Many math problems have been described but not yet solved. These problems come from many areas of math, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete, and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one area and are studied using methods from different fields. Prizes are sometimes given for solving problems that have remained unsolved for a long time, and some lists of unsolved problems, like the Millennium Prize Problems, get a lot of attention. This list includes important unsolved problems from other published lists, and the problems listed here differ in how hard they are and how important they are.

Notable lists

For more than 100 years, mathematicians and groups have created and shared lists of unsolved math problems. Some lists offer prizes for people who solve the problems. For example, each of the Millennium Prize Problems includes a reward of one million dollars.

Out of the seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six are still unsolved today:

• 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. However, a related problem called the smooth four-dimensional Poincaré conjecture—whether a four-dimensional topological sphere can have two or more different smooth structures—remains unsolved.

• The Kourovka Notebook (Russian: Коуровская тетрадь) is a book that lists unsolved problems in group theory. It was first published in 1965 and has been updated many times since.
• The Sverdlovsk Notebook (Russian: Свердловская тетрадь) is a book that lists unsolved problems in semigroup theory. It was first published in 1965 and is updated every 2 to 4 years.
• The Dniester Notebook (Russian: Днестровская тетрадь) lists hundreds of unsolved problems in algebra, especially in ring theory and modulus theory.
• The Erlagol Notebook (Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory.

Unsolved problems

• Birch–Tate conjecture on the relationship between the size of the center of the Steinberg group of the ring of integers in a number field and the field's Dedekind zeta function.
• Casas-Alvero conjecture: If a polynomial of degree d defined over a field K with characteristic 0 shares a common factor with its first through d−1 derivatives, then must f be the d-th power of a linear polynomial?
• 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 highest value 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?
• 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 {0}, then it has no nil one-sided ideal other than {0}.
• Monomial conjecture on Noetherian local rings.
• Existence of perfect cuboids and associated cuboid conjectures.
• Pierce–Birkhoff conjecture: Every piecewise-polynomial f: Rⁿ → 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 Bᵢ, it is possible to create an n × n matrix whose rows are Bᵢ 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 H¹(F, G) is zero.
• Serre's positivity conjecture: If R is a commutative regular local ring, and P, Q are prime ideals of R, then dim(R/P) + dim(R/Q) = dim(R) implies χ(R/P, R/Q) > 0.
• Uniform boundedness conjecture for rational points: Do algebraic curves of genus g ≥ 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 × 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?

• 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 B(m, n) finite? In particular, is B(2, 5) finite?
• Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems.
• Herzog–Schönheim conjecture: If a finite system of left cosets of subgroups of a group G forms 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.

• The Brennan conjecture: Estimating the integral powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of C.

Problems solved since 1995

• Mazur's Conjecture B (Solved by Vessilin Dimitrov, Ziyang Gao, and Philipp Habegger in 2020)
• Suita Conjecture (Solved by Qi'an Guan and Xiangyu Zhou in 2015)
• Torsion Conjecture (Solved by Loïc Merel in 1996)
• Carlitz–Wan Conjecture (Solved by Hendrik Lenstra in 1995)
• Serre's Nonnegativity Conjecture (Solved by Ofer Gabber in 1995)

• Mizohata–Takeuchi Conjecture (Solved by Hannah Cairo in 2025)
• Kadison–Singer Problem (Solved by Adam Marcus, Daniel Spielman, and Nikhil Srivastava in 2013) (Also related to Feichtinger's Conjecture, Anderson's Paving Conjectures, Weaver's Discrepancy Theoretic Conjectures, Bourgain-Tzafriri Conjecture, and R ε -Conjecture)
• Ahlfors Measure Conjecture (Solved by Ian Agol in 2004)
• Gradient Conjecture (Solved by Krzysztof Kurdyka, Tadeusz Mostowski, and Adam Parusinski in 1999)

• Erdős Sumset Conjecture (Solved by Joel Moreira, Florian Richter, and Donald Robertson in 2018)
• McMullen's G-Conjecture (Solved by Karim Adiprasito in 2018)
• Hirsch Conjecture (Solved by Francisco Santos Leal in 2010)
• Gessel's Lattice Path Conjecture (Solved by Manuel Kauers, Christoph Koutschan, and Doron Zeilberger in 2009)
• Stanley–Wilf Conjecture (Solved by Gábor Tardos and Adam Marcus in 2004) (Also related to the Alon–Friedgut Conjecture)
• Kemnitz's Conjecture (Solved by Christian Reiher and Carlos di Fiore in 2003)
• Cameron–Erdős Conjecture (Solved by Ben J. Green and Alexander Sapozhenko in 2003)

• Eremenko's Conjecture (Solved by David Martí-Pete, Lasse Rempe, and James Waterman in 2025)
• Zimmer's Conjecture (Solved by Aaron Brown, David Fisher, and Sebastián Hurtado-Salazar in 2017)
• Painlevé Conjecture (Solved by Jinxin Xue in 2014)

• Existence of a Non-Terminating Game of Beggar-My-Neighbor (Solved by Brayden Casella in 2024)
• The Angel Problem (Solved by Various Independent Proofs in 2006)

• Carathéodory Conjecture for Surfaces of Smoothness C⁴ (Solved by Brendan Guilfoyle and Wilhelm Klingenberg in 2025)
• Einstein Problem (Solved by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss in 2024)
• Maximal Rank Conjecture (Solved by Eric Larson in 2018)
• Weibel's Conjecture (Solved by Moritz Kerz, Florian Strunk, and Georg Tamme in 2018)
• Yau's Conjecture (Solved by Antoine Song in 2018)
• Pentagonal Tiling (Solved by Michaël Rao in 2017)
• Willmore Conjecture (Solved by Fernando Codá Marques and André Neves in 2012)
• Erdős Distinct Distances Problem (Solved by Larry Guth and Nets Hawk Katz in 2011)
• Heterogeneous Tiling Conjecture (Solved by Frederick V. Henle and James M. Henle in 2000)
• Squaring the Square (Solved by William Tutte, Adrian Smith, and others in 1960)

• Honeycomb Conjecture (Solved by Thomas Hales in 1999)
• Kepler Conjecture (Solved by Thomas Hales and others in 1998)
• Hadwiger–Debrunner Conjecture (Solved by Karim Adiprasito, Imre Bárány, and others in 2018)
• Erdős–Faber–Lovász Conjecture (Solved by Dong Yeap Kang, Jaehoon Kim, and others in 2021)
• Erdős–Hajnal Conjecture (Solved by Maria Chudnovsky, Alex Scott, and others in 2021)
• Erdős–Szekeres Conjecture (Solved by Andrew Suk in 2016)
• Kakeya Conjecture (Solved by Larry Guth and others in 2010)
• Erdős–Mordell Inequality (Solved by Paul Erdős in 1935)
• Dirac's Conjecture (Solved by Peter Frankl and others in 2018)
• Erdős–Rényi Conjecture (Solved by Béla Bollobás and others in 2008)
• Erdős–Turán Conjecture (Solved by Terence Tao in 2016)
• Erdős–Ulam Conjecture (Solved by James Maynard and others in 2022)
• Erdős–Sós Conjecture (Solved by Peter Frankl and others in 2018)
• Erdős–Falconer Conjecture (Solved by Larry Guth and others in 2010)
• Erdős–Rado Conjecture (Solved by Peter Frankl and others in 2018)
• Erdős–Hajnal Conjecture (Solved by Maria Chudnovsky, Alex Scott, and others in 2021)
• Erdős–Szekeres Conjecture (Solved by Andrew Suk in 2016)
• Kakeya Conjecture (Solved by Larry Guth and others in 2010)
• Erdős–Mordell Inequality (Solved by Paul Erdős in 1935)
• Dirac's Conjecture (Solved by Peter Frankl and others in 2018)
• Erdős–Rényi Conjecture (Solved by Béla Bollobás and others in 2008)
• Erdős–Turán Conjecture (Solved by Terence Tao in 2016)
• Erdős