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 ${\displaystyle G}$, the number of irreducible complex characters of degree not divisible by a prime number ${\displaystyle p}$ is equal to the number of irreducible complex characters of the normalizer of any Sylow ${\displaystyle p}$-subgroup within ${\displaystyle G}$.

Combinatorial games

• Sudoku:
  • How many puzzles have exactly one solution?^[41]
  • How many puzzles with exactly one solution are minimal?^[41]
  • What is the maximum number of givens for a minimal puzzle?^[41]
• 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)^[42]
• 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?
• What is the Turing completeness status of all unique elementary cellular automata?
• 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

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 ${\displaystyle A[t_{1},\ldots ,t_{n}]}$.
• Deligne conjecture: any one of numerous named for Pierre Deligne.
• 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 ${\displaystyle K_{M}\otimes L^{\otimes m}}$ constructed from a positive holomorphic line bundle ${\displaystyle L}$ on a compact complex manifold ${\displaystyle M}$ and the canonical line bundle ${\displaystyle K_{M}}$ of ${\displaystyle M}$
• General elephant problem: do general elephants have at most Du Val singularities?
• Hartshorne's conjectures^[43]
• 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.
• Manin conjecture on the distribution of rational points of bounded height in certain subsets of Fano varieties
• Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory^[44]
• 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 ${\displaystyle X}$ is a smooth algebraic surface and ${\displaystyle L}$ is an ample line bundle on ${\displaystyle X}$ of degree ${\displaystyle d}$, then for sufficiently large ${\displaystyle r}$, the Seshadri constant satisfies ${\displaystyle \varepsilon (p_{1},\ldots ,p_{r};X,L)=d/{\sqrt {r}}}$.
• 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 ${\displaystyle k}$ to the Galois group of ${\displaystyle k}$.
• 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^[45]

• Are infinite sequences of flips possible in dimensions greater than 3?
• Resolution of singularities in characteristic ${\displaystyle p}$

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?^[46]
• The Erdős–Oler conjecture: 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^[47]
• The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24^[48]
• Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets^[49]
• 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?^[50]
• Ulam's packing conjecture about the identity of the worst-packing convex solid^[51]

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.^[52]
• 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^[53]
• The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds^[54]
• Yau's conjecture: a closed Riemannian 3-manifold has an infinite number of smooth closed immersed minimal surfaces.
• Yau's conjecture on the first eigenvalue that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of ${\displaystyle S^{n+1}}$ is ${\displaystyle n}$.

Discrete geometry

In three dimensions, the kissing number 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^[55]
• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2ⁿ smaller copies^[56]
• Solving the happy ending problem for arbitrary ${\displaystyle n}$^[57]
• 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.^[58]
• The Kobon triangle problem on triangles in line arrangements^[59]
• The Kusner conjecture: at most ${\displaystyle 2d}$ points can be equidistant in ${\displaystyle L^{1}}$ spaces^[60]
• The McMullen problem on projectively transforming sets of points into convex position^[61]
• Opaque forest problem on finding opaque sets for various planar shapes

• How many unit distances can be determined by a set of n points in the Euclidean plane?^[62]

• Finding matching upper and lower bounds for k-sets and halving lines^[63]
• Tripod packing:^[64] 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 ${\displaystyle n}$-by-${\displaystyle n}$ matrix depending on ${\displaystyle n}$ points in ${\displaystyle \mathbb {R} ^{3}}$^[65]
• 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^[66]
• Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?^[67]
• Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?^[68]
• Dissection into orthoschemes – is it possible for simplices of every dimension?^[69]
• Ehrhart's volume conjecture: a convex body ${\displaystyle K}$ in ${\displaystyle n}$ dimensions containing a single lattice point in its interior as its center of mass cannot have volume greater than ${\displaystyle (n+1)^{n}/n!}$
• The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?^{[70][lower-alpha 1]}
• Falconer's conjecture: sets of Hausdorff dimension greater than ${\displaystyle d/2}$ in ${\displaystyle \mathbb {R} ^{d}}$ must have a distance set of nonzero Lebesgue measure^[72]
• The values of the Hermite constants for dimensions other than 1–8 and 24
• Inscribed square problem, also known as Toeplitz' conjecture and the square peg problem – does every Jordan curve have an inscribed square?^[73]
• 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 ${\displaystyle n}$?^[74]
• 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^[75]
• Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one^[76]
• Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar.^[77]
• Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?^[78]
• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?^[79]
• Does every convex polyhedron have Rupert's property?^[80][81]
• Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net, or simple edge-unfolding?^[82][83]
• 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 ${\displaystyle n}$ mutually-repelling particles on a unit sphere?^[84]
• Convex uniform 5-polytopes – find and classify the complete set of these shapes^[85]

Graph coloring and labeling

An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.

• The 1-factorization conjecture that if ${\displaystyle n}$ is odd or even and ${\displaystyle k\geq n,n-1}$ respectively, then a ${\displaystyle k}$-regular graph with ${\displaystyle 2n}$ 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^[86]
• The Earth–Moon problem: what is the maximum chromatic number of biplanar graphs?^[87]
• The Erdős–Faber–Lovász conjecture on coloring unions of cliques^[88]
• 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^[89]
• The Hadwiger conjecture relating coloring to clique minors^[90]
• The Hadwiger–Nelson problem on the chromatic number of unit distance graphs^[91]
• Jaeger's Petersen-coloring conjecture: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph^[92]
• The list coloring conjecture: for every graph, the list chromatic index equals the chromatic index^[93]
• The overfull conjecture that a graph with maximum degree ${\displaystyle \Delta (G)\geq n/3}$ is class 2 if and only if it has an overfull subgraph ${\displaystyle S}$ satisfying ${\displaystyle \Delta (S)=\Delta (G)}$.
• The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree^[94]

Graph drawing

• The Albertson conjecture: the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number^[95]
• Conway's thrackle conjecture^[96] that thrackles cannot have more edges than vertices
• Harborth's conjecture: every planar graph can be drawn with integer edge lengths^[97]
• Negami's conjecture on projective-plane embeddings of graphs with planar covers^[98]
• The strong Papadimitriou–Ratajczak conjecture: every polyhedral graph has a convex greedy embedding^[99]
• Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?^[100]

• Universal point sets of subquadratic size for planar graphs^[101]

Paths and cycles in graphs

• Barnette's conjecture: every cubic bipartite three-connected planar graph has a Hamiltonian cycle^[102]
• Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane that the Steiner ratio is ${\displaystyle {\sqrt {3}}/2}$
• Chvátal's toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian^[103]
• The cycle double cover conjecture: every bridgeless graph has a family of cycles that includes each edge twice^[104]
• The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs^[105]
• The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree^[106]
• The Lovász conjecture on Hamiltonian paths in symmetric graphs^[107]
• 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.^[108]
• Szymanski's conjecture: every permutation on the ${\displaystyle n}$-dimensional doubly-directed hypercube graph can be routed with edge-disjoint paths.

Word-representation of graphs

• Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter?^{[109][110][111][112]}
• Characterise (non-)word-representable planar graphs^{[109][110][111][112]}
• Characterise word-representable graphs in terms of (induced) forbidden subgraphs.^{[109][110][111][112]}
• Characterise word-representable near-triangulations containing the complete graph K₄ (such a characterisation is known for K₄-free planar graphs^[113])
• 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^[114]
• Is it true that out of all bipartite graphs, crown graphs require longest word-representants?^[115]
• Is the line graph of a non-word-representable graph always non-word-representable?^{[109][110][111][112]}
• Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?^{[109][110][111][112]}

Miscellaneous 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
• Conway's 99-graph problem: does there exist a strongly regular graph with parameters (99,14,1,2)?^[116]
• Degree diameter problem: given two positive integers ${\displaystyle d,k}$, what is the largest graph of diameter ${\displaystyle k}$ such that all vertices have degrees at most ${\displaystyle d}$?
• The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph^[117]
• The GNRS conjecture on whether minor-closed graph families have ${\displaystyle \ell _{1}}$ embeddings with bounded distortion^[118]
• Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs^[119]
• The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs^[120]
• Jørgensen's conjecture that every 6-vertex-connected K₆-minor-free graph is an apex graph^[121]
• Meyniel's conjecture that cop number is ${\displaystyle O({\sqrt {n}})}$^[122]
• Does a Moore graph with girth 5 and degree 57 exist?^[123]
• What is the largest possible pathwidth of an n-vertex cubic graph?^[124]
• The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.^[125][126]
• 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?^[127]
• Sidorenko's conjecture on homomorphism densities of graphs in graphons
• Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?^[128]
• Sumner's conjecture: does every ${\displaystyle (2n-2)}$-vertex tournament contain as a subgraph every ${\displaystyle n}$-vertex oriented tree?^[129]
• Tutte's conjectures:
  • every bridgeless graph has a nowhere-zero 5-flow^[130]
  • every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow^[131]
• Tuza's conjecture: if the maximum number of disjoint triangles is ${\displaystyle \nu }$, can all triangles be hit by a set of at most ${\displaystyle 2\nu }$ edges?^[132]
• Vizing's conjecture on the domination number of cartesian products of graphs^[133]
• 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
• 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?

General

6 is a perfect number 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.

• André–Oort conjecture: is every irreducible component of the Zariski closure of a set of special points in a Shimura variety a special subvariety?
• Beilinson's conjectures
• Brocard's problem: are there any integer solutions to ${\displaystyle n!+1=m^{2}}$ other than ${\displaystyle n=4,5,7}$?
• 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 ${\displaystyle 1}$?
• Casas-Alvero conjecture: if a polynomial of degree ${\displaystyle d}$ defined over a field ${\displaystyle K}$ of characteristic ${\displaystyle 0}$ has a factor in common with its first through ${\displaystyle d-1}$-th derivative, then must ${\displaystyle f}$ be the ${\displaystyle d}$-th power of a linear polynomial?
• Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating.
• 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 ${\displaystyle 1^{1}+2^{1}=3^{1}}$ the only solution to the Erdős–Moser equation?
• Erdős–Straus conjecture: for every ${\displaystyle n\geq 2}$, there are positive integers ${\displaystyle x,y,z}$ such that ${\displaystyle 4/n=1/x+1/y+1/z}$.
• Erdős–Ulam problem: is there a dense set of points in the plane all at rational distances from one-another?
• Exponent pair conjecture: for all ${\displaystyle \epsilon >0}$, is the pair ${\displaystyle (\epsilon ,1/2+\epsilon )}$ 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?
• Goormaghtigh conjecture on solutions to ${\displaystyle (x^{m}-1)/(x-1)=(y^{n}-1)/(y-1)}$ where ${\displaystyle x>y>1}$ and ${\displaystyle m,n>2}$.
• Grand Riemann hypothesis: do the nontrivial zeros of all automorphic L-functions lie on the critical line ${\displaystyle 1/2+it}$ with real ${\displaystyle t}$?
  • Generalized Riemann hypothesis: do the nontrivial zeros of all Dirichlet L-functions lie on the critical line ${\displaystyle 1/2+it}$ with real ${\displaystyle t}$?
    • Riemann hypothesis: do the nontrivial zeros of the Riemann zeta function lie on the critical line ${\displaystyle 1/2+it}$ with real ${\displaystyle t}$?
• 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 ${\displaystyle \epsilon >0}$, there is some constant ${\displaystyle c(\epsilon )}$ such that either ${\displaystyle y^{2}=x^{3}}$ or ${\displaystyle |y^{2}-x^{3}|>c(\epsilon )x^{1/2-\epsilon }}$.
• Hardy–Littlewood zeta-function conjectures
• Hilbert–Pólya conjecture: the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator.
• Hilbert's eleventh problem: classify quadratic forms over algebraic number fields.
• Hilbert's ninth problem: find the most general reciprocity law for the norm residues of ${\displaystyle k}$-th order in a general algebraic number field, where ${\displaystyle k}$ is a power of a prime.
• Hilbert's twelfth problem: extend the Kronecker–Weber theorem on Abelian extensions of ${\displaystyle \mathbb {Q} }$ to any base number field.
• Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function^[148]
• Lehmer's totient problem: if ${\displaystyle \phi (n)}$ divides ${\displaystyle n-1}$, must ${\displaystyle n}$ be prime?
• Leopoldt's conjecture: a p-adic analogue of the regulator of an algebraic number field does not vanish.
• Lindelöf hypothesis that for all ${\displaystyle \epsilon >0}$, ${\displaystyle \zeta (1/2+it)=o(t^{\epsilon })}$
  • The density hypothesis for zeroes of the Riemann zeta function
• Littlewood conjecture: for any two real numbers ${\displaystyle \alpha ,\beta }$, ${\displaystyle \liminf _{n\rightarrow \infty }n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0}$, where ${\displaystyle \Vert x\Vert }$ is the distance from ${\displaystyle x}$ to the nearest integer.
• Mahler's 3/2 problem that no real number ${\displaystyle x}$ has the property that the fractional parts of ${\displaystyle x(3/2)^{n}}$ are less than ${\displaystyle 1/2}$ for all positive integers ${\displaystyle n}$.
• 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.
• n conjecture: a generalization of the abc conjecture to more than three integers.
  • abc conjecture: for any ${\displaystyle \epsilon >0}$, ${\displaystyle {\text{rad}}(abc)^{1+\epsilon }<c}$ is true for only finitely many positive ${\displaystyle a,b,c}$ such that ${\displaystyle a+b=c}$.
  • Szpiro's conjecture: for any ${\displaystyle \epsilon >0}$, there is some constant ${\displaystyle C(\epsilon )}$ such that, for any elliptic curve ${\displaystyle E}$ defined over ${\displaystyle \mathbb {Q} }$ with minimal discriminant ${\displaystyle \Delta }$ and conductor ${\displaystyle f}$, we have ${\displaystyle |\Delta |\leq C(\epsilon )\cdot f^{6+\epsilon }}$.
• Newman's conjecture: the partition function satisfies any arbitrary congruence infinitely often.
• Pillai's conjecture: for any ${\displaystyle A,B,C}$, the equation ${\displaystyle Ax^{m}-By^{n}=C}$ has finitely many solutions when ${\displaystyle m,n}$ are not both ${\displaystyle 2}$.
• Piltz divisor problem on bounding ${\displaystyle \Delta _{k}(x)=D_{k}(x)-xP_{k}(log(x))}$
  • Dirichlet's divisor problem: the specific case of the Piltz divisor problem for ${\displaystyle k=1}$
• Ramanujan–Petersson conjecture: a number of related conjectures that are generalizations of the original conjecture.
• Sato–Tate conjecture: also a number of related conjectures that are generalizations of the original conjecture.
• Scholz conjecture: the length of the shortest addition chain producing ${\displaystyle 2^{n}-1}$ is at most ${\displaystyle n-1}$ plus the length of the shortest addition chain producing ${\displaystyle n}$.
• Do Siegel zeros exist?
• Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?^[149]
• The uniqueness conjecture for Markov numbers^[150] that every Markov number is the largest number in exactly one normalized solution to the Markov Diophantine equation.
• Vojta's conjecture on heights of points on algebraic varieties over algebraic number fields.

• 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 amicable numbers?
• Are there infinitely many betrothed numbers?
• Are there infinitely many Giuga numbers?
• Does every rational number with an odd denominator have an odd greedy expansion?
• Do any Lychrel numbers exist?
• 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?^[151]
• Is ${\displaystyle \pi }$ a normal number (i.e., is each digit 0–9 equally frequent)?^[152]
• Are all irrational algebraic numbers normal?
• Is 10 a solitary number?
• Can a 3×3 magic square be constructed from 9 distinct perfect square numbers?^[153]
• Which integers can be written as the sum of three perfect cubes?^[154]
• Can every integer be written as a sum of four perfect cubes?

• Find the value of the De Bruijn–Newman constant.

Additive number theory

• Beal's conjecture: for all integral solutions to ${\displaystyle A^{x}+B^{y}=C^{z}}$ where ${\displaystyle x,y,z>2}$, all three numbers ${\displaystyle A,B,C}$ must share some prime factor.
• 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–Heilbronn conjecture that ${\displaystyle |2^{\wedge }A|\geq \min\{p,2|A|-3\}}$ if ${\displaystyle p}$ is a prime and ${\displaystyle A}$ is a nonempty subset of the field ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$.
• Erdős–Turán conjecture on additive bases: if ${\displaystyle B}$ is an additive basis of order ${\displaystyle 2}$, then the number of ways that positive integers ${\displaystyle n}$ can be expressed as the sum of two numbers in ${\displaystyle B}$ must tend to infinity as ${\displaystyle n}$ tends to infinity.
• Fermat–Catalan conjecture: there are finitely many distinct solutions ${\displaystyle (a^{m},b^{n},c^{k})}$ to the equation ${\displaystyle a^{m}+b^{n}=c^{k}}$ with ${\displaystyle a,b,c}$ being positive coprime integers and ${\displaystyle m,n,k}$ being positive integers satisfying ${\displaystyle 1/m+1/n+1/k<1}$.
• 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 ${\displaystyle 2}$ is the sum of two prime numbers.
• Lander, Parkin, and Selfridge conjecture: if the sum of ${\displaystyle m}$ ${\displaystyle k}$-th powers of positive integers is equal to a different sum of ${\displaystyle n}$ ${\displaystyle k}$-th powers of positive integers, then ${\displaystyle m+n\geq k}$.
• Lemoine's conjecture: all odd integers greater than ${\displaystyle 5}$ 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 ${\displaystyle \{1,\ldots ,2n\}}$
• 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 r_k(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 ${\displaystyle n}$, to assign a sequence of natural numbers to each real number such that the sequence for ${\displaystyle x}$ is eventually periodic if and only if ${\displaystyle x}$ is algebraic of degree ${\displaystyle n}$?
• Kummer–Vandiver conjecture: primes ${\displaystyle p}$ do not divide the class number of the maximal real subfield of the ${\displaystyle p}$-th cyclotomic field.
• Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant ${\displaystyle X}$ is within a constant multiple of ${\displaystyle {\sqrt {X}}/\ln {X}}$
• Selberg's 1/4 conjecture: the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least ${\displaystyle 1/4}$.
• Stark conjectures (including Brumer–Stark conjecture)

• Characterize all algebraic number fields that have some power basis.

Computational number theory

• Can integer factorization be done in polynomial time?

Prime numbers

Goldbach's conjecture 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 ${\displaystyle p}$ is prime if and only if ${\displaystyle pB_{p-1}\equiv -1{\pmod {p}}}$
• Agrawal's conjecture that given coprime positive integers ${\displaystyle n}$ and ${\displaystyle r}$, if ${\displaystyle (X-1)^{n}\equiv X^{n}-1{\pmod {n,X^{r}-1}}}$, then either ${\displaystyle n}$ is prime or ${\displaystyle n^{2}\equiv 1{\pmod {r}}}$
• Artin's conjecture on primitive roots that if an integer is neither a perfect square nor ${\displaystyle -1}$, then it is a primitive root modulo infinitely many prime numbers ${\displaystyle p}$
• Brocard's conjecture: there are always at least ${\displaystyle 4}$ prime numbers between consecutive squares of prime numbers, aside from ${\displaystyle 2^{2}}$ and ${\displaystyle 3^{2}}$.
• Bunyakovsky conjecture: if an integer-coefficient polynomial ${\displaystyle f}$ has a positive leading coefficient, is irreducible over the integers, and has no common factors over all ${\displaystyle f(x)}$ where ${\displaystyle x}$ is a positive integer, then ${\displaystyle f(x)}$ 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 ${\displaystyle a_{1}+b_{1}n,\ldots ,a_{k}+b_{k}n}$ with each ${\displaystyle b_{i}\geq 1}$, there are infinitely many ${\displaystyle n}$ for which all forms are prime, unless there is some congruence condition preventing it.
• Dubner's conjecture: every even number greater than ${\displaystyle 4208}$ 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 ${\displaystyle p}$ and ${\displaystyle q}$, ${\displaystyle (p^{q}-1)/(p-1)}$ does not divide ${\displaystyle (q^{p}-1)/(q-1)}$
• 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 ${\displaystyle 2}$ are the sum of two prime numbers.
  • Legendre's conjecture: for every positive integer ${\displaystyle n}$, there is a prime between ${\displaystyle n^{2}}$ and ${\displaystyle (n+1)^{2}}$.
  • Twin prime conjecture: there are infinitely many twin primes.
  • Are there infinitely many primes of the form ${\displaystyle n^{2}+1}$?
• Problems associated to Linnik's theorem
• New Mersenne conjecture: for any odd natural number ${\displaystyle p}$, if any two of the three conditions ${\displaystyle p=2^{k}\pm 1}$ or ${\displaystyle p=4^{k}\pm 3}$, ${\displaystyle 2^{p}-1}$ is prime, and ${\displaystyle (2^{p}+1)/3}$ is prime are true, then the third condition is true.
• Polignac's conjecture: for all positive even numbers ${\displaystyle n}$, there are infinitely many prime gaps of size ${\displaystyle n}$.
• Schinzel's hypothesis H that for every finite collection ${\displaystyle \{f_{1},\ldots ,f_{k}\}}$ of nonconstant irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers ${\displaystyle n}$ for which ${\displaystyle f_{1}(n),\ldots ,f_{k}(n)}$ are all primes, or there is some fixed divisor ${\displaystyle m>1}$ which, for all ${\displaystyle n}$, divides some ${\displaystyle f_{i}(n)}$.
• 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 Carol 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?
• Are there infinitely many regular primes, and if so is their relative density ${\displaystyle e^{-1/2}}$?
• 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 ${\displaystyle 2^{p-1}\equiv 1{\pmod {p^{2}}}}$ and ${\displaystyle 3^{p-1}\equiv 1{\pmod {p^{2}}}}$ simultaneously?^[155]
• 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 a^{p − 1} ≡ 1 (mod p²)?^[156]
• 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 integer b which is not a perfect power and not of the form −4k⁴ for integer k, are there infinitely many repunit primes to base b?
• For any given integers ${\displaystyle k\geq 1,b\geq 2,c\neq 0}$, with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form ${\displaystyle (k\times b^{n}+c)/{\text{gcd}}(k+c,b-1)}$ with integer n ≥ 1?
• Is every Fermat number ${\displaystyle 2^{2^{n}}+1}$ composite for ${\displaystyle n>4}$?
• Is 509,203 the lowest Riesel number?

21st century

• Maximal rank conjecture (Eric Larson, 2018)^[183]
• Weibel's conjecture (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018)^[184]
• Yau's conjecture (Antoine Song, 2018)^[185][186]
• Pentagonal tiling (Michaël Rao, 2017)^[187]
• Willmore conjecture (Fernando Codá Marques and André Neves, 2012)^[188]
• Erdős distinct distances problem (Larry Guth, Nets Hawk Katz, 2011)^[189]
• Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)^[190]
• Tameness conjecture (Ian Agol, 2004)^[164]
• Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)^[191]
• Carpenter's rule problem (Robert Connelly, Erik Demaine, Günter Rote, 2003)^[192]
• Lambda g conjecture (Carel Faber and Rahul Pandharipande, 2003)^[193]
• Nagata's conjecture (Ivan Shestakov, Ualbai Umirbaev, 2003)^[194]
• Double bubble conjecture (Michael Hutchings, Frank Morgan, Manuel Ritoré, Antonio Ros, 2002)^[195]

20th century

• Honeycomb conjecture (Thomas Callister Hales, 1999)^[196]
• Lange's conjecture (Montserrat Teixidor i Bigas and Barbara Russo, 1999)^[197]
• Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)^[198][199]
• Kepler conjecture (Samuel Ferguson, Thomas Callister Hales, 1998)^[200]
• Dodecahedral conjecture (Thomas Callister Hales, Sean McLaughlin, 1998)^[201]

21st century

• Duffin-Schaeffer conjecture (Dimitris Koukoulopoulos, James Maynard, 2019)
• Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)^[224]
• Goldbach's weak conjecture (Harald Helfgott, 2013)^{[225][226][227]}
• Existence of bounded gaps between primes (Yitang Zhang, Polymath8, James Maynard, 2013)^{[228][229][230]}
• Sidon set problem (Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa, 2010)^[231]
• Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)^{[232][233][234]}
• Green–Tao theorem (Ben J. Green and Terence Tao, 2004)^[235]
• Catalan's conjecture (Preda Mihăilescu, 2002)^[236]
• Erdős–Graham problem (Ernest S. Croot III, 2000)^[237]

20th century

• Lafforgue's theorem (Laurent Lafforgue, 1998)^[238]
• Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)^[239][240]

2010s

• Erdős discrepancy problem (Terence Tao, 2015)^[256]
• Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)^[257]
• Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (Jeff Cheeger, Aaron Naber, 2014)^[258]
• Gaussian correlation inequality (Thomas Royen, 2014)^[259]
• Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman, Aleksandar Nikolov, 2011)^[260]
• Bloch–Kato conjecture (Vladimir Voevodsky, 2011)^[261] (and Quillen–Lichtenbaum conjecture and by work of Thomas Geisser and Marc Levine (2001) also Beilinson–Lichtenbaum conjecture^{[262][263]: 359 [264]})

2000s

• Kauffman–Harary conjecture (Thomas Mattman, Pablo Solis, 2009)^[265]
• Surface subgroup conjecture (Jeremy Kahn, Vladimir Markovic, 2009)^[266]
• Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Zhiqin Lu, 2007)^[267]
• Nirenberg–Treves conjecture (Nils Dencker, 2005)^[268][269]
• Lax conjecture (Adrian Lewis, Pablo Parrilo, Motakuri Ramana, 2005)^[270]
• The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)^[271]
• Milnor conjecture (Vladimir Voevodsky, 2003)^[272]
• Kirillov's conjecture (Ehud Baruch, 2003)^[273]
• Kouchnirenko's conjecture (Bertrand Haas, 2002)^[274]
• n! conjecture (Mark Haiman, 2001)^[275] (and also Macdonald positivity conjecture)
• Kato's conjecture (Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philipp Tchamitchian, 2001)^[276]
• Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)^[277]
• Modularity theorem (Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, 2001)^[278]
• Erdős–Stewart conjecture (Florian Luca, 2001)^[279]
• Berry–Robbins problem (Michael Atiyah, 2000)^[280]