10.4171/dms/6/13 |doi-access=free |isbn=978-3-936609-58-5 | url=https://www.math.uni-bielefeld.de/documenta/vol-ismp/22_ziegler-guenter.html}}
• Gessel's lattice path conjecture (Manuel Kauers, Christoph Koutschan, and Doron Zeilberger, 2009){{cite journal | last1=Kauers | first1=Manuel | author1-link=Manuel Kauers | last2=Koutschan | first2=Christoph | author2-link=Christoph Koutschan | last3=Zeilberger | first3=Doron | author3-link=Doron Zeilberger | title=Proof of Ira Gessel's lattice path conjecture | journal=Proceedings of the National Academy of Sciences | volume=106 | issue=28 | date=2009-07-14 | issn=0027-8424 | doi=10.1073/pnas.0901678106 | pages=11502–11505 | pmc=2710637 | arxiv=0806.4300 | bibcode=2009PNAS..10611502K | doi-access=free }}
• Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004){{cite journal |last1=Chung |first1=Fan |last2=Greene |first2=Curtis |last3=Hutchinson |first3=Joan |date=April 2015 |title=Herbert S. Wilf (1931–2012) |journal=Notices of the AMS |volume=62 |issue=4 |page=358 |issn=1088-9477 |oclc=34550461 |quote=The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004. |doi=10.1090/noti1247 |doi-access=free }} (and also the Alon–Friedgut conjecture)
• Kemnitz's conjecture (Christian Reiher, 2003, Carlos di Fiore, 2003){{cite journal|title=Kemnitz' conjecture revisited | doi=10.1016/j.disc.2005.02.018 |doi-access=free| volume=297|issue=1–3 |journal=Discrete Mathematics|pages=196–201|year=2005 | last1 = Savchev | first1 = Svetoslav}}
• Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003){{cite journal | last = Green | first = Ben | author-link = Ben J. Green | arxiv = math.NT/0304058 | doi = 10.1112/S0024609304003650 | issue = 6 | journal = The Bulletin of the London Mathematical Society | mr = 2083752 | pages = 769–778 | title = The Cameron–Erdős conjecture | volume = 36 | year = 2004| s2cid = 119615076 }}{{cite web |url=https://www.ams.org/news?news_id=155 |title=News from 2007 |author= |date=31 December 2007 |website=American Mathematical Society |publisher=AMS |access-date=2015-11-13 |quote=The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..." |archive-url=https://web.archive.org/web/20151117030726/http://www.ams.org/news?news_id=155 |archive-date=17 November 2015 |url-status=live }}

• Zimmer's conjecture (Aaron Brown, David Fisher, and Sebastián Hurtado-Salazar, 2017){{cite arXiv

| last1=Brown

| first1=Aaron

| last2=Fisher

| first2=David

| last3=Hurtado

| first3=Sebastian

| date=2017-10-07

| title=Zimmer's conjecture for actions of {{not a typo|SL(𝑚,ℤ)}}

| eprint=1710.02735

| class=math.DS}}

• Painlevé conjecture (Jinxin Xue, 2014){{Cite arXiv|title=Noncollision Singularities in a Planar Four-body Problem|last=Xue|first=Jinxin|date=2014|class=math.DS |eprint = 1409.0048}}{{Cite journal|title=Non-collision singularities in a planar 4-body problem|last=Xue|first=Jinxin|date=2020|journal=Acta Mathematica|volume=224|issue=2|pages=253–388|doi=10.4310/ACTA.2020.v224.n2.a2|s2cid=226420221}}

• Existence of a non-terminating game of beggar-my-neighbour (Brayden Casella, 2024)

{{cite web | url= https://richardpmann.com/beggar-my-neighbour-records.html | title= Known Historical Beggar-My-Neighbour Records |author= Richard P Mann |access-date= 2024-02-10 }}

• The angel problem (Various independent proofs, 2006){{Cite web |url=http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf |title=The angel game in the plane |first=Brian H. |last=Bowditch|date=2006|location=School of Mathematics, University of Southampton |publisher=warwick.ac.uk Warwick University|access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160304185616/http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf |archive-date=2016-03-04 |url-status=live }}{{Cite web |url=http://home.broadpark.no/~oddvark/angel/Angel.pdf |title=A Solution to the Angel Problem |first=Oddvar |last=Kloster |publisher=SINTEF ICT |location=Oslo, Norway|access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160107125925/http://home.broadpark.no/~oddvark/angel/Angel.pdf |archive-date=2016-01-07 }}{{Cite journal |url=http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf |title=The Angel of power 2 wins |first=Andras |last=Mathe |date=2007|journal=Combinatorics, Probability and Computing |volume=16 |number=3|pages= 363–374|doi=10.1017/S0963548306008303 |doi-broken-date=1 November 2024 |s2cid=16892955 |access-date=2016-03-18 |archive-url=https://web.archive.org/web/20161013034302/http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf |archive-date=2016-10-13 |url-status=live }}{{Cite web |last=Gacs |first=Peter |date=June 19, 2007 |title=THE ANGEL WINS |url=http://www.cs.bu.edu/~gacs/papers/angel.pdf |archive-url=https://web.archive.org/web/20160304030433/http://www.cs.bu.edu/~gacs/papers/angel.pdf |archive-date=2016-03-04 |access-date=2016-03-18}}

• Einstein problem (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2024){{Cite journal |last1=Smith |first1=David |last2=Myers |first2=Joseph Samuel |last3=Kaplan |first3=Craig S. |last4=Goodman-Strauss |first4=Chaim |date=2024 |title=An aperiodic monotile |url=https://escholarship.org/uc/item/3317z9z9 |journal=Combinatorial Theory |language=en |volume=4 |issue=1 |doi=10.5070/C64163843 |issn=2766-1334}}
• Maximal rank conjecture (Eric Larson, 2018){{Cite arXiv| eprint=1711.04906 | last1=Larson | first1=Eric | title=The Maximal Rank Conjecture | year=2017 | class=math.AG }}
• Weibel's conjecture (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018){{citation

| first1=Moritz

| last1=Kerz

| first2=Florian

| last2=Strunk

| first3=Georg

| last3=Tamme

| title=Algebraic K-theory and descent for blow-ups

| journal=Inventiones Mathematicae

| volume=211

| year=2018

| issue=2

| pages=523–577

| mr=3748313

| doi=10.1007/s00222-017-0752-2

| arxiv=1611.08466| bibcode=2018InMat.211..523K

| s2cid=253741858

}}

• Yau's conjecture (Antoine Song, 2018){{cite web

| url = https://www.ams.org/amsmtgs/2251_abstracts/1147-53-499.pdf

| title = Existence of infinitely many minimal hypersurfaces in closed manifolds.

| author = Song, Antoine

| work = www.ams.org

| quote = "..I will present a solution of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves.."

| access-date = 19 June 2021}}

{{Cite web

| url=https://www.claymath.org/people/antoine-song

| title = Antoine Song | Clay Mathematics Institute

| quote="...Building on work of Codá Marques and Neves, in 2018 Song proved Yau's conjecture in complete generality"}}

• Pentagonal tiling (Michaël Rao, 2017){{citation|url=https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/|magazine=Quanta Magazine|title=Pentagon Tiling Proof Solves Century-Old Math Problem|first=Natalie|last=Wolchover|date=July 11, 2017|access-date=July 18, 2017|archive-url=https://web.archive.org/web/20170806093353/https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/|archive-date=August 6, 2017}}
• Willmore conjecture (Fernando Codá Marques and André Neves, 2012){{cite journal|last1=Marques |first1=Fernando C.|first2=André|last2=Neves|title=Min-max theory and the Willmore conjecture|journal=Annals of Mathematics |year=2013|arxiv=1202.6036|doi=10.4007/annals.2014.179.2.6|volume=179|issue=2|pages=683–782|s2cid=50742102}}
• Erdős distinct distances problem (Larry Guth, Nets Hawk Katz, 2011){{cite journal

| arxiv=1011.4105

| last1=Guth | first1=Larry

| last2=Katz | first2=Nets Hawk

| title=On the Erdos distinct distance problem in the plane

| journal=Annals of Mathematics

| pages=155–190

| volume=181

| date=2015

| issue=1

| doi=10.4007/annals.2015.181.1.2 | doi-access=free}}

• Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008){{Cite web |url=http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf |title=Squaring the Plane |first1=Frederick V. |last1=Henle |first2=James M. |last2=Henle |access-date=2016-03-18 |publisher=www.maa.org Mathematics Association of America|archive-url=https://web.archive.org/web/20160324074609/http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf |archive-date=2016-03-24 |url-status=live }}
• Tameness conjecture (Ian Agol, 2004)
• Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004){{Cite journal

| arxiv=math/0412006

| last1=Brock | first1=Jeffrey F.

| last2=Canary | first2=Richard D.

| last3=Minsky | first3=Yair N. | author-link3=Yair Minsky

| title=The classification of Kleinian surface groups, II: The Ending Lamination Conjecture

| date=2012

| journal=Annals of Mathematics

| volume=176

| issue=1

| pages=1–149

| doi=10.4007/annals.2012.176.1.1 | doi-access=free}}

• Carpenter's rule problem (Robert Connelly, Erik Demaine, Günter Rote, 2003){{citation

| last1 = Connelly | first1 = Robert | author1-link = Robert Connelly

| last2 = Demaine | first2 = Erik D. | author2-link = Erik Demaine

| last3 = Rote | first3 = Günter

| doi = 10.1007/s00454-003-0006-7 | doi-access = free

| issue = 2

| journal = Discrete & Computational Geometry

| mr = 1931840

| pages = 205–239

| title = Straightening polygonal arcs and convexifying polygonal cycles

| url = http://page.mi.fu-berlin.de/~rote/Papers/pdf/Straightening+polygonal+arcs+and+convexifying+polygonal+cycles-DCG.pdf

| volume = 30

| year = 2003| s2cid = 40382145 }}

• Lambda g conjecture (Carel Faber and Rahul Pandharipande, 2003){{Citation

| first1=C.

| last1=Faber

| first2=R.

| last2=Pandharipande

| author2-link=Rahul Pandharipande

| title=Hodge integrals, partition matrices, and the $\backslash lambda\_g$ conjecture

| journal=Ann. of Math.

| series= 2

| volume=157

| issue=1

| pages=97–124

| year=2003

| arxiv=math.AG/9908052

| doi=10.4007/annals.2003.157.97}}

• Nagata's conjecture (Ivan Shestakov, Ualbai Umirbaev, 2003){{cite journal

| last1 = Shestakov | first1 = Ivan P.

| last2 = Umirbaev | first2 = Ualbai U.

| doi = 10.1090/S0894-0347-03-00440-5

| issue = 1

| journal = Journal of the American Mathematical Society

| mr = 2015334

| pages = 197–227

| title = The tame and the wild automorphisms of polynomial rings in three variables

| volume = 17

| year = 2004}}

• Double bubble conjecture (Michael Hutchings, Frank Morgan, Manuel Ritoré, Antonio Ros, 2002){{cite journal

| last1 = Hutchings | first1 = Michael

| last2 = Morgan | first2 = Frank

| last3 = Ritoré | first3 = Manuel

| last4 = Ros | first4 = Antonio

| doi = 10.2307/3062123

| issue = 2

| journal = Annals of Mathematics

| mr = 1906593

| pages = 459–489

| series = Second Series

| title = Proof of the double bubble conjecture

| volume = 155

| year = 2002| jstor = 3062123

| arxiv = math/0406017

| hdl = 10481/32449

}}

• Honeycomb theorem (Thomas Callister Hales, 1999){{Cite journal

| arxiv=math/9906042

| last1=Hales | first1=Thomas C. | author-link1=Thomas Callister Hales

| title=The Honeycomb Conjecture

| journal=Discrete & Computational Geometry

| volume=25

| pages=1–22

| date=2001

| doi=10.1007/s004540010071 | doi-access=free}}

• Lange's conjecture (Montserrat Teixidor i Bigas and Barbara Russo, 1999){{cite journal

| last1=Teixidor i Bigas

| first1=Montserrat

| author1-link=Montserrat Teixidor i Bigas

| first2=Barbara

| last2=Russo

| title=On a conjecture of Lange

| arxiv=alg-geom/9710019

| mr=1689352

| year=1999

| journal=Journal of Algebraic Geometry

| issn=1056-3911

| volume=8

| issue=3

| pages=483–496

| bibcode=1997alg.geom.10019R }}

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| journal=Forum of Mathematics, Pi

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• Kahn–Kalai conjecture (Jinyoung Park and Huy Tuan Pham, 2022){{cite arXiv |last1=Park |first1=Jinyoung |last2=Pham |first2=Huy Tuan |date=2022-03-31 |title=A Proof of the Kahn-Kalai Conjecture |class=math.CO |eprint=2203.17207 }}
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| arxiv = 2011.04195

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| journal = Combinatorica

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• Ringel's conjecture that the complete graph $K\_\{2n+1\}$ can be decomposed into $2n+1$ copies of any tree with $n$ edges (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020){{cite journal|last1=Huang |first1=C.|title=Further results on tree labellings |journal=Utilitas Mathematica |volume=21 |pages=31–48 |year=1982|mr=668845|last2=Kotzig|first2=A.|last3=Rosa|first3=A.|author2-link=Anton Kotzig}}.{{Cite web |url=https://www.quantamagazine.org/mathematicians-prove-ringels-graph-theory-conjecture-20200219/|title=Rainbow Proof Shows Graphs Have Uniform Parts|last=Hartnett |first=Kevin|website=Quanta Magazine|date=19 February 2020|language=en|access-date=2020-02-29}}
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| url=http://www.sciencedirect.com/science/article/pii/S0095895619301212

| journal=Journal of Combinatorial Theory, Series B

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{{Cite journal

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| s2cid=119175309}}

• Goldberg–Seymour conjecture (Guantao Chen, Guangming Jing, and Wenan Zang, 2019){{Cite arXiv

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| title=Proof of the Goldberg–Seymour Conjecture on Edge-Colorings of Multigraphs

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• Babai's problem (Alireza Abdollahi, Maysam Zallaghi, 2015){{cite journal | first= Zallaghi M.|last= Abdollahi A. | year = 2015 | journal = Communications in Algebra | title = Character sums for Cayley graphs | volume = 43| issue = 12| pages = 5159–5167 | doi = 10.1080/00927872.2014.967398 |s2cid= 117651702 }}
• Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
• Alon–Saks–Seymour conjecture (Hao Huang, Benny Sudakov, 2012)
• Read–Hoggar conjecture (June Huh, 2009){{cite journal

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| title=Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

| arxiv=1008.4749

| journal=Journal of the American Mathematical Society

| volume=25

| date=2012

| issue=3

| pages=907–927

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| doi-access=free}}

• Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009){{cite conference

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| editor-last = Mitzenmacher | editor-first = Michael

| contribution = Every planar graph is the intersection graph of segments in the plane: extended abstract

| doi = 10.1145/1536414.1536500

| pages = 631–638

| publisher = ACM

| title = Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 – June 2, 2009

| year = 2009}}

• Erdős–Menger conjecture (Ron Aharoni, Eli Berger 2007){{Cite journal

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| last2=Berger | first2=Eli

| title = Menger's theorem for infinite graphs

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| volume=176

| pages=1–62

| date=2009

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• Road coloring conjecture (Avraham Trahtman, 2007){{cite news |last=Seigel-Itzkovich |first=Judy |title=Russian immigrant solves math puzzle |newspaper=The Jerusalem Post |date=2008-02-08

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• Robertson–Seymour theorem (Neil Robertson, Paul Seymour, 2004){{cite book |last=Diestel |first=Reinhard |year=2005 |chapter=Minors, Trees, and WQO |edition=Electronic Edition 2005 |pages=326–367 |publisher=Springer |title=Graph Theory |chapter-url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/preview/Ch12.pdf}}
• Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002){{cite journal |url=https://annals.math.princeton.edu/2006/164-1/p02 |title=The strong perfect graph theorem |last1=Chudnovsky |first1=Maria |last2=Robertson |first2=Neil |last3=Seymour |first3=Paul |last4=Thomas |first4=Robin |journal=Annals of Mathematics |year=2002 |volume=164 |pages=51–229 |arxiv=math/0212070 |doi=10.4007/annals.2006.164.51 |bibcode=2002math.....12070C |s2cid=119151552}}
• Toida's conjecture (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001)Klin, M. H., M. Muzychuk and R. Poschel: The isomorphism problem for circulant graphs via Schur ring theory, Codes and Association Schemes, American Math. Society, 2001.
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| volume=160

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| year=1996

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• Hanna Neumann conjecture (Joel Friedman, 2011, Igor Mineyev, 2011){{Cite journal |last=Friedman |first=Joel |date=January 2015 |title=Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: with an Appendix by Warren Dicks |url=https://www.cs.ubc.ca/~jf/pubs/web_stuff/shnc_memoirs.pdf |journal=Memoirs of the American Mathematical Society |language=en |volume=233 |issue=1100 |page=0 |doi=10.1090/memo/1100 |s2cid=117941803 |issn=0065-9266}}{{cite journal

| last = Mineyev | first = Igor

| doi = 10.4007/annals.2012.175.1.11

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| journal = Annals of Mathematics

| mr = 2874647

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| series = Second Series

| title = Submultiplicativity and the Hanna Neumann conjecture

| volume = 175

| year = 2012}}

• Density theorem (Hossein Namazi, Juan Souto, 2010){{Cite journal |url=https://www.researchgate.net/publication/228365532 |doi=10.1007/s11511-012-0088-0|title=Non-realizability and ending laminations: Proof of the density conjecture|journal=Acta Mathematica|volume=209|issue=2|pages=323–395|year=2012|last1=Namazi|first1=Hossein|last2=Souto|first2=Juan|doi-access=free}}
• Full classification of finite simple groups (Koichiro Harada, Ronald Solomon, 2008)

• André–Oort conjecture (Jonathan Pila, Ananth Shankar, Jacob Tsimerman, 2021){{cite arXiv |last1=Pila |first1=Jonathan |last2=Shankar |first2=Ananth |last3=Tsimerman |first3=Jacob |last4=Esnault |first4=Hélène |last5=Groechenig |first5=Michael |date=2021-09-17 |title=Canonical Heights on Shimura Varieties and the André-Oort Conjecture |class=math.NT |eprint=2109.08788}}
• Duffin–Schaeffer theorem (Dimitris Koukoulopoulos, James Maynard, 2019)
• Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015){{cite journal|last1=Bourgain |first1=Jean|first2=Demeter|last2=Ciprian|first3=Guth|last3=Larry|title=Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three|journal=Annals of Mathematics |year=2015|doi=10.4007/annals.2016.184.2.7|volume=184|issue=2|pages=633–682|hdl=1721.1/115568|bibcode=2015arXiv151201565B|arxiv=1512.01565|s2cid=43929329}}
• Goldbach's weak conjecture (Harald Helfgott, 2013){{cite arXiv |eprint=1305.2897 |title = Major arcs for Goldbach's theorem|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}{{cite arXiv |eprint=1205.5252 |title = Minor arcs for Goldbach's problem |last = Helfgott|first = Harald A.|class=math.NT |year=2012}}{{cite arXiv |eprint=1312.7748 |title = The ternary Goldbach conjecture is true|last = Helfgott|first = Harald A. |class=math.NT |year=2013}}
• Existence of bounded gaps between arbitrarily large primes (Yitang Zhang, Polymath8, James Maynard, 2013){{Cite journal|last=Zhang|first=Yitang|date=2014-05-01|title=Bounded gaps between primes|journal=Annals of Mathematics|volume=179|issue=3|pages=1121–1174|doi=10.4007/annals.2014.179.3.7|issn=0003-486X}}{{Cite web|title=Bounded gaps between primes – Polymath Wiki|url=https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes|access-date=2021-08-27|website=asone.ai|archive-date=2020-12-08|archive-url=https://web.archive.org/web/20201208045925/https://asone.ai/polymath/index.php?title=Bounded_gaps_between_primes}}{{Cite journal|last=Maynard|first=James|date=2015-01-01|title=Small gaps between primes|journal=Annals of Mathematics|pages=383–413|doi=10.4007/annals.2015.181.1.7|arxiv=1311.4600|s2cid=55175056|issn=0003-486X}}
• Sidon set problem (Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa, 2010){{cite journal|title=Generalized Sidon sets|doi=10.1016/j.aim.2010.05.010 | volume=225|issue=5|journal=Advances in Mathematics|pages=2786–2807|year=2010 | last1 = Cilleruelo | first1 = Javier|hdl=10261/31032|s2cid=7385280|doi-access=free|hdl-access=free}}
• Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008){{Citation |last1=Khare |first1=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre's modularity conjecture (I) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=485–504 |doi=10.1007/s00222-009-0205-7 |bibcode=2009InMat.178..485K |citeseerx=10.1.1.518.4611 |s2cid=14846347 }}{{Citation |last1=Khare |first1=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre's modularity conjecture (II) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=505–586 |doi=10.1007/s00222-009-0206-6 |bibcode=2009InMat.178..505K |citeseerx=10.1.1.228.8022 |s2cid=189820189 }}{{cite journal |author= |title=2011 Cole Prize in Number Theory |url=https://www.ams.org/notices/201104/rtx110400610p.pdf |journal=Notices of the AMS |volume=58 |issue=4 |pages=610–611 |issn=1088-9477 |oclc=34550461 |access-date=2015-11-12 |archive-url=https://web.archive.org/web/20151106051835/http://www.ams.org/notices/201104/rtx110400610p.pdf |archive-date=2015-11-06 |url-status=live }}
• Green–Tao theorem (Ben J. Green and Terence Tao, 2004){{cite journal |author= |date=May 2010 |title=Bombieri and Tao Receive King Faisal Prize |url=https://www.ams.org/notices/201005/rtx100500642p.pdf |journal=Notices of the AMS |volume=57 |issue=5 |pages=642–643 |issn=1088-9477 |oclc=34550461 |quote=Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem. |access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160304063504/http://www.ams.org/notices/201005/rtx100500642p.pdf |archive-date=2016-03-04 |url-status=live }}
• Catalan's conjecture (Preda Mihăilescu, 2002){{cite journal |last=Metsänkylä |first=Tauno |date=5 September 2003 |title=Catalan's conjecture: another old diophantine problem solved |url=https://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf |journal=Bulletin of the American Mathematical Society |volume=41 |issue=1 |pages=43–57 |issn=0273-0979 |quote=The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu. |doi=10.1090/s0273-0979-03-00993-5 |access-date=13 November 2015 |archive-url=https://web.archive.org/web/20160304082755/http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf |archive-date=4 March 2016 |url-status=live }}
• Erdős–Graham problem (Ernest S. Croot III, 2000){{cite book | last = Croot | first = Ernest S. III | author-link = Ernest S. Croot III | publisher = University of Georgia, Athens | series = Ph.D. thesis | title = Unit Fractions | year = 2000}} {{cite journal | last = Croot | first = Ernest S. III | author-link = Ernest S. Croot III | arxiv = math.NT/0311421 | doi = 10.4007/annals.2003.157.545 | issue = 2 | journal = Annals of Mathematics | pages = 545–556 | title = On a coloring conjecture about unit fractions | volume = 157 | year = 2003| bibcode = 2003math.....11421C | s2cid = 13514070 }}

• Lafforgue's theorem (Laurent Lafforgue, 1998){{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 }}
• Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995){{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}}{{cite journal |author=Taylor R, Wiles A |year=1995 |title=Ring theoretic properties of certain Hecke algebras |url=http://www.math.harvard.edu/~rtaylor/hecke.ps |journal=Annals of Mathematics |volume=141 |issue=3 |pages=553–572 |citeseerx=10.1.1.128.531 |doi=10.2307/2118560 |jstor=2118560 |oclc=37032255 |archive-url=https://web.archive.org/web/20000916161311/http://www.math.harvard.edu/~rtaylor/hecke.ps |archive-date=16 September 2000}}

• Burr–Erdős conjecture (Choongbum Lee, 2017){{cite journal | last1 = Lee | first1 = Choongbum | year = 2017 | title = Ramsey numbers of degenerate graphs | journal = Annals of Mathematics | volume = 185 | issue = 3| pages = 791–829 | doi = 10.4007/annals.2017.185.3.2 | arxiv = 1505.04773 | s2cid = 7974973 }}
• Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor W. Marek, 2016){{cite journal |last=Lamb |first=Evelyn |date=26 May 2016 |title=Two-hundred-terabyte maths proof is largest ever |journal=Nature |doi=10.1038/nature.2016.19990 |volume=534 |issue=7605 |pages=17–18 |pmid=27251254 |bibcode=2016Natur.534...17L|doi-access=free }}{{cite book

| last1 = Heule | first1 = Marijn J. H. | author1-link=Marijn Heule

| last2 = Kullmann | first2 = Oliver

| last3 = Marek | first3 = Victor W. | author3-link=Victor W. Marek

| editor-last1 = Creignou | editor-first1 = N.

| editor-last2 = Le Berre | editor-first2 = D.

| arxiv = 1605.00723

| chapter = Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer

| doi = 10.1007/978-3-319-40970-2_15

| mr = 3534782

| pages = 228–245

| publisher = Springer, [Cham]

| series = Lecture Notes in Computer Science

| title = Theory and Applications of Satisfiability Testing – SAT 2016

| volume = 9710

| year = 2016| isbn = 978-3-319-40969-6

| s2cid = 7912943

}}

• Sensitivity conjecture for Boolean functions (Hao Huang, 2019){{cite web |author=Linkletter, David |date=27 December 2019 |title=The 10 Biggest Math Breakthroughs of 2019 |url=https://www.popularmechanics.com/science/math/g30346822/biggest-math-breakthroughs-2019/ |access-date=20 June 2021 |work=Popular Mechanics}}

• Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020){{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=Annals of Mathematics |volume=191 |issue=2 |pages=581–591 |doi=10.4007/annals.2020.191.2.5|s2cid=52398890 }}{{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=Quanta Magazine |language=en}}
• Virtual Haken conjecture (Ian Agol, Daniel Groves, Jason Manning, 2012){{Cite journal

| arxiv = 1204.2810v1

| last1 = Agol | first1 = Ian

| title = The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves, and Jason Manning)

| journal=Documenta Mathematica

| volume=18

| date=2013

| pages=1045–1087

| doi = 10.4171/dm/421 | doi-access = free | s2cid = 255586740 | url=https://www.math.uni-bielefeld.de/documenta/vol-18/33.pdf}} (and by work of Daniel Wise also virtually fibered conjecture)

• Hsiang–Lawson's conjecture (Simon Brendle, 2012){{Cite journal

| arxiv=1203.6597

| last1 = Brendle | first1 = Simon | author1-link=Simon Brendle

| title = Embedded minimal tori in $S^3$ and the Lawson conjecture

| journal=Acta Mathematica

| volume=211

| issue=2

| pages=177–190

| date=2013

| doi=10.1007/s11511-013-0101-2 | doi-access=free}}

• Ehrenpreis conjecture (Jeremy Kahn, Vladimir Markovic, 2011){{Cite journal

| arxiv=1101.1330

| last1=Kahn | first1=Jeremy | author1-link=Jeremy Kahn

| last2=Markovic | first2=Vladimir | author2-link=Vladimir Markovic

| title=The good pants homology and the Ehrenpreis conjecture

| journal=Annals of Mathematics

| pages=1–72

| volume=182

| date=2015

| issue=1

| doi=10.4007/annals.2015.182.1.1 | doi-access=free}}

• Atiyah conjecture for groups with finite subgroups of unbounded order (Austin, 2009){{cite journal

| arxiv = 0909.2360

| last1 = Austin |first1 = Tim

| title = Rational group ring elements with kernels having irrational dimension

| journal = Proceedings of the London Mathematical Society

| volume = 107

| issue = 6

| pages = 1424–1448

| date = December 2013

| doi = 10.1112/plms/pdt029 | bibcode = 2009arXiv0909.2360A|s2cid = 115160094}}

• Cobordism hypothesis (Jacob Lurie, 2008){{cite journal | last1 = Lurie | first1 = Jacob | year = 2009 | title = On the classification of topological field theories | journal = Current Developments in Mathematics | volume = 2008 | pages = 129–280 | doi=10.4310/cdm.2008.v2008.n1.a3| bibcode = 2009arXiv0905.0465L | arxiv = 0905.0465 | s2cid = 115162503 }}
• Spherical space form conjecture (Grigori Perelman, 2006)
• Poincaré conjecture (Grigori Perelman, 2002){{cite press release | publisher=Clay Mathematics Institute | date=March 18, 2010 | title=Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman | url=http://www.claymath.org/sites/default/files/millenniumprizefull.pdf | access-date=November 13, 2015 | quote=The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. | archive-url=https://web.archive.org/web/20100322192115/http://www.claymath.org/poincare/ | archive-date=March 22, 2010 | url-status=live }}
• Geometrization conjecture, (Grigori Perelman, series of preprints in 2002–2003){{Cite arXiv |eprint = 0809.4040|last1 = Morgan |first1 = John |title = Completion of the Proof of the Geometrization Conjecture|last2 = Tian|first2 = Gang|class = math.DG|year = 2008}}
• Nikiel's conjecture (Mary Ellen Rudin, 1999){{cite journal

| first1=M.E. | last1=Rudin | author-link1=Mary Ellen Rudin

| title=Nikiel's Conjecture

| journal=Topology and Its Applications

| volume=116

| year=2001

| issue=3 | pages=305–331

| doi=10.1016/S0166-8641(01)00218-8 | doi-access=free}}

• Disproof of the Ganea conjecture (Iwase, 1997){{cite web|url=https://www.researchgate.net/publication/220032558|title=Ganea's Conjecture on Lusternik-Schnirelmann Category|author=Norio Iwase|date=1 November 1998|work=ResearchGate}}

• Erdős discrepancy problem (Terence Tao, 2015){{Cite arXiv |eprint = 1509.05363v5|last1 = Tao|first1 = Terence | author-link1=Terence Tao|title = The Erdős discrepancy problem|class = math.CO|year = 2015}}
• Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015){{cite journal|title=Proof of the umbral moonshine conjecture|first1=John F. R.|last1=Duncan|first2=Michael J.|last2=Griffin|first3=Ken|last3=Ono|date=1 December 2015|journal=Research in the Mathematical Sciences|volume=2|issue=1|page=26|doi=10.1186/s40687-015-0044-7|bibcode=2015arXiv150301472D|arxiv=1503.01472|s2cid=43589605 |doi-access=free }}
• Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (Jeff Cheeger, Aaron Naber, 2014){{cite journal

| arxiv=1406.6534

| last1=Cheeger | first1=Jeff

| last2=Naber | first2=Aaron

| title=Regularity of Einstein Manifolds and the Codimension 4 Conjecture

| journal=Annals of Mathematics

| pages=1093–1165

| volume=182

| issue=3

| date=2015

| doi=10.4007/annals.2015.182.3.5 | doi-access=free}}

• Gaussian correlation inequality (Thomas Royen, 2014){{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=Quanta Magazine |archive-url=https://web.archive.org/web/20170424133433/https://www.quantamagazine.org/20170328-statistician-proves-gaussian-correlation-inequality/ |archive-date=April 24, 2017 |access-date=May 2, 2017}}
• Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman, Aleksandar Nikolov, 2011){{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}}
• Bloch–Kato conjecture (Vladimir Voevodsky, 2011){{Cite web |url=https://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf |title=On motivic cohomology with Z/l-coefficients |last=Voevodsky |first=Vladimir |access-date=2016-03-18 |archive-url=https://web.archive.org/web/20160327035457/http://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf |location=Princeton, NJ |website=annals.math.princeton.edu |publisher=Princeton University |date=1 July 2011|volume=174|issue=1|pages=401–438|archive-date=2016-03-27 |url-status=live }} (and Quillen–Lichtenbaum conjecture and by work of Thomas Geisser and Marc Levine (2001) also Beilinson–Lichtenbaum conjecture{{cite journal

| last1 = Geisser | first1 = Thomas

| last2 = Levine | first2 = Marc

| doi = 10.1515/crll.2001.006

| journal = Journal für die Reine und Angewandte Mathematik

| mr = 1807268

| pages = 55–103

| title = The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky

| volume = 2001

| year = 2001| issue = 530

}}{{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}}{{Rp|page=359}}{{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 }})

• Kauffman–Harary conjecture (Thomas Mattman, Pablo Solis, 2009){{Cite journal

| arxiv = 0906.1612

| last1 = Mattman |first1 = Thomas W.

| last2 = Solis | first2 = Pablo

| title = A proof of the Kauffman-Harary Conjecture

| journal = Algebraic & Geometric Topology

| volume = 9

| issue = 4

| pages = 2027–2039

| year = 2009

| doi = 10.2140/agt.2009.9.2027 | bibcode = 2009arXiv0906.1612M | s2cid = 8447495}}

• Surface subgroup conjecture (Jeremy Kahn, Vladimir Markovic, 2009){{cite journal

| arxiv=0910.5501

| last1 = Kahn | first1 = Jeremy

| last2 = Markovic | first2 = Vladimir

| title = Immersing almost geodesic surfaces in a closed hyperbolic three manifold

| journal = Annals of Mathematics

| pages=1127–1190

| volume=175

| issue=3

| year=2012

| doi=10.4007/annals.2012.175.3.4 | doi-access=free}}

• Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Zhiqin Lu, 2007){{cite journal

| first=Zhiqin | last=Lu

| orig-date=2007

| title=Normal Scalar Curvature Conjecture and its applications

| arxiv=0711.3510

| journal=Journal of Functional Analysis

| volume=261

| issue=5

| date=September 2011

| pages=1284–1308

| doi=10.1016/j.jfa.2011.05.002 | doi-access=free}}

• Nirenberg–Treves conjecture (Nils Dencker, 2005){{citation |last=Dencker |first=Nils |author-link=Nils Dencker |title=The resolution of the Nirenberg–Treves conjecture |journal=Annals of Mathematics |volume=163 |issue=2 |year=2006 |pages=405–444 |url=https://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf |doi=10.4007/annals.2006.163.405 |s2cid=16630732 |access-date=2019-04-07 |archive-url=https://web.archive.org/web/20180720145723/http://annals.math.princeton.edu/wp-content/uploads/annals-v163-n2-p02.pdf |archive-date=2018-07-20 |url-status=live }}{{cite web |url=https://www.claymath.org/research |title=Research Awards |website=Clay Mathematics Institute |access-date=2019-04-07 |archive-url=https://web.archive.org/web/20190407160116/https://www.claymath.org/research |archive-date=2019-04-07 |url-status=live }}
• Lax conjecture (Adrian Lewis, Pablo Parrilo, Motakuri Ramana, 2005){{cite journal

| last1 = Lewis | first1 = A. S.

| last2 = Parrilo | first2 = P. A.

| last3 = Ramana | first3 = M. V.

| doi = 10.1090/S0002-9939-05-07752-X

| issue = 9

| journal = Proceedings of the American Mathematical Society

| mr = 2146191

| pages = 2495–2499

| title = The Lax conjecture is true

| volume = 133

| year = 2005| s2cid = 17436983

}}

• The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004){{cite web |url=http://www.icm2010.in/prize-winners-2010/fields-medal-ngo-bao-chau |title=Fields Medal – Ngô Bảo Châu |author= |date=19 August 2010 |website=International Congress of Mathematicians 2010 |publisher=ICM |access-date=2015-11-12 |quote=Ngô Bảo Châu is being awarded the 2010 Fields Medal for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods. |archive-url=https://web.archive.org/web/20150924032610/http://www.icm2010.in/prize-winners-2010/fields-medal-ngo-bao-chau |archive-date=24 September 2015 |url-status=live }}
• Milnor conjecture (Vladimir Voevodsky, 2003){{cite journal |title=Reduced power operations in motivic cohomology |pages=1–57|journal=Publications Mathématiques de l'IHÉS |volume=98 |year=2003 |last1=Voevodsky |first1=Vladimir |doi=10.1007/s10240-003-0009-z |citeseerx=10.1.1.170.4427 |url=http://archive.numdam.org/item/PMIHES_2003__98__1_0/ |access-date=2016-03-18 |url-status=live |archive-url=https://web.archive.org/web/20170728114725/http://archive.numdam.org/item/PMIHES_2003__98__1_0 |archive-date=2017-07-28 |arxiv=math/0107109 |s2cid=8172797}}
• Kirillov's conjecture (Ehud Baruch, 2003){{cite journal

| last = Baruch | first = Ehud Moshe

| doi = 10.4007/annals.2003.158.207

| issue = 1

| journal = Annals of Mathematics

| mr = 1999922

| pages = 207–252

| series = Second Series

| title = A proof of Kirillov's conjecture

| volume = 158

| year = 2003}}

• Kouchnirenko's conjecture (Bertrand Haas, 2002){{Cite journal |last=Haas |first=Bertrand |date=2002 |title=A Simple Counterexample to Kouchnirenko's Conjecture |url=https://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf |url-status=live |journal=Beiträge zur Algebra und Geometrie |volume=43 |issue=1 |pages=1–8 |archive-url=https://web.archive.org/web/20161007091417/http://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf |archive-date=2016-10-07 |access-date=2016-03-18}}
• n! conjecture (Mark Haiman, 2001){{cite journal

| last = Haiman | first = Mark

| doi = 10.1090/S0894-0347-01-00373-3

| issue = 4

| journal = Journal of the American Mathematical Society

| mr = 1839919

| pages = 941–1006

| title = Hilbert schemes, polygraphs and the Macdonald positivity conjecture

| volume = 14

| year = 2001| s2cid = 9253880

}} (and also Macdonald positivity conjecture)

• Kato's conjecture (Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philipp Tchamitchian, 2001){{cite journal

| last1 = Auscher | first1 = Pascal

| last2 = Hofmann | first2 = Steve

| last3 = Lacey | first3 = Michael

| last4 = McIntosh | first4 = Alan

| last5 = Tchamitchian | first5 = Ph.

| doi = 10.2307/3597201

| issue = 2

| journal = Annals of Mathematics

| mr = 1933726

| pages = 633–654

| series = Second Series

| title = The solution of the Kato square root problem for second order elliptic operators on $\backslash mathbb\{R\}^n$

| volume = 156

| year = 2002| jstor = 3597201

}}

• Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001){{cite journal

| arxiv=math/0102150

| last1=Barbieri-Viale |first1=Luca

| last2=Rosenschon | first2=Andreas

| last3=Saito | first3=Morihiko

| title = Deligne's Conjecture on 1-Motives

| journal=Annals of Mathematics

| pages=593–633

| volume=158

| date=2003

| issue=2

| doi=10.4007/annals.2003.158.593 | doi-access=free}}

• Modularity theorem (Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, 2001){{Citation | last1=Breuil | first1=Christophe | last2=Conrad | first2=Brian | last3=Diamond | first3=Fred | last4=Taylor | first4=Richard | title=On the modularity of elliptic curves over Q: wild 3-adic exercises | doi=10.1090/S0894-0347-01-00370-8 | mr=1839918 | year=2001 | journal=Journal of the American Mathematical Society | issn=0894-0347 | volume=14 | issue=4 | pages=843–939| doi-access=free }}
• Erdős–Stewart conjecture (Florian Luca, 2001){{Cite journal|url=https://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf|doi=10.1090/s0025-5718-00-01178-9|title=On a conjecture of Erdős and Stewart|journal=Mathematics of Computation|volume=70|issue=234|pages=893–897|year=2000|last1=Luca|first1=Florian|access-date=2016-03-18|archive-url=https://web.archive.org/web/20160402030443/http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf|archive-date=2016-04-02|url-status=live|bibcode=2001MaCom..70..893L}}
• Berry–Robbins problem (Michael Atiyah, 2000){{cite book

| last = Atiyah | first = Michael | author-link = Michael Atiyah

| editor-last = Yau | editor-first = Shing-Tung | editor-link = Shing-Tung Yau

| contribution = The geometry of classical particles

| doi = 10.4310/SDG.2002.v7.n1.a1

| mr = 1919420

| pages = 1–15

| publisher = International Press | location = Somerville, Massachusetts

| series = Surveys in Differential Geometry

| title = Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer

| volume = 7

| year = 2000}}

• List of conjectures
• List of unsolved problems in statistics
• List of unsolved problems in computer science
• List of unsolved problems in physics
• Lists of unsolved problems
• Open Problems in Mathematics
• The Great Mathematical Problems
• Scottish Book

{{notelist}}

{{reflist|colwidth=30em}}

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

• {{cite book |first1=Fan|last1= Chung|author-link1=Fan Chung |last2=Graham |first2=Ron |author-link2=Ronald Graham| title=Erdös on Graphs: His Legacy of Unsolved Problems|title-link= Erdős on Graphs |publisher=AK Peters |year=1999 |isbn=978-1-56881-111-6}}
• {{cite book |last1=Croft |first1=Hallard T. |last2=Falconer |first2=Kenneth J. |last3=Guy |first3=Richard K. |author-link2=Kenneth Falconer (mathematician) |author-link3=Richard K. Guy |date=1994 |title=Unsolved Problems in Geometry |publisher=Springer |isbn=978-0-387-97506-1 |url-access=registration |url=https://archive.org/details/unsolvedproblems0000crof }}
• {{cite book |last=Guy |first=Richard K. |author-link=Richard K. Guy |date=2004 |title=Unsolved Problems in Number Theory |publisher=Springer |isbn=978-0-387-20860-2}}
• {{cite book |last1=Klee |first1=Victor |author-link1=Victor Klee |last2=Wagon |first2=Stan |author-link2=Stan Wagon |date=1996 |title=Old and New Unsolved Problems in Plane Geometry and Number Theory |url=https://archive.org/details/oldnewunsolvedpr0000klee |url-access=registration |publisher=The Mathematical Association of America |isbn=978-0-88385-315-3}}
• {{cite book |last=du Sautoy |first=Marcus |author-link=Marcus du Sautoy |date=2003 |title=The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics |publisher=Harper Collins |isbn=978-0-06-093558-0 |url-access=registration |url=https://archive.org/details/musicofprimes00marc }}
• {{cite book |last=Derbyshire |first=John |author-link=John Derbyshire |date=2003 |title=Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics |publisher=Joseph Henry Press |isbn=978-0-309-08549-6 |url-access=registration |url=https://archive.org/details/primeobsessionbe00derb_0 }}
• {{cite book |last=Devlin |first=Keith |author-link=Keith Devlin |date=2006 |title=The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time |publisher=Barnes & Noble |isbn=978-0-7607-8659-8}}
• {{cite book |last1=Blondel |first1=Vincent D. |last2=Megrestski |first2=Alexandre |author-link1=Vincent Blondel |date=2004 |title=Unsolved problems in mathematical systems and control theory |publisher=Princeton University Press |isbn=978-0-691-11748-5}}
• {{cite book |first1=Lizhen|last1= Ji|author-link1=Lizhen Ji |first2=Yat-Sun|last2= Poon |first3=Shing-Tung|last3= Yau|author-link3=Shing-Tung Yau |date=2013 |title=Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics) |publisher=International Press of Boston |isbn=978-1-57146-278-7}}
• {{cite journal |last=Waldschmidt |first=Michel |author-link=Michel Waldschmidt |date=2004 |title=Open Diophantine Problems |journal=Moscow Mathematical Journal |issn=1609-3321 |zbl=1066.11030 |volume=4 |number=1 |pages=245–305 |url=http://www.math.jussieu.fr/~miw/articles/pdf/odp.pdf |doi=10.17323/1609-4514-2004-4-1-245-305 |arxiv=math/0312440 |s2cid=11845578 }}
• {{cite arXiv |last1=Mazurov |first1=V. D. |author-link1=Victor Mazurov |last2=Khukhro |first2=E. I. |eprint=1401.0300v6 |title= Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version) |date= 1 Jun 2015|class=math.GR }}

• [http://faculty.evansville.edu/ck6/integer/unsolved.html 24 Unsolved Problems and Rewards for them]
• [http://www.openproblems.net/ List of links to unsolved problems in mathematics, prizes and research]
• [http://garden.irmacs.sfu.ca/ Open Problem Garden]
• [http://aimpl.org/ AIM Problem Lists]
• [http://cage.ugent.be/~hvernaev/problems/archive.html Unsolved Problem of the Week Archive]. MathPro Press.
• {{cite web|last1=Ball|first1=John M.|author-link=John M. Ball|title=Some Open Problems in Elasticity|url=https://people.maths.ox.ac.uk/ball/Articles%20in%20Conference%20Proceedings%20and%20Books/JMB%202002%20re%20Marsden%2060th.pdf}}
• {{cite web|last1=Constantin|first1=Peter|author-link=Peter Constantin|title=Some open problems and research directions in the mathematical study of fluid dynamics|url=https://web.math.princeton.edu/~const/2k.pdf}}
• {{cite web|last1=Serre|first1=Denis|author-link=Denis Serre|title=Five Open Problems in Compressible Mathematical Fluid Dynamics|url=http://www.umpa.ens-lyon.fr/~serre/DPF/Ouverts.pdf}}
• [http://unsolvedproblems.org/ Unsolved Problems in Number Theory, Logic and Cryptography]
• [http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html 200 open problems in graph theory] {{Webarchive|url=https://web.archive.org/web/20170515145908/http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/oproblems.html |date=2017-05-15 }}
• [http://cs.smith.edu/~orourke/TOPP/ The Open Problems Project (TOPP)], discrete and computational geometry problems
• [http://math.berkeley.edu/~kirby/problems.ps.gz Kirby's list of unsolved problems in low-dimensional topology]
• [http://www.math.ucsd.edu/~erdosproblems/ Erdös' Problems on Graphs]
• [https://arxiv.org/abs/1409.2823 Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory]
• [http://www.sciencedirect.com/science/article/pii/S0165011414003194 Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications]
• [http://wwwmath.uni-muenster.de/logik/Personen/rds/list.html List of open problems in inner model theory]
• {{cite web|last1=Aizenman|first1=Michael|author-link=Michael Aizenman|title=Open Problems in Mathematical Physics|url=https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/OPlist.html}}
• Barry Simon's [http://math.caltech.edu/SimonPapers/R27.pdf 15 Problems in Mathematical Physics]
• Alexandre Eremenko. [https://www.math.purdue.edu/~eremenko/uns1.html Unsolved problems in Function Theory]
• [https://www.erdosproblems.com/start Erdos Problems collection]

{{unsolved problems}}

{{DEFAULTSORT:Unsolved problems in mathematics}}

Mathematics

Category:Lists of problems