List of unsolved problems in mathematics

Lists of unsolved problems in mathematics

Over the course of time, several lists of unsolved mathematical problems have appeared.

Millennium Prize Problems

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved, as of 2017:

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. 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.

Algebra

Homological conjectures in commutative algebra

Hilbert's sixteenth problem

Hilbert's fifteenth problem

Hadamard conjecture

Jacobson'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

Uniformity conjecture

Kaplansky's conjecture

Kummer–Vandiver conjecture

Serre's multiplicity conjectures

Pierce–Birkhoff conjecture

Eilenberg–Ganea conjecture

Green's conjecture

Grothendieck–Katz p-curvature conjecture

Sendov's conjecture

Algebraic geometry

Bass conjecture

Deligne conjecture

Fröberg conjecture

Fujita conjecture

Hartshorne conjectures

The Jacobian conjecture

Manin conjecture

Nakai conjecture

Resolution of singularities in characteristic p

Standard conjectures on algebraic cycles

Section conjecture

Tate conjecture

Virasoro conjecture

Zariski multiplicity conjecture

Analysis

Schanuel's conjecture and four exponentials conjecture

Lehmer's conjecture

Pompeiu problem

Are γ (the Euler–Mascheroni constant), π + e, π − e, πe, π/e, π^e, π^√2, π^π, e^π2, ln π, 2^e, e^e, Catalan's constant or Khinchin's constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?

Khabibullin’s conjecture on integral inequalities

Hilbert's thirteenth problem

Vitushkin's conjecture

Combinatorics

Number of magic squares (sequence A006052 in the OEIS)

Number of magic tori (sequence A270876 in the OEIS)

Finding a formula for the probability that two elements chosen at random generate the symmetric group S n

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 k + 1 runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance 1 / ( k + 1 ) from each other runner) at some time?

Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?

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?

unicity conjecture for Markov numbers

Give a combinatorial interpretation of the Kronecker coefficients.

Discrete geometry

Solving the happy ending problem for arbitrary 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 2ⁿ smaller copies

The Kobon triangle problem on triangles in line arrangements

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

Ulam's packing conjecture about the identity of the worst-packing convex solid

Filling area conjecture

Hopf conjecture

Kakeya conjecture

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? (see Counting unit distances)

Euclidean geometry

The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?

Inscribed square problem – does every Jordan curve have an inscribed square?

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?

The Thomson problem – what is the minimum energy configuration of N particles bound to the surface of a unit sphere that repel each other with a 1/r potential (or any potential in general)?

Pentagonal tiling – 15 types of convex pentagons are known to monohedrally tile the plane, and it is not known whether this list is complete.

Falconer's conjecture

g-conjecture

Circle packing in an equilateral triangle

Circle packing in an isosceles right triangle

Lebesgue's universal covering problem – what is the convex shape in the plane of least area which provides an isometric cover for any shape of diameter one?

Bellman's lost in a forest problem – for a given shape of forest find the shortest escape path which will intersect the edge of the forest at some point for any given starting point and direction inside the forest.

Find the complete set of uniform 5-polytopes

Covering problem of Rado

The strong bellows conjecture – must the Dehn invariant of a flexible polyhedron stay constant as it flexes?

Dynamical systems

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 × 2 , × 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?

Is every reversible cellular automaton in three or more dimensions locally reversible?

Many problems concerning an outer billiard, for example show that outer billiards relative to almost every convex polygon has unbounded orbits.

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

Graph coloring and labeling

The Erdős–Faber–Lovász conjecture on coloring unions of cliques

The Hadwiger conjecture relating coloring to clique minors

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

Hedetniemi's conjecture on the chromatic number of tensor products of 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 Ringel–Kotzig conjecture on graceful labeling of trees

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

Miscellaneous graph theory

The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph

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

Deriving a closed-form expression for the percolation threshold values, especially p c (square site)

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.

Sumner's conjecture: does every ( 2 n − 2 ) -vertex tournament contain as a subgraph every 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?

Model theory

Vaught's conjecture

The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ℵ 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 ℵ 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 Z p decidable? of the field of polynomials over 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 ℵ ω 1 does it have a model of cardinality continuum?

Shelah's eventual Categority conjecture: For every cardinal λ there exists a cardinal μ ( λ ) such that If an AEC K with LS(K)<= λ is categorical in a cardinal above μ ( λ ) then it is categorical in all cardinals above μ ( λ ) .

Shelah's categoricity conjecture for L ω 1 , ω : 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 ℵ n , is it categorical in every cardinal?

Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)

Kueker's conjecture

Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?

Lachlan's decision problem

Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?

Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)

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?

General

Grand Riemann hypothesis

Generalized Riemann hypothesis

Riemann hypothesis

n conjecture

abc conjecture (Proof claimed in 2012, currently under review.)

Hilbert's ninth problem

Hilbert's eleventh problem

Hilbert's twelfth problem

Carmichael's totient function conjecture

Erdős–Straus conjecture

Pillai's conjecture

Hall's conjecture

Lindelöf hypothesis

Montgomery's pair correlation conjecture

Hilbert–Pólya conjecture

Grimm's conjecture

Leopoldt's conjecture

Do any odd perfect numbers exist?

Are there infinitely many perfect numbers?

Do quasiperfect numbers exist?

Do any odd weird numbers exist?

Do any Lychrel numbers exist?

Is 10 a solitary number?

Catalan–Dickson conjecture on aliquot sequences

Do any Taxicab(5, 2, n) exist for n > 1?

Brocard's problem: existence of integers, (n,m), such that n! + 1 = m² other than n = 4, 5, 7

Beilinson conjecture

Littlewood conjecture

Szpiro's 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 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?

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 1¹ + 2¹ = 3¹ the only solution to the Erdős–Moser equation?

Additive number theory

Beal's conjecture

Fermat–Catalan conjecture

Goldbach's conjecture

The values of g(k) and G(k) in Waring's problem

Collatz conjecture (3n + 1 conjecture)

Lander, Parkin, and Selfridge conjecture

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

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

Combinatorial number theory

Singmaster's conjecture: Is there a finite upper bound on the number of times that a number other than 1 can appear in Pascal's triangle?

Prime numbers

Catalan's Mersenne conjecture

Agoh–Giuga 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

Are there infinitely many prime quadruplets?

Are there infinitely many cousin primes?

Are there infinitely many sexy primes?

Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even 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 e − 1 / 2 ?

Are there infinitely many repunit primes to every base except perfect power and numbers of the form −4k⁴?

Are there infinitely many Cullen primes?

Are there infinitely many Woodall primes?

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 all Mersenne numbers of prime index square-free?

Are there infinitely many Wieferich primes?

Are there any Wieferich primes in base 47?

Are there any composite c satisfying 2^{c − 1} ≡ 1 (mod c²)?

For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p²)?

Can a prime p satisfy 2^p − 1 ≡ 1 (mod p²) and 3^p − 1 ≡ 1 (mod p²) simultaneously?

Are there infinitely many Wilson primes?

Are there infinitely many Wolstenholme primes?

Are there any Wall–Sun–Sun primes?

Is every Fermat number 2²ⁿ + 1 composite for n > 4 ?

Are all Fermat numbers square-free?

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?

Artin's conjecture on primitive roots

Is 78,557 the lowest Sierpiński number (so-called Selfridge's conjecture)?

Is 509,203 the lowest Riesel number?

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

Partial differential equations

Regularity of solutions of Vlasov–Maxwell equations

Regularity of solutions of Euler equations

Ramsey theory

The values of the Ramsey numbers, particularly R ( 5 , 5 )

The values of the Van der Waerden numbers

Erdős–Burr conjecture

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_ω < 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 ♢ ( E cf ⁡ ( λ ) λ + ) for every singular cardinal λ ?

Does the Generalized Continuum Hypothesis imply the existence of an ℵ₂-Suslin tree?

Other

List of unsolved problems in statistics

List of unsolved problems in computer science

Problems in loop theory and quasigroup theory

Problems in Latin squares

Invariant subspace problem

Kaplansky's conjectures on groups rings

Painlevé conjecture

Dixmier conjecture

Baum–Connes conjecture

Novikov conjecture

Prove Turing completeness for all unique elementary cellular automaton

Generalized star height problem

Assorted sphere packing problems, e.g. the densest irregular hypersphere packings

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.

Keating–Snaith conjecture

Kung–Traub conjecture

Atiyah conjecture on configurations

Toeplitz' conjecture (open since 1911)

Carathéodory conjecture

Church–Turing thesis

Weight-monodromy conjecture

Berry–Tabor conjecture

Birkhoff conjecture

Guralnick–Thompson conjecture

Hilbert–Smith conjecture

MNOP conjecture

Mazur's conjectures

Rendezvous problem

Scholz conjecture

Nirenberg–Treves conjecture

Quantum unique ergodicity conjecture

Density hypothesis

Zhou conjecture

Borel conjecture

Zeeman conjecture

Erdös–Ulam problem

Problems solved since 1995

Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)

Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (A. Abdollahi, M. Zallaghi, 2015)

Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)

Erdős discrepancy problem (Terence Tao, 2015)

Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)

Anderson conjecture (Cheeger, Naber, 2014)

Goldbach's weak conjecture (Harald Helfgott, 2013)

Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013) (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic K S r and K S r ′ conjectures, Bourgain-Tzafriri conjecture and R ϵ -conjecture)

Virtual Haken conjecture (Agol, Groves, Manning, 2012) (and by work of Wise also virtually fibered conjecture)

Hsiang–Lawson's conjecture (Brendle, 2012)

Willmore conjecture (Fernando Codá Marques and André Neves, 2012)

Ehrenpreis conjecture (Kahn, Markovic, 2011)

Hanna Neumann conjecture (Mineyev, 2011)

Bloch–Kato conjecture (Voevodsky, 2011) (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture)

Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)

Density theorem (Namazi, Souto, 2010)

Hirsch conjecture (Francisco Santos Leal, 2010)

Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)

Atiyah conjecture (Austin, 2009)

Kauffman–Harary conjecture (Matmann, Solis, 2009)

Surface subgroup conjecture (Kahn, Markovic, 2009)

Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)

Full classification of finite simple groups (Harada, Solomon, 2008)

Geometrization conjecture (proof was completed by Morgan and Tian in 2008 and it is based mostly on work of Grigori Perelman, 2002)

Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)

Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)

Erdős–Menger conjecture (Aharoni, Berger 2007)

Road coloring conjecture (Avraham Trahtman, 2007)

The angel problem (Various independent proofs, 2006)

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)