Algebra representation of a
qubit, a
SIC-POVM forms a
regular tetrahedron. Zauner conjectured that analogous structures exist in complex
Hilbert spaces of all finite dimensions. •
Birch–Tate conjecture on the relation between the order of the
center of the
Steinberg group of the
ring of integers of a
number field to the field's
Dedekind zeta function. •
Casas-Alvero conjecture: if a polynomial of degree d defined over a
field K of
characteristic 0 has a factor in common with its first through d - 1-th derivative, then must f be the d-th power of a linear polynomial? •
Connes embedding problem in
Von Neumann algebra theory •
Crouzeix's conjecture: the
matrix norm of a complex function f applied to a complex matrix A is at most twice the
supremum of |f(z)| over the
field of values of A. •
Determinantal conjecture on the
determinant of the sum of two
normal matrices. •
Eilenberg–Ganea conjecture: a group with
cohomological dimension 2 also has a 2-dimensional
Eilenberg–MacLane space K(G, 1). •
Farrell–Jones conjecture on whether certain
assembly maps are
isomorphisms. •
Bost conjecture: a specific case of the Farrell–Jones conjecture •
Finite lattice representation problem: is every finite
lattice isomorphic to the
congruence lattice of some finite
algebra? •
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:\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 n with n disjoint bases B_{i}, it is possible to create an n \times n matrix whose rows are B_{i} and whose columns are also bases. •
Serre's conjecture II: if G is a
simply connected semisimple algebraic group over a perfect
field of
cohomological dimension at most 2, then the
Galois cohomology set H^{1}(F, G) is zero. •
Serre's positivity conjecture that if R is a commutative
regular local ring, and P, Q are
prime ideals of R, then \dim (R/P) + \dim (R/Q) = \dim (R) implies \chi(R/P, R/Q) > 0. •
Uniform boundedness conjecture for rational points: do
algebraic curves of
genus g \geq 2 over
number fields K have at most some bounded number N(K, g) of K-
rational points? •
Wild problems: problems involving classification of pairs of n\times n matrices under simultaneous conjugation. •
Zariski–Lipman conjecture: for a
complex algebraic variety V with
coordinate ring R, if the
derivations of R are a
free module over R, then V is
smooth. • Zauner's conjecture: do
SIC-POVMs exist in all dimensions?
Group theory B(2,3) is finite; in its
Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups B(m,n) are finite remains open. •
Andrews–Curtis conjecture: every balanced
presentation of the
trivial group can be transformed into a trivial presentation by a sequence of
Nielsen transformations on
relators and conjugations of relators •
Bounded Burnside problem: for which positive integers
m,
n is the free Burnside group finite? In particular, is 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 form a partition of G, then the finite indices of said subgroups cannot be distinct. • The
inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals? •
Isomorphism problem of Coxeter groups • Are there an infinite number of
Leinster groups? • Does
generalized moonshine exist? • Is every
finitely presented periodic group finite? • Is every group
surjunctive? • Is every discrete, countable group
sofic? •
Problems in loop theory and quasigroup theory consider generalizations of groups
Representation theory •
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.
Analysis • 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 \mathbb{C} •
Fuglede's conjecture on whether nonconvex sets in \mathbb{R} and \mathbb{R}^{2} are spectral if and only if they tile by
translation. •
Goodman's conjecture on the coefficients of
multivalued functions •
Invariant subspace problem – does every
bounded operator on a complex
Banach space send some non-trivial
closed subspace to itself? • Kung–Traub conjecture on the optimal order of a multipoint iteration without memory •
Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials • The
mean value problem: given a
complex polynomial f of
degree d \ge 2 and a complex number z, is there a
critical point c of f such that |f(z)-f(c)| \le |f'(z)||z-c|? • The
Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy •
Sendov's conjecture: if a complex polynomial with degree at least 2 has all roots in the closed
unit disk, then each root is within distance 1 from some
critical point. •
Vitushkin's conjecture on compact subsets of \mathbb{C} with
analytic capacity 0 • What is the exact value of
Landau's constants, including
Bloch's constant? • Regularity of solutions of
Euler equations • Convergence of Flint Hills series • Regularity of solutions of
Vlasov–Maxwell equations • Are there infinitely many
Lehmer pairs?
Combinatorics • The
1/3–2/3 conjecture – does every finite
partially ordered set that is not
totally ordered contain two elements
x and
y such that the probability that
x appears before
y in a random
linear extension is between 1/3 and 2/3? • The
Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition •
Problems in Latin squares – open questions concerning
Latin squares • The
lonely runner conjecture – if k runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance 1/k from each other runner) at some time? • The
sunflower conjecture – can the number of k size sets required for the existence of a sunflower of r sets be bounded by an exponential function in k for every fixed r>2? • Frankl's
union-closed sets conjecture – for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets • Give a combinatorial interpretation of the
Kronecker coefficients • The size m(n) of the smallest collection of n-uniform sets without
Property B for n \ge 5 • The values of the
Dedekind numbers M(n) for n \ge 10 • The values of the
Ramsey numbers, particularly R(5, 5) • The values of the
Van der Waerden numbers • Finding a function to model n-step
self-avoiding walks
Dynamical systems . It is not known whether the Mandelbrot set is
locally connected or not. •
Arnold–Givental conjecture and
Arnold conjecture – relating symplectic geometry to Morse theory. •
Berry–Tabor conjecture in
quantum chaos •
Banach's problem – is there an
ergodic system with simple Lebesgue spectrum? •
Birkhoff conjecture – if a
billiard table is strictly convex and integrable, is its boundary necessarily an ellipse? •
Collatz conjecture (also known as the 3n + 1 conjecture) •
Eden's conjecture that the
supremum of the local
Lyapunov dimensions on the global
attractor is achieved on a stationary point or an unstable periodic orbit embedded into the attractor. •
Fatou conjecture that a quadratic family of maps from the
complex plane to itself is hyperbolic for an open dense set of parameters. •
Furstenberg conjecture – is every invariant and
ergodic measure for the \times 2,\times 3 action on the circle either Lebesgue or atomic? •
Kaplan–Yorke conjecture on the dimension of an
attractor in terms of its
Lyapunov exponents •
Margulis conjecture – measure classification for diagonalizable actions in higher-rank groups. •
Hilbert–Arnold problem – is there a
uniform bound on
limit cycles in generic finite-parameter families of
vector fields on a sphere? •
MLC conjecture – is the Mandelbrot set locally connected? • Many problems concerning an
outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits. • Quantum unique ergodicity conjecture on the distribution of large-frequency
eigenfunctions of the
Laplacian on a
negatively-curved manifold •
Rokhlin's multiple mixing problem – are all
strongly mixing systems also strongly 3-mixing? •
Triangular billiards – does every triangle have a periodic billiards path? •
Weinstein conjecture – does a regular compact
contact type level set of a
Hamiltonian on a
symplectic manifold carry at least one periodic orbit of the Hamiltonian flow? • Does every positive integer generate a
juggler sequence terminating at 1? •
Lyapunov function: Lyapunov's second method for stability – For what classes of
ODEs, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion? • Is every
reversible cellular automaton in three or more dimensions locally reversible?
Games and puzzles Combinatorial games •
Sudoku: • How many puzzles have exactly one solution? •
Tic-tac-toe variants: • Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also
Hales–Jewett theorem and
nd game) •
Chess: • What is the outcome of a perfectly played game of chess? (See also
first-move advantage in chess) •
Go: • What is the perfect value of
Komi? •
Set: • What is the largest possible
cap set, as a function of n in the n-dimensional
affine space over the
three-element field? • Are the nim-sequences of all finite
octal games eventually periodic? • Is the nim-sequence of
Grundy's game eventually periodic?
Games with imperfect information •
Rendezvous problem Geometry Algebraic geometry •
Abundance conjecture: if the
canonical bundle of a
projective variety with
Kawamata log terminal singularities is
nef, then it is semiample. •
Bass conjecture on the
finite generation of certain
algebraic K-groups. •
Bass–Quillen conjecture relating
vector bundles over a
regular Noetherian ring and over the
polynomial ring A[t_{1}, \ldots, t_{n}]. •
Deligne conjecture: any one of numerous named for
Pierre Deligne. •
Deligne's conjecture on Hochschild cohomology about the
operadic structure on
Hochschild cochain complex. •
Dixmier conjecture: any
endomorphism of a
Weyl algebra is an
automorphism. •
Fröberg conjecture on the
Hilbert functions of a set of forms. •
Fujita conjecture regarding the line bundle K_{M} \otimes L^{\otimes m} constructed from a
positive holomorphic line bundle L on a
compact complex manifold M and the
canonical line bundle K_{M} of M •
General elephant problem: do
general elephants have at most
Du Val singularities? • Hartshorne's conjectures • In
spherical or
hyperbolic geometry, must polyhedra with the same volume and
Dehn invariant be
scissors-congruent? •
Jacobian conjecture: if a
polynomial mapping over a
characteristic-0 field has a constant nonzero
Jacobian determinant, then it has a
regular (i.e. with polynomial components) inverse function. •
Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between
Gromov–Witten theory and
Donaldson–Thomas theory •
Nagata's conjecture on curves, specifically the minimal degree required for a
plane algebraic curve to pass through a collection of very general points with prescribed
multiplicities. •
Nagata–Biran conjecture that if X is a smooth
algebraic surface and L is an
ample line bundle on X of degree d, then for sufficiently large r, the
Seshadri constant satisfies \varepsilon(p_1,\ldots,p_r;X,L) = d/\sqrt{r}. •
Nakai conjecture: if a
complex algebraic variety has a ring of
differential operators generated by its contained
derivations, then it must be
smooth. •
Parshin's conjecture: the higher
algebraic K-groups of any
smooth projective variety defined over a
finite field must vanish up to torsion. •
Section conjecture on splittings of
group homomorphisms from
fundamental groups of complete
smooth curves over finitely-generated
fields k to the
Galois group of k. •
Standard conjectures on algebraic cycles •
Tate conjecture on the connection between
algebraic cycles on
algebraic varieties and
Galois representations on
étale cohomology groups. •
Virasoro conjecture: a certain
generating function encoding the
Gromov–Witten invariants of a
smooth projective variety is fixed by an action of half of the
Virasoro algebra. • Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of
varieties at
singular points • Are infinite sequences of
flips possible in dimensions greater than 3? •
Resolution of singularities in characteristic p
Covering and packing •
Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a
bounded n-dimensional set. • The
covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be? • The
Erdős–Oler conjecture: when n is a
triangular number, packing n-1 circles in an equilateral triangle requires a triangle of the same size as packing n circles. • The
disk covering problem about finding the smallest
real number r(n) such that n
disks of radius r(n) can be arranged in such a way as to cover the
unit disk. • The
kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24 •
Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets •
Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions. •
Square packing in a square: what is the asymptotic growth rate of wasted space? •
Ulam's packing conjecture about the identity of the worst-packing convex solid • The
Tammes problem for numbers of nodes greater than 14 (except 24).
Differential geometry • The
spherical Bernstein's problem, a generalization of
Bernstein's problem •
Carathéodory conjecture: any convex, closed, and twice-differentiable surface in three-dimensional
Euclidean space admits at least two
umbilical points. •
Cartan–Hadamard conjecture: can the classical
isoperimetric inequality for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as
Cartan–Hadamard manifolds? •
Chern's conjecture (affine geometry) that the
Euler characteristic of a
compact affine manifold vanishes. •
Chern's conjecture for hypersurfaces in spheres, a number of closely related conjectures. • Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed. • 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 •
Osserman conjecture: that every
Osserman manifold is either
flat or locally
isometric to a rank-one
symmetric space •
Yau's conjecture on the first eigenvalue that the first
eigenvalue for the
Laplace–Beltrami operator on an embedded
minimal hypersurface of S^{n+1} is n.
Discrete geometry is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a
regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24. • The
big-line-big-clique conjecture on the existence of either many collinear points or many mutually visible points in large planar point sets • The
Dirac–Motzkin conjecture on the minimum number of ordinary lines for
n points in the Euclidean plane • The
Hadwiger conjecture on covering
n-dimensional convex bodies with at most 2
n smaller copies • Solving the
happy ending problem for arbitrary n • Improving lower and upper bounds for the
Heilbronn triangle problem. •
Kalai's 3d conjecture on the least possible number of faces of
centrally symmetric polytopes. • The
Kobon triangle problem on triangles in line arrangements • The
Kusner conjecture: at most 2d points can be equidistant in L^1 spaces • The
McMullen problem on projectively transforming sets of points into
convex position •
Opaque forest problem on finding
opaque sets for various planar shapes •
Orchard-planting problem on the maximum number of 3-point lines attainable by a configuration of
n points in the plane •
How many unit distances can be determined by a set of points in the Euclidean plane? • Finding matching upper and lower bounds for
k-sets and halving lines • For each arrangement of points in which the
rectilinear crossing number is minimized, is the number of halving lines maximized? •
Tripod packing: how many tripods can have their apexes packed into a given cube?
Euclidean geometry • The
Atiyah conjecture on configurations on the invertibility of a certain n-by-n matrix depending on n points in \mathbb{R}^{3} •
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? •
Connelly’s blooming conjecture: Does every net of a convex polyhedron have a
blooming? • 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? •
Ehrhart's volume conjecture: a convex body K in n dimensions containing a single lattice point in its interior as its
center of mass cannot have volume greater than (n+1)^{n}/n! •
Falconer's conjecture: sets of Hausdorff dimension greater than d/2 in \mathbb{R}^d must have a distance set of nonzero
Lebesgue measure • The values of the
Hermite constants for dimensions other than 1–8 and 24 • What is the lowest number of faces possible for a
holyhedron? •
Inscribed square problem, also known as
Toeplitz' conjecture and the square peg problem – does every
Jordan curve have an inscribed square? • The
Kakeya conjecture – do n-dimensional sets that contain a unit line segment in every direction necessarily have
Hausdorff dimension and
Minkowski dimension equal to n? • 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 •
Mahler's conjecture on the product of the volumes of a
centrally symmetric convex body and its
polar. • In
Meissner body): :* Are the two Meissner tetrahedra the minimum-volume three-dimensional shapes of constant width? •
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? • In
parallelohedron: • Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combinatorial structure as a parallelohedron? • Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a
Voronoi diagram? •
Ropelength problems: • Is there a general expression for the minimum ropelength of an arbitrary closed knot? • What constant 1.1 governs the lower bound of a closed knot K's minimum ropelength L(K) \geq a\operatorname{Cr}(K)^{3/4}? • Is the upper bound of a closed knot's minimum ropelength linear to its crossing number? • Is there a general expression for how much the ends of a long rope of radius 1 get closer when a tight open knot is tied into it? • Does every convex polyhedron have
Rupert's property? •
Shephard's problem (a.k.a. Dürer's conjecture) – does every
convex polyhedron have a
net, or simple edge-unfolding? • Is there a non-convex polyhedron without self-intersections with
more than seven faces, all of which share an edge with each other? • The
Thomson problem – what is the minimum energy configuration of n mutually-repelling particles on a unit sphere? • Convex
uniform 5-polytopes – find and classify the complete set of these shapes
Non-Euclidean geometry •
Hilbert's third problem for non-Euclidean geometries: in spherical or hyperbolic geometry, must polyhedra with the same volume and
Dehn invariant be scissors-congruent?
Graph theory Algebraic graph theory •
Babai's problem: which groups are Babai invariant groups? •
Brouwer's conjecture on upper bounds for sums of
eigenvalues of
Laplacians of graphs in terms of their number of edges
Games on graphs • Does there exist a graph G such that the
dominating number \gamma(G) equals the
eternal dominating number \gamma∞(G) of G and \gamma(G) is less than the
clique covering number of G? •
Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs • Meyniel's conjecture that
cop number is O(\sqrt n) • Suppose Alice has a winning strategy for the
vertex coloring game on a graph G with k colors. Does she have one for k+1 colors?
Graph coloring and labeling • The
1-factorization conjecture that if n is odd or even and k \geq n, n - 1, respectively, then a k-
regular graph with 2n vertices is
1-factorable. • The
perfect 1-factorization conjecture that every
complete graph on an even number of vertices admits a
perfect 1-factorization. •
Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs • The
Earth–Moon problem: what is the maximum chromatic number of biplanar graphs? • The
Erdős–Faber–Lovász conjecture on coloring unions of cliques • The
graceful tree conjecture that every tree admits a graceful labeling •
Rosa's conjecture that all
triangular cacti are graceful or nearly-graceful • The
Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree • 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: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph • The
list coloring conjecture: for every graph, the list chromatic index equals the chromatic index • The
overfull conjecture that a graph with maximum degree \Delta(G) \geq n/3 is
class 2 if and only if it has an
overfull subgraph S satisfying \Delta(S) = \Delta(G). • The
total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree
Graph drawing and embedding • The
Albertson conjecture: the crossing number can be lower-bounded by the crossing number of a
complete graph with the same
chromatic number •
Conway's thrackle conjecture that
thrackles cannot have more edges than vertices • The
GNRS conjecture on whether minor-closed graph families have \ell_1 embeddings with bounded distortion •
Harborth's conjecture: 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: 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? • Guy's conjecture on the
crossing number for
complete graphs – Is there a drawing of any complete graph with fewer crossings than the number given by his upper bound? •
Universal point sets of subquadratic size for planar graphs
Restriction of graph parameters • Does there exist a
conference graph for every number of vertices v>1 where v \equiv 1 \bmod 4 and v is an odd sum of two squares? •
Conway's 99-graph problem: does there exist a
strongly regular graph with parameters (99,14,1,2)? •
Degree diameter problem: given two positive integers d, k, what is the largest graph of diameter k such that all vertices have degrees at most d? • Jørgensen's conjecture that every 6-vertex-connected K_6-minor-free graph is an
apex graph • Does a
Moore graph with girth 5 and degree 57 exist? • Do there exist infinitely many
strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?
Subgraphs •
Barnette's conjecture: every cubic bipartite three-connected planar graph has a Hamiltonian cycle •
Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane that the Steiner ratio is \sqrt{3}/2 •
Chvátal's toughness conjecture, that there is a number such that every -tough graph is Hamiltonian • The
cycle double cover conjecture: 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
Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph • The
Grünbaum–Nash-Williams conjecture on whether every
4-vertex-connected toroidal graph has a
Hamiltonian cycle. • 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. • What is the largest possible
pathwidth of an -vertex
cubic graph? • The
reconstruction conjecture and
new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs. • The
snake-in-the-box problem: what is the longest possible
induced path in an n-dimensional
hypercube graph? •
Sumner's conjecture: does every (2n-2)-vertex tournament contain as a subgraph every n-vertex oriented tree? •
Szymanski's conjecture: every
permutation on the n-dimensional doubly-
directed hypercube graph can be routed with edge-disjoint
paths. •
Tuza's conjecture: if the maximum number of disjoint triangles is \nu, can all triangles be hit by a set of at most 2\nu edges? • The
unfriendly partition conjecture: Does every countable graph have an unfriendly partition into two parts? •
Vizing's conjecture on the
domination number of
cartesian products of graphs •
Walescki's theorem for hypergraphs: Do complete k-uniform hypergraphs admit
Hamiltonian decompositions into tight cycles? •
Zarankiewicz problem: how many edges can there be in a
bipartite graph on a given number of vertices with no
complete bipartite subgraphs of a given size?
Word-representation of graphs • Are there any graphs on
n vertices whose
representation requires more than floor(
n/2) copies of each letter? • Characterise (non-)
word-representable 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 it true that out of all
bipartite graphs,
crown graphs require longest word-representants? • Is the
line graph of a non-
word-representable graph always non-
word-representable? • The
imbalance conjecture: If the imbalance for each edge of a graph is at least 1, is the multiset of all edge imbalances always
graphic? • The
implicit graph conjecture on the existence of implicit representations for slowly-growing
hereditary families of graphs •
Ryser's conjecture relating the maximum
matching size and minimum
transversal size in
hypergraphs • The
second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one? •
Sidorenko's conjecture on
homomorphism densities of graphs in
graphons • Teschner's
bondage number conjecture: is the bondage number of a graph always less than or equal to 3/2 times its maximum
degree? • Tutte's conjectures: • every bridgeless graph has a
nowhere-zero 5-flow • every
Petersen-
minor-free bridgeless graph has a nowhere-zero 4-flow •
Woodall's conjecture that the minimum number of edges in a
dicut of a
directed graph is equal to the maximum number of disjoint
dijoins.
Model theory and formal languages • The
Cherlin–Zilber conjecture: A simple group whose first-order theory is
stable in \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 • The main gap conjecture, e.g. for uncountable
first order theories, for
AECs, and for \aleph_1-saturated models of a countable theory. • Shelah's categoricity conjecture for L_{\omega_1,\omega}: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number. • The stable field conjecture: every infinite field with a
stable first-order theory is separably closed. • The stable forking conjecture for simple theories •
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? • The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum? •
Vaught conjecture: the number of
countable models of a
first-order complete theory in a countable
language is either finite, \aleph_0, or 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 \aleph_{\omega_1} does it have a model of cardinality continuum? • 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 \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.) • Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable? • Is the theory of the field of Laurent series over \mathbb{Z}_p
decidable? of the field of polynomials over \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? • Determine the structure of Keisler's order. • 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?
Probability theory •
Ibragimov–Iosifescu conjecture for φ-mixing sequences Number theory General because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them is odd. •
Büchi's problem on sufficiently large sequences of square numbers with constant second difference. •
Carmichael's totient function conjecture: do all values of
Euler's totient function have
multiplicity greater than 1? •
Catalan–Dickson conjecture on aliquot sequences: no
aliquot sequences are infinite but non-repeating. •
Exponent pair conjecture: for all \varepsilon > 0, is the pair (\varepsilon, 1/2 + \varepsilon) an
exponent pair? • The
Gauss circle problem: how far can the number of integer points in a circle centered at the origin be from the area of the circle? •
Grimm's conjecture: each element of a set of consecutive
composite numbers can be assigned a distinct
prime number that divides it. •
Hall's conjecture: for any \varepsilon > 0, there is some constant c(\varepsilon) such that either y^2 = x^3 or |y^2 - x^3| > c(\varepsilon)x^{1/2 - \varepsilon}. •
Lehmer's totient problem: if \phi(n) divides n - 1, must n be prime? •
Magic square of squares: is there a 3x3 magic square composed of distinct perfect squares? •
Mahler's 3/2 problem that no real number x has the property that the fractional parts of x(3/2)^n are less than 1/2 for all positive integers n. •
Newman's conjecture: the
partition function satisfies any arbitrary congruence infinitely often. •
Scholz conjecture: the length of the shortest
addition chain producing 2^n - 1 is at most n - 1 plus the length of the shortest addition chain producing n. •
Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in
Pascal's triangle? • Are there infinitely many
perfect numbers? • Do any
odd perfect numbers exist? • Do
quasiperfect numbers exist? • Do any non-power of 2
almost perfect numbers exist? • Are there 65, 66, or 67
idoneal numbers? • Are there any pairs of
amicable numbers which have opposite parity? • Are there any pairs of
betrothed numbers which have same parity? • Are there any pairs of
relatively prime amicable numbers? • Are there infinitely many pairs of
amicable numbers? • Are there infinitely many
betrothed numbers? • Are there infinitely many
Giuga numbers? • Do any
Lychrel numbers exist in base 10? • Do any odd
noncototients exist? • Do any odd
weird numbers exist? • Do any
(2, 5)-perfect numbers exist? • Do any
Taxicab(5, 2, n) exist for
n > 1? • Is there a
covering system with odd distinct moduli? • Is \pi a
normal number (i.e., is each digit 0–9 equally frequent)? • Are all
irrational algebraic numbers normal? • Is 10 a
solitary number?
Additive number theory •
Erdős conjecture on arithmetic progressions that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long
arithmetic progressions. •
Erdős–Turán conjecture on additive bases: if B is an
additive basis of order 2, then the number of ways that positive integers n can be expressed as the sum of two numbers in B must tend to infinity as n tends to infinity. •
Gilbreath's conjecture on consecutive applications of the unsigned
forward difference operator to the sequence of
prime numbers. •
Goldbach's conjecture: every even natural number greater than 2 is the sum of two
prime numbers. •
Lander, Parkin, and Selfridge conjecture: if the sum of m k-th powers of positive integers is equal to a different sum of n k-th powers of positive integers, then m + n \geq k. •
Lemoine's conjecture: all odd integers greater than 5 can be represented as the sum of an odd
prime number and an even
semiprime. •
Minimum overlap problem of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set \{1, \ldots, 2n\} •
Pollock's conjectures • Does every nonnegative integer appear in
Recamán's sequence? •
Skolem problem: can an algorithm determine if a
constant-recursive sequence contains a zero? • The values of
g(
k) and
G(
k) in
Waring's problem • Do the
Ulam numbers have a positive density? • Determine growth rate of
rk(
N) (see
Szemerédi's theorem)
Algebraic number theory •
Class number problem: are there infinitely many
real quadratic number fields with
unique factorization? •
Fontaine–Mazur conjecture: actually numerous conjectures, all proposed by
Jean-Marc Fontaine and
Barry Mazur. •
Gan–Gross–Prasad conjecture: a
restriction problem in
representation theory of real or p-adic Lie groups. •
Greenberg's conjectures •
Hermite's problem: is it possible, for any natural number n, to assign a sequence of
natural numbers to each
real number such that the sequence for x is eventually
periodic if and only if x is
algebraic of degree n? •
Hilbert's 9th problem: find the most general
reciprocity law for the
norm residues of k-th order in a general
algebraic number field, where k is a power of a prime. •
Hilbert's 11th problem: classify
quadratic forms over
algebraic number fields. •
Hilbert's 12th problem: extend the
Kronecker–Weber theorem on
Abelian extensions of \mathbb{Q} to any base number field. •
Kummer–Vandiver conjecture: primes p do not divide the
class number of the maximal real
subfield of the p-th
cyclotomic field. • Lang and Trotter's conjecture on
supersingular primes that the number of
supersingular primes less than a constant X is within a constant multiple of \sqrt{X}/\ln{X} •
Leopoldt's conjecture: a
p-adic analogue of the
regulator of an
algebraic number field does not vanish. •
Stark conjectures (including
Brumer–Stark conjecture) • Characterize all algebraic number fields that have some
power basis.
Analytic number theory • Beilinson's conjectures about
special values of L-functions. • Do
Siegel zeros exist? • Find the value of the
De Bruijn–Newman constant. • Is
Selberg class of Dirichlet series equal to class of
automorphic L-functions? •
Hardy–Littlewood zeta function conjectures • Keating–Snaith conjecture concerning the asymptotics of an integral involving the
Riemann zeta function •
Hilbert–Pólya conjecture: the nontrivial zeros of the
Riemann zeta function correspond to
eigenvalues of a
self-adjoint operator. •
Lindelöf hypothesis that for all \varepsilon > 0, \zeta(1/2 + it) = o(t^\varepsilon) • The
density hypothesis for zeroes of the Riemann zeta function. • Location of nontrivial zeros of L-functions, especially conjectures concerning their critical lines: •
Grand Riemann hypothesis: do the nontrivial zeros of all automorphic L-functions lie on the critical line 1/2 + it with real t? •
Generalized Riemann hypothesis for Selberg class: do the nontrivial zeros of all functions in Selberg class lie on the critical line 1/2 + it with real t? •
Extended Riemann Hypothesis: do the nontrivial zeros of all
Dedekind zeta functions lie on the critical line 1/2 + it with real t? •
Generalized Riemann hypothesis: do the nontrivial zeros of all
Dirichlet L-functions lie on the critical line 1/2 + it with real t? •
Riemann hypothesis: do the nontrivial zeros of the
Riemann zeta function lie on the critical line 1/2 + it with real t? •
Montgomery's pair correlation conjecture: the normalized pair
correlation function between pairs of zeros of the Riemann zeta function is the same as the pair correlation function of
random Hermitian matrices. •
Piltz divisor problem on bounding \Delta_k(x) = D_k(x) - xP_k(\log(x)) •
Dirichlet's divisor problem: the specific case of the Piltz divisor problem for: k = 1 •
Ramanujan–Petersson conjecture: a number of generalizations of the original conjecture, concerning growth rate of coefficients of automorphic forms. •
Selberg's 1/4 conjecture: the
eigenvalues of the
Laplace operator on
Maass wave forms of
congruence subgroups are at least 1/4. •
Selberg's orthogonality conjecture: generalization of
Mertens' theorem for functions in Selberg class.
Arithmetic geometry •
Bombieri–Lang conjecture: K -rational points on a
variety of general type over a number field K are not a dense set in Zariski topology. •
Erdős–Ulam problem: Is there a
dense set of points in the plane all at rational distances from one another? •
Manin conjecture: if K-rational points on
Fano variety are Zariski-dense subset, then the distribution of points of
height: H(x)\leq B in any Zariski-open subset U is proportional to B \log (B)^{r-1}, where r is rank of
Picard group of that variety. •
Sato–Tate conjecture: also a number of related conjectures that are generalizations of the original conjecture. • In
Unit square: •
Rational dense problem: Is there a point in the plane at a rational distance from all four corners of a unit square? In particular: Are \pi and e
algebraically independent? Which nontrivial combinations of
transcendental numbers (such as e + \pi, e\pi, \pi^e, \pi^{\pi}, e^e) are themselves transcendental? • The
four exponentials conjecture: the transcendence of at least one of four exponentials of combinations of irrationals • Which transcendental numbers are
(exponential) periods? • How well can
non-quadratic irrational numbers be approximated? What is the
irrationality measure of specific (suspected) transcendental numbers such as \pi and \gamma? •
Hartmanis–Stearns conjecture Diophantine equations •
Beal's conjecture: for all integral solutions to A^x + B^y = C^z where x, y, z > 2, all three numbers A, B, C must share some prime factor. •
Brocard's problem: are there any integer solutions to n! + 1 = m^{2} other than n = 4, 5, 7? •
Congruent number problem (a corollary to
Birch and Swinnerton-Dyer conjecture, per
Tunnell's theorem): determine precisely what rational numbers are
congruent numbers. • Erdős–Moser problem: is 1^1 + 2^1 = 3^1 the only solution to the
Erdős–Moser equation? •
Erdős–Straus conjecture: for every n \geq 2, there are positive integers x, y, z such that 4/n = 1/x + 1/y + 1/z. •
Fermat–Catalan conjecture: there are finitely many distinct solutions (a^m, b^n, c^k) to the equation a^m + b^n = c^k with a, b, c being positive
coprime integers and m, n, k being positive integers satisfying 1/m + 1/n + 1/k . •
Goormaghtigh conjecture on solutions to (x^m - 1)/(x - 1) = (y^n - 1)/(y - 1) where x > y > 1 and m, n > 2. • The
uniqueness conjecture for Markov numbers that every
Markov number is the largest number in exactly one normalized solution to the Markov
Diophantine equation. •
Pillai's conjecture: for any A, B, C, the equation Ax^m - By^n = C has finitely many solutions when m, n are not both 2. • Which integers can be written as the
sum of three perfect cubes? •
Can every integer be written as a sum of four perfect cubes? Prime numbers states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28. •
Agoh–Giuga conjecture on the
Bernoulli numbers that p is prime if and only if pB_{p-1} \equiv -1 \pmod p •
Agrawal's conjecture that given
coprime positive integers n and r, if (X - 1)^n \equiv X^n - 1 \pmod{n, X^r - 1}, then either n is prime or n^{2} \equiv 1 \pmod{r} •
Artin's conjecture on primitive roots that if an integer is neither a perfect square nor -1, then it is a
primitive root modulo infinitely many
prime numbers p •
Brocard's conjecture: there are always at least 4
prime numbers between consecutive squares of prime numbers, aside from 2^{2} and 3^{2}. •
Bunyakovsky conjecture: if an integer-coefficient polynomial f has a positive leading coefficient, is irreducible over the integers, and has no common factors over all f(x) where x is a positive integer, then f(x) is prime infinitely often. •
Catalan's Mersenne conjecture: some
Catalan–Mersenne number is composite and thus all Catalan–Mersenne numbers are composite after some point. •
Dickson's conjecture: for a finite set of linear forms a_1 + b_1 n, \ldots, a_k + b_k n with each b_i \geq 1, there are infinitely many n for which all forms are
prime, unless there is some
congruence condition preventing it. • Dubner's conjecture: every even number greater than 4208 is the sum of two
primes which both have a
twin. •
Elliott–Halberstam conjecture on the distribution of
prime numbers in
arithmetic progressions. •
Erdős–Mollin–Walsh conjecture: no three consecutive numbers are all
powerful. •
Feit–Thompson conjecture: for all distinct
prime numbers p and q, (p^q - 1)/(p - 1) does not divide (q^p - 1)/(q - 1) • Fortune's conjecture that no
Fortunate number is composite. • The
Gaussian moat problem: is it possible to find an infinite sequence of distinct
Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded? •
Gillies' conjecture on the distribution of
prime divisors of
Mersenne numbers. •
Landau's problems •
Goldbach conjecture: all even
natural numbers greater than 2 are the sum of two
prime numbers. •
Legendre's conjecture: for every positive integer n, there is a prime between n^{2} and (n+1)^{2}. •
Twin prime conjecture: there are infinitely many
twin primes. • Are there infinitely many primes of the form n^{2} + 1? • Problems associated to
Linnik's theorem •
New Mersenne conjecture: for any odd
natural number p, if any two of the three conditions p = 2^k \pm 1 or p = 4^k \pm 3, 2^p - 1 is prime, and (2^{p} + 1)/3 is prime are true, then the third condition is also true. •
Polignac's conjecture: for all positive even numbers n, there are infinitely many
prime gaps of size n. •
Schinzel's hypothesis H that for every finite collection \{f_1, \ldots, f_k\} of nonconstant
irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers n for which f_1(n), \ldots, f_k(n) are all
primes, or there is some fixed divisor m > 1 which, for all n, divides some f_i(n). •
Selfridge's conjecture: is 78,557 the lowest
Sierpiński number? • Does the
converse of Wolstenholme's theorem hold for all natural numbers? • Are all
Euclid numbers
square-free? • Are all
Fermat numbers
square-free? • Are all
Mersenne numbers of prime index
square-free? • Are there any composite
c satisfying 2
c − 1 ≡ 1 (mod
c2)? • Are there any
Wall–Sun–Sun primes? • Are there any
Wieferich primes in base 47? • Are there infinitely many
balanced primes? • Are there infinitely many
cluster primes? • Are there infinitely many
cousin primes? • Are there infinitely many
Cullen primes? • Are there infinitely many
Euclid primes? • Are there infinitely many
Fibonacci primes? • Are there infinitely many
Kummer primes? • Are there infinitely many Kynea primes? • Are there infinitely many
Lucas primes? • Are there infinitely many
Mersenne primes (
Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even
perfect numbers? • Are there infinitely many
Newman–Shanks–Williams primes? • Are there infinitely many
palindromic primes to every base? • Are there infinitely many
Pell primes? • Are there infinitely many
Pierpont primes? • Are there infinitely many
prime quadruplets? • Are there infinitely many
prime triplets? •
Siegel's conjecture: are there infinitely many regular primes, and if so is their
natural density as a subset of all primes e^{-1/2}? • Are there infinitely many
sexy primes? • Are there infinitely many
safe and Sophie Germain primes? • Are there infinitely many
Wagstaff primes? • Are there infinitely many
Wieferich primes? • Are there infinitely many
Wilson primes? • Are there infinitely many
Wolstenholme primes? • Are there infinitely many
Woodall primes? • Can a prime
p satisfy 2^{p-1}\equiv 1\pmod{p^2} and 3^{p-1}\equiv 1\pmod{p^2} simultaneously? • Does every prime number appear in the
Euclid–Mullin sequence? • What is the smallest
Skewes's number? • For any given integer
a > 0, are there infinitely many
Lucas–Wieferich primes associated with the pair (
a, −1)? (Specially, when
a = 1, this is the Fibonacci-Wieferich primes, and when
a = 2, this is the Pell-Wieferich primes) • For any given integer
a > 0, are there infinitely many primes
p such that
ap − 1 ≡ 1 (mod
p2)? • For any given integer
b which is not a perfect power and not of the form −4
k4 for integer
k, are there infinitely many
repunit primes to base
b? • For any given integers k\geq 1, b\geq 2, c\neq 0, with and are there infinitely many primes of the form (k\times b^n+c)/\gcd(k+c,b-1) with integer
n ≥ 1? • Is every
Fermat number 2^{2^n} + 1 composite for n > 4? • Is 509,203 the lowest
Riesel number?
Set theory Note: 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. • Does the
partition principle (PP) imply the
axiom of choice (AC)? • (
Woodin) Does the
generalized continuum hypothesis below a
strongly compact cardinal imply the
generalized continuum hypothesis everywhere? • Does the
generalized continuum hypothesis entail
{\diamondsuit(E^{\lambda^+}_{\operatorname{cf}(\lambda)}}) for every
singular cardinal \lambda? • Does the
generalized continuum hypothesis imply the existence of an
ℵ2-Suslin tree? • If ℵω is a strong limit cardinal, is 2^{\aleph_\omega} (see
Singular cardinals hypothesis)? The best bound, ℵω4, was obtained by
Shelah using his
PCF theory. • The problem of finding the ultimate
core model, one that contains all
large cardinals. •
Woodin's Ω-conjecture: if there is a
proper class of
Woodin cardinals, then
Ω-logic satisfies an analogue of
Gödel's completeness theorem. • Does the
consistency of the existence of a
strongly compact cardinal imply the consistent existence of a
supercompact cardinal? • Does there exist a
Jónsson algebra on ℵω? • Is OCA (the
open coloring axiom) consistent with 2^{\aleph_{0}}>\aleph_{2}? •
Reinhardt cardinals: Without assuming the
axiom of choice, can a
nontrivial elementary embedding V→
V exist?
Topology asks whether there is an efficient algorithm to identify when the shape presented in a
knot diagram is actually the
unknot. •
Baum–Connes conjecture: the
assembly map is an
isomorphism. •
Berge conjecture that the only
knots in the
3-sphere which admit
lens space surgeries are
Berge knots. •
Bing–Borsuk conjecture: every n-dimensional
homogeneous absolute neighborhood retract is a
topological manifold. •
Borel conjecture:
aspherical closed manifolds are determined up to
homeomorphism by their
fundamental groups. •
Halperin conjecture on rational
Serre spectral sequences of certain
fibrations. •
Hilbert–Smith conjecture: if a
locally compact topological group has a
continuous,
faithful group action on a
topological manifold, then the group must be a
Lie group. • Mazur's conjectures •
Novikov conjecture on the
homotopy invariance of certain
polynomials in the
Pontryagin classes of a
manifold, arising from the
fundamental group. •
Quadrisecants of
wild knots: it has been conjectured that wild knots always have infinitely many quadrisecants. •
Telescope conjecture: the last of
Ravenel's conjectures in
stable homotopy theory to be resolved. •
Unknotting problem: can
unknots be recognized in
polynomial time? •
Volume conjecture relating
quantum invariants of
knots to the
hyperbolic geometry of their
knot complements. •
Whitehead conjecture: every
connected subcomplex of a two-dimensional
aspherical CW complex is aspherical. •
Zeeman conjecture: given a finite
contractible two-dimensional
CW complex K, is the space K \times [0, 1]
collapsible? •
Nearby Lagrangian conjecture: prove or find a counter-example to the statement: Any closed exact
Lagrangian submanifold of the
cotangent bundle of a
closed manifold is
Hamiltonian isotopic to the
zero section. == Problems solved since 1995 ==