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Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, yet many mathematical problems, both major and minor, still remain unsolved. These unsolved problems occur in multiple domains, including physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and more. Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention.

This article is a composite of unsolved problems derived from many sources, including but not limited to lists considered authoritative. It does not claim to be comprehensive, it may not always be quite up to date, and it includes problems which are considered by the mathematical community to be widely varying in both difficulty and centrality to the science as a whole.

Lists of unsolved problems in mathematics

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

Millennium Prize Problems

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved as of July, 2020:

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

The seventh problem, the Poincaré conjecture, has been solved; 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 still unsolved.

Unsolved problems

Algebra

• Homological conjectures in commutative algebra
• Finite lattice representation problem
• Hilbert's sixteenth problem
• Hilbert's fifteenth problem
• Hadamard conjecture
• Jacobson's conjecture
• Crouzeix's conjecture
• Existence of perfect cuboids and associated cuboid conjectures
• Zauner's conjecture: existence of SIC-POVMs in all dimensions
• Wild problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems
• Köthe conjecture
• Birch–Tate conjecture
• Serre's conjecture II
• Bombieri–Lang conjecture
• Farrell–Jones conjecture
• Bost conjecture
• Rota's basis conjecture
• Uniformity conjecture
• Kaplansky's conjectures
• Kummer–Vandiver conjecture
• Serre's multiplicity conjectures
• Pierce–Birkhoff conjecture
• Eilenberg–Ganea conjecture
• Green's conjecture
• Grothendieck–Katz p-curvature conjecture
• Sendov's conjecture
• Zariski–Lipman conjecture
• The Dneister Notebook (Dnestrovskaya Tetrad) collects several hundred unresolved problems in algebra, particularly ring theory and modulus theory.
• The Erlagol Notebook (Erlagolskaya Tetrad) collects unresolved problems in algebra and model theory.

Analysis

• The four exponentials conjecture on the transcendence of at least one of four exponentials of combinations of irrationals
• Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials
• The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy
• Schanuel's conjecture on the transcendence degree of exponentials of linearly independent irrationals
• Are ${\displaystyle \gamma }$ (the Euler–Mascheroni constant), π + e, π − e, πe, π/e, π, π, π, e, ln π, 2, e, Catalan's constant, or Khinchin's constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?
• Vitushkin's conjecture
• Invariant subspace problem
• Kung–Traub conjecture
• Regularity of solutions of Vlasov–Maxwell equations
• Regularity of solutions of Euler equations
• Convergence of Flint Hills series

Combinatorics

• 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
• The lonely runner conjecture: if ${\displaystyle k+1}$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance ${\displaystyle 1/(k+1)}$ from each other runner) at some time?
• Finding a function to model n-step self-avoiding walks.
• 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?
• Give a combinatorial interpretation of the Kronecker coefficients.
• Open questions concerning Latin squares
• The values of the Dedekind numbers ${\displaystyle M(n)}$ for ${\displaystyle n\geq 9}$.
• The values of the Ramsey numbers, particularly ${\displaystyle R(5,5)}$
• The values of the Van der Waerden numbers

Dynamical systems

• Collatz conjecture (3n + 1 conjecture)
• Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does the Lyapunov’s second method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion?
• Furstenberg conjecture – Is every invariant and ergodic measure for the ${\displaystyle \times 2,\times 3}$ action on the circle either Lebesgue or atomic?
• Margulis conjecture – Measure classification for diagonalizable actions in higher-rank groups
• MLC conjecture – Is the Mandelbrot set locally connected?
• 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?
• Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theory
• Eremenko's conjecture that every component of the escaping set of an entire transcendental function is unbounded
• Is every reversible cellular automaton in three or more dimensions locally reversible?
• Birkhoff conjecture: if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse?
• 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
• Berry–Tabor conjecture
• Painlevé conjecture

Games and puzzles

Combinatorial games

• Sudoku:
  • What is the maximum number of givens for a minimal puzzle?
  • How many puzzles have exactly one solution?
  • How many puzzles with exactly one solution are minimal?
• Tic-tac-toe variants:
  • Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy?
• What is the Turing completeness status of all unique elementary cellular automata?

Games with imperfect information

• Rendezvous problem

Geometry

Algebraic geometry

• Abundance conjecture
• Bass conjecture
• Deligne conjecture
• Dixmier conjecture
• Fröberg conjecture
• Fujita conjecture
• Hartshorne conjectures
• The Jacobian conjecture
• Manin conjecture
• Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory
• Nakai conjecture
• Resolution of singularities in characteristic ${\displaystyle p}$
• Standard conjectures on algebraic cycles
• Section conjecture
• Tate conjecture
• Termination of flips
• Virasoro conjecture
• Weight-monodromy conjecture
• Zariski multiplicity conjecture

Differential geometry

• 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
• The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds
• The spherical Bernstein's problem, a possible generalization of the original Bernstein's problem
• 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?
• Carathéodory conjecture
• Chern's conjecture (affine geometry)
• Chern's conjecture for hypersurfaces in spheres
• Yau's conjecture
• Yau's conjecture on the first eigenvalue
• 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.

Discrete geometry

• Solving the happy ending problem for arbitrary ${\displaystyle n}$
• Finding matching upper and lower bounds for k-sets and halving lines
• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2 smaller copies
• Find lower and upper bounds for Borsuk's problem on the number of smaller-diameter subsets needed to cover a bounded n-dimensional set.
• The Kobon triangle problem on triangles in line arrangements
• The McMullen problem on projectively transforming sets of points into convex position
• Tripod packing
• Ulam's packing conjecture about the identity of the worst-packing convex solid
• 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.
• What is the asymptotic growth rate of wasted space for packing unit squares into a half-integer square?
• Kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24
• How many unit distances can be determined by a set of n points in the Euclidean plane?
• Opaque forest problem
• 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.

Euclidean geometry

• 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
• Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?
• Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?
• Dissection into orthoschemes – is it possible for simplices of every dimension?
• The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?
• The Erdős–Oler conjecture that when ${\displaystyle n}$ is a triangular number, packing ${\displaystyle n-1}$ circles in an equilateral triangle requires a triangle of the same size as packing ${\displaystyle n}$ circles
• Falconer's conjecture that sets of Hausdorff dimension greater than ${\displaystyle d/2}$ in ${\displaystyle \mathbb {R} ^{d}}$ must have a distance set of nonzero Lebesgue measure
• Inscribed square problem, also known as Toeplitz' conjecture – does every Jordan curve have an inscribed square?
• The Kakeya conjecture – do ${\displaystyle n}$-dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to ${\displaystyle n}$?
• 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
• Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one
• Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?
• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?
• Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net, or simple edge-unfolding?
• The Thomson problem – what is the minimum energy configuration of ${\displaystyle n}$ mutually-repelling particles on a unit sphere?
• Uniform 5-polytopes – find and classify the complete set of these shapes
• 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?
• Atiyah conjecture on configurations
• Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar.

Graph theory

Paths and cycles in graphs

• Barnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle
• Chvátal's toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian
• The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice
• The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs
• The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree
• The Lovász conjecture on Hamiltonian paths in symmetric graphs
• 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.

Graph coloring and labeling

• Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs
• The Erdős–Faber–Lovász conjecture on coloring unions of cliques
• The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree
• The Hadwiger conjecture relating coloring to clique minors
• The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
• Jaeger's Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph
• The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index
• The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree

Graph drawing

• The Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number
• The Blankenship–Oporowski conjecture on the book thickness of subdivisions
• Conway's thrackle conjecture
• Harborth's conjecture that every planar graph can be drawn with integer edge lengths
• Negami's conjecture on projective-plane embeddings of graphs with planar covers
• The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding
• Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?
• Universal point sets of subquadratic size for planar graphs

Word-representation of graphs

• Characterise (non-)word-representable planar graphs
• Characterise word-representable near-triangulations containing the complete graph K₄ (such a characterisation is known for K₄-free planar graphs )
• 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
• Is the line graph of a non-word-representable graph always non-word-representable?
• Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter?
• Is it true that out of all bipartite graphs crown graphs require longest word-representants?
• Characterise word-representable graphs in terms of (induced) forbidden subgraphs.
• Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?

Miscellaneous graph theory

• Conway's 99-graph problem: does there exist a strongly regular graph with parameters (99,14,1,2)?
• The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph
• The GNRS conjecture on whether minor-closed graph families have ${\displaystyle \ell _{1}}$ embeddings with bounded distortion
• Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs
• The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs
• Jørgensen's conjecture that every 6-vertex-connected K₆-minor-free graph is an apex graph
• Meyniel's conjecture that cop number is ${\displaystyle O({\sqrt {n}})}$
• Does a Moore graph with girth 5 and degree 57 exist?
• What is the largest possible pathwidth of an n-vertex cubic graph?
• The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.
• 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?
• Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?
• Sumner's conjecture: does every ${\displaystyle (2n-2)}$-vertex tournament contain as a subgraph every ${\displaystyle n}$-vertex oriented tree?
• Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow
• Vizing's conjecture on the domination number of cartesian products of graphs

Group theory

• Is every finitely presented periodic group finite?
• The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
• For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
• Is every group surjunctive?
• Andrews–Curtis conjecture
• Herzog–Schönheim conjecture
• Does generalized moonshine exist?
• Are there an infinite number of Leinster groups?
• Guralnick–Thompson conjecture
• Problems in loop theory and quasigroup theory consider generalizations of groups
• The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.

Model theory and formal languages

• Vaught's conjecture
• 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.
• The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for ${\displaystyle \aleph _{1}}$-saturated models of a countable theory.
• Determine the structure of Keisler's order
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• Is the theory of the field of Laurent series over ${\displaystyle \mathbb {Z} _{p}}$ decidable? of the field of polynomials over ${\displaystyle \mathbb {C} }$?
• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?
• The Stable Forking Conjecture for simple theories
• For which number fields does Hilbert's tenth problem hold?
• 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?
• 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)<= ${\displaystyle \lambda }$ is categorical in a cardinal above ${\displaystyle \mu (\lambda )}$ then it is categorical in all cardinals above ${\displaystyle \mu (\lambda )}$.
• 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.
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?
• 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?
• 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.)
• Kueker's conjecture
• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
• Do the Henson graphs have the finite model property?
• The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?
• Generalized star height problem

Number theory

General

• Grand Riemann hypothesis
  • Generalized Riemann hypothesis
    • Riemann hypothesis
• n conjecture
  • abc conjecture
  • Szpiro's conjecture
• Hilbert's ninth problem
• Hilbert's eleventh problem
• Hilbert's twelfth problem
• Carmichael's totient function conjecture
• Erdős–Straus conjecture
• Erdős–Ulam problem
• Pillai's conjecture
• Hall's conjecture
• Lindelöf hypothesis and its consequence, the density hypothesis for zeroes of the Riemann zeta function (see Bombieri–Vinogradov theorem)
• Montgomery's pair correlation conjecture
• Hilbert–Pólya conjecture
• Grimm's conjecture
• Leopoldt's conjecture
• Scholz conjecture
• Are there any 3×3 magic squares with 9 positive square numbers?
• Do any Lychrel numbers exist?
• Find the set of solitary numbers, exceptionally, is 10 a solitary number?
• Catalan–Dickson conjecture on aliquot sequences
• Do any Taxicab(5, 2, n) exist for n > 1?
• Do any Taxicab(k, 2, n) exist for k > 4 and n > 1?
• Lander, Parkin, and Selfridge conjecture
• Beal conjecture
• Fermat–Catalan conjecture: Are there any solutions other than these?

${\displaystyle 1^{m}+2^{3}=3^{2}\;}$

${\displaystyle 2^{5}+7^{2}=3^{4}\;}$

${\displaystyle 7^{3}+13^{2}=2^{9}\;}$

${\displaystyle 2^{7}+17^{3}=71^{2}\;}$

${\displaystyle 3^{5}+11^{4}=122^{2}\;}$

${\displaystyle 33^{8}+1549034^{2}=15613^{3}\;}$

${\displaystyle 1414^{3}+2213459^{2}=65^{7}\;}$

${\displaystyle 9262^{3}+15312283^{2}=113^{7}\;}$

${\displaystyle 17^{7}+76271^{3}=21063928^{2}\;}$

${\displaystyle 43^{8}+96222^{3}=30042907^{2}\;}$

• Brocard's problem: existence of integers, (n,m), such that n! + 1 = m other than n = 4, 5, 7
• Beilinson conjecture
• Littlewood conjecture
• Vojta's conjecture
• Goormaghtigh conjecture
• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem)
• Lehmer's totient problem: if φ(n) divides n−1, must n be prime?
• Are there any number n such that there is exactly one integer m such that φ(m) = n?
• Are there infinitely many perfect numbers?
• Are there any odd perfect numbers?
• Are there any almost perfect numbers other than powers of 2?
• Are there any quasiperfect numbers?
• Are there any odd weird numbers?
• Are there infinitely many amicable numbers?
• Are there any pairs of amicable numbers which have opposite parity?
• Are there any pairs of relatively prime amicable numbers?
• Are there infinitely many betrothed numbers?
• Are there any pairs of betrothed numbers which have same parity?
• Are there infinitely many sociable number cycles?
• Are there any sociable number cycles with length 3?
• Are there any sociable number cycles such that not all numbers have same parity?
• Are there any quasi-sociable number cycles with odd length?
• 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?
• Piltz divisor problem, especially Dirichlet's divisor problem
• Exponent pair conjecture
• Is π a normal number (its digits are "random")?
• Casas-Alvero conjecture
• Sato–Tate conjecture
• Find value of De Bruijn–Newman constant
• Which integers can be written as the sum of three perfect cubes?
• Erdős–Moser problem: is 1 + 2 = 3 the only solution to the Erdős–Moser equation?
• Is there a covering system with odd distinct moduli?
• Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
• The uniqueness conjecture for Markov numbers
• Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function

Additive number theory

• The values of g(k) and G(k) in Waring's problem
• Gilbreath's conjecture
• Erdős conjecture on arithmetic progressions
• Erdős–Turán conjecture on additive bases
• Pollock octahedral numbers conjecture
• Skolem problem
• Determine growth rate of r_k(N) (see Szemerédi's theorem)
• Minimum overlap problem
• Do the Ulam numbers have a positive density?

Algebraic number theory

• Are there infinitely many real quadratic number fields with unique factorization (Class number problem)?
• Characterize all algebraic number fields that have some power basis.
• Stark conjectures (including Brumer–Stark conjecture)
• Kummer–Vandiver conjecture
• Greenberg's conjectures

Computational number theory

• Integer factorization: Can integer factorization be done in polynomial time?

Prime numbers

• Goldbach conjecture
• Twin prime conjecture
• Polignac's conjecture
• Brocard's Conjecture
• Catalan's Mersenne conjecture
• Agoh–Giuga conjecture
• Dubner's conjecture
• 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?
• New Mersenne conjecture
• Erdős–Mollin–Walsh conjecture
• First Hardy–Littlewood conjecture
• Second Hardy–Littlewood conjecture
• Bunyakovsky conjecture
• Dickson's conjecture
• Schinzel's hypothesis H
• Are there infinitely many prime quadruplets?
• Are there infinitely many cousin primes?
• Are there infinitely many sexy primes?
• For any given positive even integer k, are there infinitely many primes p such that p+k is also prime?
• Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
• Are there any odd perfect numbers?
• Are there infinitely many Wagstaff primes?
• Are there infinitely many Sophie Germain primes?
• Are there infinitely many Pierpont primes?
• Are there infinitely many regular primes, and if so is their relative density ${\displaystyle e^{-1/2}}$?
• Are there infinitely many repunit primes?
• For any given (positive or negative) integer b which is not a perfect power and not of the form −4k for integer k, are there infinitely many generalized repunit primes to base b?
• Are there infinitely many Cullen primes?
• Are there infinitely many Woodall primes?
• For any given integer b ≥ 2, are there infinitely many generalized Cullen primes base b (primes of the form n×b+1 with n ≥ b−1)?
• For any given integer b ≥ 2, are there infinitely many generalized Woodall primes base b (primes of the form n×b−1 with n ≥ b−1)?
• Are there any primes p such that p×2+1 is also prime?
• Are there any generalized Cullen primes base 11 other than 10×11+1?
• Are there any generalized Cullen primes base 13?
• Are there infinitely many Carol primes?
• Are there infinitely many Kynea primes?
• For any given even integer b ≥ 2, are there infinitely many generalized Carol primes base b?
• For any given even integer b ≥ 2, are there infinitely many generalized Kynea primes base b?
• Are there infinitely many palindromic primes to every base?
• Are there infinitely many Fibonacci primes?
• Are there infinitely many Lucas primes?
• Are there infinitely many Pell primes?
• Are there infinitely many Newman–Shanks–Williams primes?
• Are there infinitely many Perrin primes?
• Are there infinitely many Padovan primes?
• Are there infinitely many partition primes?
• Are there infinitely many Bell primes?
• Do upper bound of lengths of Cunningham chain of the 1st kind exists?
• Do upper bound of lengths of Cunningham chain of the 2nd kind exists?
• Are all Mersenne numbers of prime index square-free?
• Are there infinitely many Wieferich primes?
• Are there any Wieferich primes in bases 47, 72, 186, 187, 200, 203, 222, 231, ...?
• Are there any composite c satisfying 2 ≡ 1 (mod c)?
• For any given integer a > 0, are there infinitely many primes p such that a ≡ 1 (mod p)?
• Can a prime p satisfy 2 ≡ 1 (mod p) and 3 ≡ 1 (mod p) simultaneously?
• Are there infinitely many Wilson primes?
• Are there infinitely many Wolstenholme primes?
• Are there any Wall–Sun–Sun primes?
• Are there infinitely many Wieferich-non-Wilson primes?
• Are there infinitely many non-general primes?
• 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)
• Is every Fermat number 2 + 1 composite for ${\displaystyle n>4}$?
• Is 10 + 1 composite for ${\displaystyle n>1}$?
• Is (3 + 1)/2 composite for ${\displaystyle n>6}$?
• Are all Fermat numbers square-free?
• Are there any primes of the form “concatenate first n numbers in base 10”? Also for bases 4, 13, 18, 19, 22, 25.
• Are there infinitely may primes of the form “concatenate first n primes in base 10” (Smarandache–Wellin primes)?
• For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?
• For any given positive integer a which is not a perfect power, are there infinitely many primes with a as smallest positive primitive root? Especially, are there any primes with 108 or 150 as smallest positive primitive root?
• For any given negative integer a which is not a perfect power, are there infinitely many primes with a as largest negative primitive root?
• Artin's conjecture on primitive roots
• Is 991 the largest non-repunit permutable prime?
• Is 999331 the largest non-repunit circular prime?
• Find the set of left-truncatable primes in bases 30, 36, 40, 42, 44-46, 48, 50, 52, 54, 56-58, 60, 62-64, 66, 68-70, 72, 74-78, 80-82, 84-88, 90-100, ...
• Find the set of minimal primes in bases 17, 19, 21, 25-29, 31-41, 43-59, 61-100, ...
• Find the smallest weakly prime in bases 30 and 36.
• Is 78,557 the lowest Sierpiński number (so-called Selfridge's conjecture)?
• Is 509,203 the lowest Riesel number?
• Is 271,129 the lowest Sierpiński number which is prime?
• Is 125,050,976,086 the lowest Sierpiński number to base 3?
• Is 63,064,644,938 the lowest Riesel number to base 3?
• Is 66,741 the lowest Sierpiński number to base 4?
• Is 39,939 the lowest Riesel number to base 4 which is not square?
• Is 159,986 the lowest Sierpiński number to base 5?
• Is 346,802 the lowest Riesel number to base 5?
• Is 174,308 the lowest Sierpiński number to base 6?
• Is 1,597 Riesel number to base 6? (Equivalently, is 84,687 the lowest Riesel number to base 6?)
• Is 237 the smallest k divisible by 3 such that k×2±1 are not twin primes for all n?
• Is 3−1 the largest number n such that all of n and n±1 have primitive roots?
• For any given integers k ≥ 1, b ≥ 2, c ≠ 0, with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form (k×b+c)/gcd(k+c,b−1) with integer n ≥ 1?
• Fortune's conjecture (that no Fortunate number is composite)
• Landau's problems
• Feit–Thompson conjecture
• Does every prime number appear in the Euclid–Mullin sequence?
• Does the converse of Wolstenholme's theorem hold for all natural numbers?
• Elliott–Halberstam conjecture
• Problems associated to Linnik's theorem
• Find the smallest Skewes' number

Set theory

• The problem of finding the ultimate core model, one that contains all large cardinals.
• If ℵ_ω is a strong limit cardinal, then 2 < ℵ_ω1 (see Singular cardinals hypothesis). The best bound, ℵ_ω4, was obtained by Shelah using his pcf theory.
• Woodin's Ω-hypothesis.
• Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
• (Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesis everywhere?
• Does there exist a Jónsson algebra on ℵ_ω?
• Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
• Does the Generalized Continuum Hypothesis entail ${\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})}$ for every singular cardinal ${\displaystyle \lambda }$?
• Does the Generalized Continuum Hypothesis imply the existence of an ℵ₂-Suslin tree?
• Is OCA (Open coloring axiom) consistent with ${\displaystyle 2^{\aleph _{0}}>\aleph _{2}}$?

Topology

• Baum–Connes conjecture
• Borel conjecture
• Hilbert–Smith conjecture
• Mazur's conjectures
• Novikov conjecture
• Telescope conjectures
• Unknotting problem
• Volume conjecture
• Whitehead conjecture
• Zeeman conjecture

Problems solved since 1995

Algebra

• Connes embedding problem (Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen, 2020)

Analysis

• Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013) (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic ${\displaystyle KS_{r}}$ and ${\displaystyle KS'_{r}}$ conjectures, Bourgain-Tzafriri conjecture and ${\displaystyle R_{\epsilon }}$-conjecture)

Combinatorics

• Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)
• 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)
• Hirsch conjecture (Francisco Santos Leal, 2010)

Game theory

• The angel problem (Various independent proofs, 2006)

Geometry

• Yau's conjecture (Antoine Song, 2018)
• Pentagonal tiling (Michaël Rao, 2017)
• Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)
• Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)

Graph theory

• Ringel's conjecture on graceful labeling of trees (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)
• Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)
• Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (Alireza Abdollahi, Maysam Zallaghi, 2015)
• Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
• Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)
• Erdős–Menger conjecture (Aharoni, Berger 2007)
• Road coloring conjecture (Avraham Trahtman, 2007)

Group theory

• Hanna Neumann conjecture (Mineyev, 2011)
• Density theorem (Namazi, Souto, 2010)
• Full classification of finite simple groups (Harada, Solomon, 2008)

Number theory

• Duffin-Schaeffer conjecture (Dimitris Koukoulopoulos, James Maynard, 2019)
• Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)
• Goldbach's weak conjecture (Harald Helfgott, 2013)
• Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)

Ramsey theory

• Burr–Erdős conjecture (Choongbum Lee, 2017)
• Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)

Topology

• Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020)
• Virtual Haken conjecture (Agol, Groves, Manning, 2012) (and by work of Wise also virtually fibered conjecture)
• Hsiang–Lawson's conjecture (Brendle, 2012)
• Ehrenpreis conjecture (Kahn, Markovic, 2011)
• Atiyah conjecture (Austin, 2009)
• Cobordism hypothesis (Jacob Lurie, 2008)
• Geometrization conjecture, proven by Grigori Perelman in a series of preprints in 2002–2003.
• Spherical space form conjecture (Grigori Perelman, 2006)

Uncategorised

• Erdős discrepancy problem (Terence Tao, 2015)
• Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)
• Anderson conjecture (Cheeger, Naber, 2014)
• Gaussian correlation inequality (Thomas Royen, 2014)
• Willmore conjecture (Fernando Codá Marques and André Neves, 2012)
• Beck's 3-permutations conjecture (Newman, Nikolov, 2011)
• Bloch–Kato conjecture (Voevodsky, 2011) (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture)
• Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)
• Kauffman–Harary conjecture (Matmann, Solis, 2009)
• Surface subgroup conjecture (Kahn, Markovic, 2009)
• Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Lu, 2007)
• Nirenberg–Treves conjecture (Nils Dencker, 2005)
• Lax conjecture (Lewis, Parrilo, Ramana, 2005)
• The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)
• Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)
• Robertson–Seymour theorem (Robertson, Seymour, 2004)
• Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004) (and also Alon–Friedgut conjecture)
• Green–Tao theorem (Ben J. Green and Terence Tao, 2004)
• Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)
• Carpenter's rule problem (Connelly, Demaine, Rote, 2003)
• Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)
• Milnor conjecture (Vladimir Voevodsky, 2003)
• Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)
• Nagata's conjecture (Shestakov, Umirbaev, 2003)
• Kirillov's conjecture (Baruch, 2003)
• Poincaré conjecture (Grigori Perelman, 2002)
• Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)
• Kouchnirenko’s conjecture (Haas, 2002)
• Vaught conjecture (Knight, 2002)
• Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)
• Catalan's conjecture (Preda Mihăilescu, 2002)
• n! conjecture (Haiman, 2001) (and also Macdonald positivity conjecture)
• Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)
• Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)
• Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)
• Erdős–Stewart conjecture (Florian Luca, 2001)
• Berry–Robbins problem (Atiyah, 2000)
• Erdős–Graham problem (Croot, 2000)
• Honeycomb conjecture (Thomas Hales, 1999)
• Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)
• Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)
• Lafforgue's theorem (Laurent Lafforgue, 1998)
• Kepler conjecture (Ferguson, Hales, 1998)
• Dodecahedral conjecture (Hales, McLaughlin, 1998)
• Ganea conjecture (Iwase, 1997)
• Torsion conjecture (Merel, 1996)
• Harary's conjecture (Chen, 1996)
• Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)

See also

• 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

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Further reading

Books discussing problems solved since 1995

• Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate. ISBN 978-1-84115-791-7.
• O'Shea, Donal (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-84614-012-9.
• Szpiro, George G. (2003). Kepler's Conjecture. Wiley. ISBN 978-0-471-08601-7.
• Ronan, Mark (2006). Symmetry and the Monster. Oxford. ISBN 978-0-19-280722-9.

Books discussing unsolved problems

• Chung, Fan; Graham, Ron (1999). Erdös on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 978-1-56881-111-6.
• Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 978-0-387-97506-1.
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 978-0-387-20860-2.
• Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 978-0-88385-315-3.
• du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 978-0-06-093558-0.
• Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 978-0-309-08549-6.
• Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8.
• Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 978-0-691-11748-5.
• Ji, Lizhen; Poon, Yat-Sun; Yau, Shing-Tung (2013). Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics). International Press of Boston. ISBN 978-1-57146-278-7.
• Waldschmidt, Michel (2004). "Open Diophantine Problems" (PDF). Moscow Mathematical Journal. 4 (1): 245–305. arXiv:math/0312440. doi:10.17323/1609-4514-2004-4-1-245-305. ISSN 1609-3321. S2CID 11845578. Zbl 1066.11030.
• Mazurov, V. D.; Khukhro, E. I. (1 Jun 2015). "Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version)". arXiv:1401.0300v6 .
• The Sverdlovsk Notebook is a collection of unsolved problems in semigroup theory.
• Formulation of ${\displaystyle 50}$ unresloved problems for infinite Abelian groups are depicted in the book
• The list of ${\displaystyle 17}$ unresolved problems for Combinatorial Geometry are depicted in the book.
• Several dozens of unresolved problems for Combinatorial Geometry are depicted in the book.
• Many unresolved problems for Graph theory are depicted in the article.
• The list of several unresolved problems converning Maler Conjecture are depicted in the book .

External links

• 24 Unsolved Problems and Rewards for them
• List of links to unsolved problems in mathematics, prizes and research
• Open Problem Garden The collection of open problems in mathematics build on the principle of user editable ("wiki") site
• AIM Problem Lists
• Unsolved Problem of the Week Archive. MathPro Press.
• Ball, John M. "Some Open Problems in Elasticity" (PDF).
• Constantin, Peter. "Some open problems and research directions in the mathematical study of fluid dynamics" (PDF).
• Serre, Denis. "Five Open Problems in Compressible Mathematical Fluid Dynamics" (PDF).
• Unsolved Problems in Number Theory, Logic and Cryptography
• 200 open problems in graph theory
• The Open Problems Project (TOPP), discrete and computational geometry problems
• Kirby's list of unsolved problems in low-dimensional topology
• Erdös' Problems on Graphs
• Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory
• Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications
• List of open problems in inner model theory
• Aizenman, Michael. "Open Problems in Mathematical Physics".
• Barry Simon's 15 Problems in Mathematical Physics

1. The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1979
2. The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1989
3. Fuks 1974, p. 47, 88, 116, 134, 158, 159, 186, 210, 242, 243, 292, 318. sfn error: no target: CITEREFFuks1974 (help)
4. Boltiansky 1965, p. 83. sfn error: no target: CITEREFBoltiansky1965 (help)
5. Grunbaum 1971, p. 6. sfn error: no target: CITEREFGrunbaum1971 (help)
6. V. G. Vizing Some unresolved problems for Graph theory // Russian Mathematical Surveys, 23:6(144) (1968), 117–134; Russian Math. Surveys, 23:6 (1968), 125–141
7. Sprinjuk 1967, p. 150—154. sfn error: no target: CITEREFSprinjuk1967 (help)

• CS1 maint: DOI inactive as of October 2020
• Unsolved problems in mathematics
• Conjectures
• Lists of unsolved problems
• Mathematics-related lists