List of unsolved problems in mathematics

Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:^[6]

• 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.^[13] 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.^[14]

• The Kourovka Notebook (Russian: Коуровская тетрадь) is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.^[15]
• The Sverdlovsk Notebook (Russian: Свердловская тетрадь) is a collection of unsolved problems in semigroup theory, first published in 1965 and updated every 2 to 4 years since.^[16][17][18]
• The Dniester Notebook (Russian: Днестровская тетрадь) lists several hundred unsolved problems in algebra, particularly ring theory and modulus theory.^[19][20]
• The Erlagol Notebook (Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory.^[21]

• 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.
• Casas-Alvero conjecture: if a polynomial of degree ${\displaystyle d}$ defined over a field ${\displaystyle K}$ of characteristic ${\displaystyle 0}$ has a factor in common with its first through ${\displaystyle d-1}$-th derivative, then must ${\displaystyle f}$ be the ${\displaystyle d}$-th power of a linear polynomial?
• Connes embedding problem in Von Neumann algebra theory
• Crouzeix's conjecture: the matrix norm of a complex function ${\displaystyle f}$ applied to a complex matrix ${\displaystyle A}$ is at most twice the supremum of ${\displaystyle |f(z)|}$ over the field of values of ${\displaystyle A}$.
• 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 ${\displaystyle K(G,1)}$.
• 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?^[22]
• 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 ${\displaystyle k}$, a Hadamard matrix of order ${\displaystyle 4k}$ exists.
• 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 ${\displaystyle \{0\}}$, then it has no nil one-sided ideal other than ${\displaystyle \{0\}}$.
• Monomial conjecture on Noetherian local rings
• Existence of perfect cuboids and associated cuboid conjectures
• Pierce–Birkhoff conjecture: every piecewise-polynomial ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ is the maximum of a finite set of minimums of finite collections of polynomials.
• Rota's basis conjecture: for matroids of rank ${\displaystyle n}$ with ${\displaystyle n}$ disjoint bases ${\displaystyle B_{i}}$, it is possible to create an ${\displaystyle n\times n}$ matrix whose rows are ${\displaystyle B_{i}}$ and whose columns are also bases.
• Serre's conjecture II: if ${\displaystyle G}$ is a simply connected semisimple algebraic group over a perfect field of cohomological dimension at most ${\displaystyle 2}$, then the Galois cohomology set ${\displaystyle H^{1}(F,G)}$ is zero.
• Serre's positivity conjecture that if ${\displaystyle R}$ is a commutative regular local ring, and ${\displaystyle P,Q}$ are prime ideals of ${\displaystyle R}$, then ${\displaystyle \dim(R/P)+\dim(R/Q)=\dim(R)}$ implies ${\displaystyle \chi (R/P,R/Q)>0}$.
• Uniform boundedness conjecture for rational points: do algebraic curves of genus ${\displaystyle g\geq 2}$ over number fields ${\displaystyle K}$ have at most some bounded number ${\displaystyle N(K,g)}$ of ${\displaystyle K}$-rational points?
• Wild problems: problems involving classification of pairs of ${\displaystyle n\times n}$ matrices under simultaneous conjugation.
• Zariski–Lipman conjecture: for a complex algebraic variety ${\displaystyle V}$ with coordinate ring ${\displaystyle R}$, if the derivations of ${\displaystyle R}$ are a free module over ${\displaystyle R}$, then ${\displaystyle V}$ is smooth.
• Zauner's conjecture: do SIC-POVMs exist in all dimensions?

• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ${\displaystyle \aleph _{0}}$ is a simple algebraic group over an algebraically closed field.
• Generalized star height problem: can all regular languages be expressed using generalized regular expressions with limited nesting depths of Kleene stars?
• For which number fields does Hilbert's tenth problem hold?
• Kueker's conjecture^[157]
• The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for ${\displaystyle \aleph _{1}}$-saturated models of a countable theory.^[158]
• Shelah's categoricity conjecture for ${\displaystyle L_{\omega _{1},\omega }}$: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.^[158]
• Shelah's eventual categoricity conjecture: For every cardinal ${\displaystyle \lambda }$ there exists a cardinal ${\displaystyle \mu (\lambda )}$ such that if an AEC K with LS(K)${\displaystyle {}\leq \lambda }$ is categorical in a cardinal above ${\displaystyle \mu (\lambda )}$ then it is categorical in all cardinals above ${\displaystyle \mu (\lambda )}$.^[158][159]
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• The stable forking conjecture for simple theories^[160]
• 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?^[161]
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?^[162]
• Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is either finite, ${\displaystyle \aleph _{0}}$, or ${\displaystyle 2^{\aleph _{0}}}$.
• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ${\displaystyle \aleph _{\omega _{1}}}$ does it have a model of cardinality continuum?^[163]
• 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 ${\displaystyle \aleph _{n}}$, is it categorical in every cardinal?^[164][165]
• 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?^[166]
• Is the theory of the field of Laurent series over ${\displaystyle \mathbb {Z} _{p}}$ decidable? of the field of polynomials over ${\displaystyle \mathbb {C} }$?
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?^[167]
• Determine the structure of Keisler's order.^[168][169]
• What is the nature of the proof-theoretic ordinal (the smallest ordinal a theory cannot prove well-founded) for second-order arithmetic, ZFC, or stronger theories?^[170]

• Ibragimov–Iosifescu conjecture for φ-mixing sequences

The following conjectures are expressed in the first-order language of axiomatic set theory and, unless stated otherwise, are here taken to be over Zermelo-Frankel set theory, possibly with Choice. In particular, the conjecture's independence may not be open in set theories with a wider or conflicting class of models, such as the various constructive resp. non-wellfounded set theories, etc.

• Jacobian conjecture for more than 2 dimensions (Levent Alpöge, 2026 using Claude Fable 5).^{[191][192][193]}
  • Dixmier conjecture for ${\displaystyle n\geq 3}$, stably equivalent to the Jacobian conjecture.
• Mazur's conjecture B (Vessilin Dimitrov, Ziyang Gao, and Philipp Habegger, 2020).^[194]
• Suita conjecture (Qi'an Guan and Xiangyu Zhou, 2015).^[195]

• Sendov's conjecture and Phelps–Rodriguez conjecture (Lech Mazur using GPT-5.6 Pro, 2026).^[198][199]
• Heil–Ramanathan–Topiwala conjecture (Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven, Felix Voigtlaender using GPT-5.6 Pro, 2026).^[200]
• Mizohata–Takeuchi conjecture (Hannah Cairo, 2025).^[201][202]

• Erdős unit distance conjecture (published in an anonymized paper by OpenAI, 2026).^[203][204]
• Bunkbed conjecture (Nikita Gladkov, Igor Pak, Alexander Zimin, 2025).^[205]
• Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018).^[206]
• 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).^[207][208]
• Upper bound for the cap set problem (Jordan Ellenberg, Dion Gijswijt, 2016).^[209]
• Erdős discrepancy problem (Terence Tao, 2015)^[210]

• Eremenko's conjecture: every component of the escaping set of an entire transcendental function is unbounded. (David Martí-Pete, Lasse Rempe, and James Waterman, 2025).^[211]
• Zimmer's conjecture (Aaron Brown, David Fisher, and Sebastián Hurtado-Salazar, 2017).^[212]

• Existence of a non-terminating game of beggar-my-neighbour (Brayden Casella, 2024).^[213][214]

• Talagrand's convexity conjecture (Dongming Hua, Antoine Song, Stefan Tudose using GPT-5.5 Pro, 2026).^[215][216]
• Carathéodory conjecture for surfaces of smoothness ${\displaystyle C^{4}}$ (Brendan Guilfoyle and Wilhelm Klingenberg, 2025).^[217]
• Einstein problem (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2024).^[218]
• Egan conjecture (Sergei Drozdov, 2023).^[219]
• Existence of a complex structure on ${\displaystyle S^{6}}$ (Levent Alpöge, 2026 using Claude Fable 5).^{[220][221][222]}
• Keller's conjecture (Joshua Brakensiek, Marijn Heule, John Mackey, David Narvaez, 2019).^[223]
• Maximal rank conjecture (Eric Larson, 2018).^[224]
• Weibel's conjecture (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018).^[225]
• Yau's conjecture (Antoine Song, 2018).^[226][227]
• Pentagonal tiling (Michaël Rao, 2017).^[228]

• Dinitz–Garg–Goemans conjecture (Dmitry Rybin using GPT-5.6 Pro, 2026).^[229]
• Kahn–Kalai conjecture (Jinyoung Park and Huy Tuan Pham, 2022).^[230]
• Blankenship–Oporowski conjecture on the book thickness of subdivisions (Vida Dujmović, David Eppstein, Robert Hickingbotham, Pat Morin, and David Wood, 2021).^[231]
• Ringel's conjecture that the complete graph ${\displaystyle K_{2n+1}}$ can be decomposed into ${\displaystyle 2n+1}$ copies of any tree with ${\displaystyle n}$ edges (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020).^[232][233]
• Disproof of Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019).^[234]
• Kelmans–Seymour conjecture (Dawei He, Yan Wang, and Xingxing Yu, 2020).^{[235][236][237][238]}
• Goldberg–Seymour conjecture (Guantao Chen, Guangming Jing, and Wenan Zang, 2019).^[239]
• Babai's problem (Alireza Abdollahi, Maysam Zallaghi, 2015).^[240]

• André–Oort conjecture (Jonathan Pila, Ananth Shankar, Jacob Tsimerman, 2021).^[241]
• Duffin–Schaeffer theorem (Dimitris Koukoulopoulos, James Maynard, 2019).^[242]
• Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015).^[243]

• Burr–Erdős conjecture (Choongbum Lee, 2017).^[244]
• Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor W. Marek, 2016).^[245][246]

• Sensitivity conjecture for Boolean functions (Hao Huang, 2019).^[247]

• Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020).^[248][249]