Atlas of open mathematics

Conjecture Fieldbook separates 621 revision-pinned source placements from 16 reviewed canonical dossiers. Source-list placement is not independent verification.

Reviewed problem dossiers

• Riemann hypothesis — Number theory
• P versus NP — Computer science
• Navier–Stokes existence — Analysis
• Birch and Swinnerton-Dyer — Arithmetic geometry
• Hodge conjecture — Algebraic geometry
• Yang–Mills and mass gap — Mathematical physics
• Collatz conjecture — Discrete mathematics
• Goldbach conjecture — Number theory
• Twin prime conjecture — Number theory
• Hadamard matrix conjecture — Algebra
• Perfect cuboid — Diophantine equations
• Invariant subspace problem — Analysis
• Smooth 4D Poincaré conjecture — Topology
• Two-dimensional Jacobian conjecture — Algebra
• Erdős–Straus conjecture — Number theory
• Hadwiger conjecture — Graph theory

Revision-pinned source records

561 substantive, representative source placements have dedicated indexable records. Thin, duplicate-title, or pending-review rows remain directly addressable but are excluded from search indexes.

• Birch–Tate conjecture — Algebra
• Casas-Alvero conjecture — Algebra
• Connes embedding problem in Von Neumann algebra theory — Algebra
• Crouzeix's conjecture — Algebra
• Determinantal conjecture on the determinant of the sum of two normal matrices. — Algebra
• Eilenberg–Ganea conjecture — Algebra
• Farrell–Jones conjecture on whether certain assembly maps are isomorphisms. — Algebra
• Bost conjecture — Algebra
• Finite lattice representation problem — Algebra
• Goncharov conjecture on the cohomology of certain motivic complexes. — Algebra
• Green's conjecture — Algebra
• Grothendieck–Katz p-curvature conjecture — Algebra
• Hadamard conjecture — Algebra
• Williamson conjecture — Algebra
• Hadamard's maximal determinant problem — Algebra
• Hilbert's fifteenth problem — Algebra
• Hilbert's sixteenth problem — Algebra
• Homological conjectures in commutative algebra — Algebra
• Jacobson's conjecture — Algebra
• Köthe conjecture — Algebra
• Monomial conjecture on Noetherian local rings — Algebra
• Existence of perfect cuboids and associated cuboid conjectures — Algebra
• Pierce–Birkhoff conjecture — Algebra
• Rota's basis conjecture — Algebra
• Serre's conjecture II — Algebra
• Serre's positivity conjecture — Algebra
• Uniform boundedness conjecture for rational points — Algebra
• Wild problems — Algebra
• Zariski–Lipman conjecture — Algebra
• Zauner's conjecture — Algebra
• Andrews–Curtis conjecture — Algebra · Group theory
• Bounded Burnside problem — Algebra · Group theory
• Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems — Algebra · Group theory
• Herzog–Schönheim conjecture — Algebra · Group theory
• The inverse Galois problem — Algebra · Group theory
• Leinster groups — Algebra · Group theory
• finitely presented — Algebra · Group theory
• sofic — Algebra · Group theory
• Problems in loop theory and quasigroup theory consider generalizations of groups — Algebra · Group theory
• Dade's conjecture — Algebra · Representation theory
• Demazure conjecture on representations of algebraic groups over the integers. — Algebra · Representation theory
• Kazhdan–Lusztig conjectures — Algebra · Representation theory
• McKay conjecture — Algebra · Representation theory
• The Brennan conjecture — Analysis
• Fuglede's conjecture — Analysis
• Goodman's conjecture on the coefficients of multivalued functions — Analysis
• Invariant subspace problem — Analysis
• Kung–Traub conjecture on the optimal order of a multipoint iteration without memory — Analysis
• Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials — Analysis
• The mean value problem — Analysis
• The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy — Analysis
• Sendov's conjecture — Analysis
• Vitushkin's conjecture — Analysis
• Landau's constants — Analysis
• Regularity of solutions of Euler equations — Analysis
• Regularity of solutions of Vlasov–Maxwell equations — Analysis
• The 1/3–2/3 conjecture — Combinatorics
• Dittert conjecture — Combinatorics
• Problems in Latin squares — Combinatorics
• The lonely runner conjecture — Combinatorics
• Superpermutations - smallest possible string of n digits that contains all possible permutations of n — Combinatorics
• Map folding — Combinatorics
• No-three-in-line problem — Combinatorics
• Rudin's conjecture on the number of squares in finite arithmetic progressions — Combinatorics
• The sunflower conjecture — Combinatorics
• Frankl's union-closed sets conjecture — Combinatorics
• Give a combinatorial interpretation of the Kronecker coefficients — Combinatorics
• Property B — Combinatorics
• Dedekind numbers — Combinatorics
• The values of the Ramsey numbers, particularly R ( 5 , 5 ) {\displaystyle R(5,5)} {\displaystyle R(5,5)} — Combinatorics
• The values of the Van der Waerden numbers — Combinatorics
• Finding a function to model n-step self-avoiding walks — Combinatorics
• Arnold–Givental conjecture and Arnold conjecture — Dynamical systems
• Banach's problem — Dynamical systems
• Birkhoff conjecture — Dynamical systems
• Collatz conjecture (also known as the 3 n + 1 {\displaystyle 3n+1} {\displaystyle 3n+1} conjecture) — Dynamical systems
• Eden's conjecture — Dynamical systems
• Fatou conjecture — Dynamical systems
• Furstenberg conjecture — Dynamical systems
• Kaplan–Yorke conjecture on the dimension of an attractor in terms of its Lyapunov exponents — Dynamical systems
• Margulis conjecture — Dynamical systems
• Hilbert–Arnold problem — Dynamical systems
• MLC conjecture — Dynamical systems
• outer billiard — Dynamical systems
• eigenfunctions — Dynamical systems
• Rokhlin's multiple mixing problem — Dynamical systems
• Triangular billiards — Dynamical systems
• Weinstein conjecture — Dynamical systems
• juggler sequence — Dynamical systems
• Lyapunov function — Dynamical systems
• reversible cellular automaton — Dynamical systems
• How many puzzles have exactly one solution? — Games and puzzles · Combinatorial games
• minimal — Games and puzzles · Combinatorial games
• maximum number of givens — Games and puzzles · Combinatorial games
• Hales–Jewett theorem — Games and puzzles · Combinatorial games
• first-move advantage in chess — Games and puzzles · Combinatorial games
• cap set — Games and puzzles · Combinatorial games
• octal games — Games and puzzles · Combinatorial games
• Grundy's game — Games and puzzles · Combinatorial games
• Abundance conjecture — Geometry · Algebraic geometry
• Bass conjecture on the finite generation of certain algebraic K-groups. — Geometry · Algebraic geometry
• Bass–Quillen conjecture — Geometry · Algebraic geometry
• Deligne conjecture — Geometry · Algebraic geometry
• Deligne's conjecture on Hochschild cohomology about the operadic structure on Hochschild cochain complex. — Geometry · Algebraic geometry
• Fröberg conjecture on the Hilbert functions of a set of forms. — Geometry · Algebraic geometry
• Fujita conjecture — Geometry · Algebraic geometry
• General elephant problem — Geometry · Algebraic geometry
• spherical — Geometry · Algebraic geometry
• Jacobian conjecture for two-dimensional spaces — Geometry · Algebraic geometry
• Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory — Geometry · Algebraic geometry
• Nagata's conjecture on curves — Geometry · Algebraic geometry
• Nagata–Biran conjecture — Geometry · Algebraic geometry
• Nakai conjecture — Geometry · Algebraic geometry
• Parshin's conjecture — Geometry · Algebraic geometry
• Section conjecture — Geometry · Algebraic geometry
• Standard conjectures on algebraic cycles — Geometry · Algebraic geometry
• Tate conjecture — Geometry · Algebraic geometry
• Virasoro conjecture — Geometry · Algebraic geometry
• Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points — Geometry · Algebraic geometry
• flips — Geometry · Algebraic geometry
• Resolution of singularities in characteristic p {\displaystyle p} {\displaystyle p} — Geometry · Algebraic geometry
• Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set. — Geometry · Covering and packing
• The covering problem of Rado — Geometry · Covering and packing
• The Erdős–Oler conjecture — Geometry · Covering and packing
• disk covering problem — Geometry · Covering and packing
• The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24 — Geometry · Covering and packing
• Reinhardt's conjecture — Geometry · Covering and packing
• Sphere packing — Geometry · Covering and packing
• Square packing in a square — Geometry · Covering and packing
• Ulam's packing conjecture about the identity of the worst-packing convex solid — Geometry · Covering and packing
• The Tammes problem for numbers of nodes greater than 14 (except 24). — Geometry · Covering and packing
• The spherical Bernstein's problem, a generalization of Bernstein's problem — Geometry · Differential geometry
• Carathéodory conjecture — Geometry · Differential geometry
• Cartan–Hadamard conjecture — Geometry · Differential geometry
• Chern's conjecture (affine geometry) that the Euler characteristic of a compact affine manifold vanishes. — Geometry · Differential geometry
• Chern's conjecture for hypersurfaces in spheres, a number of closely related conjectures. — Geometry · Differential geometry
• Closed curve problem — Geometry · Differential geometry
• filling area conjecture — Geometry · Differential geometry
• The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds — Geometry · Differential geometry
• Osserman conjecture — Geometry · Differential geometry
• Yau's conjecture on the first eigenvalue — Geometry · Differential geometry
• big-line-big-clique conjecture — Geometry · Discrete geometry
• The Dirac–Motzkin conjecture on the minimum number of ordinary lines for n points in the Euclidean plane — Geometry · Discrete geometry
• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2 smaller copies — Geometry · Discrete geometry
• Solving the happy ending problem for arbitrary n {\displaystyle n} {\displaystyle n} — Geometry · Discrete geometry
• Improving lower and upper bounds for the Heilbronn triangle problem. — Geometry · Discrete geometry
• Kalai's 3 conjecture on the least possible number of faces of centrally symmetric polytopes. — Geometry · Discrete geometry
• The Kobon triangle problem on triangles in line arrangements — Geometry · Discrete geometry
• The Kusner conjecture — Geometry · Discrete geometry
• The McMullen problem on projectively transforming sets of points into convex position — Geometry · Discrete geometry
• Opaque forest problem on finding opaque sets for various planar shapes — Geometry · Discrete geometry
• Orchard-planting problem on the maximum number of 3-point lines attainable by a configuration of n points in the plane — Geometry · Discrete geometry
• How many unit distances — Geometry · Discrete geometry
• Finding matching upper and lower bounds for k-sets and halving lines — Geometry · Discrete geometry
• crossing number — Geometry · Discrete geometry
• Tripod packing — Geometry · Discrete geometry
• Atiyah conjecture on configurations — Geometry · Euclidean geometry
• Bellman's lost-in-a-forest problem — Geometry · Euclidean geometry
• Borromean rings — Geometry · Euclidean geometry
• Connelly’s blooming conjecture — Geometry · Euclidean geometry
• Danzer's problem and Conway's dead fly problem — Geometry · Euclidean geometry
• Dissection into orthoschemes — Geometry · Euclidean geometry
• Ehrhart's volume conjecture — Geometry · Euclidean geometry
• Falconer's conjecture — Geometry · Euclidean geometry
• The values of the Hermite constants for dimensions other than 1–8 and 24 — Geometry · Euclidean geometry
• holyhedron — Geometry · Euclidean geometry
• Inscribed square problem, also known as Toeplitz' conjecture and the square peg problem — Geometry · Euclidean geometry
• The Kakeya conjecture — Geometry · Euclidean geometry
• Weaire–Phelan structure — Geometry · Euclidean geometry
• Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one — Geometry · Euclidean geometry
• Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar. — Geometry · Euclidean geometry
• Are the two Meissner tetrahedra the minimum-volume three-dimensional shapes of constant width? — Geometry · Euclidean geometry
• Moser's worm problem — Geometry · Euclidean geometry
• The moving sofa problem — Geometry · Euclidean geometry
• Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combin — Geometry · Euclidean geometry
• Voronoi diagram — Geometry · Euclidean geometry
• Is there a general expression for the minimum ropelength of an arbitrary closed knot? — Geometry · Euclidean geometry
• What constant 1 — Geometry · Euclidean geometry
• Is the upper bound of a closed knot's minimum ropelength linear to its crossing number? — Geometry · Euclidean geometry
• 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? — Geometry · Euclidean geometry
• Rupert's property — Geometry · Euclidean geometry
• Shephard's problem (a.k.a. Dürer's conjecture) — Geometry · Euclidean geometry
• more than seven faces — Geometry · Euclidean geometry
• The Thomson problem — Geometry · Euclidean geometry
• Convex uniform 5-polytopes — Geometry · Euclidean geometry
• Hilbert's third problem for non-Euclidean geometries — Geometry · Non-Euclidean geometry
• Babai's problem — Graph theory · Algebraic graph theory
• Brouwer's conjecture on upper bounds for sums of eigenvalues of Laplacians of graphs in terms of their number of edges — Graph theory · Algebraic graph theory
• γ–θ conjecture — Graph theory · Games on graphs
• Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs — Graph theory · Games on graphs
• Meyniel's conjecture that cop number is O ( n ) {\displaystyle O({\sqrt {n}})} {\displaystyle O({\sqrt {n}})} — Graph theory · Games on graphs
• vertex coloring game — Graph theory · Games on graphs
• 1-factorization conjecture — Graph theory · Graph coloring and labeling
• The perfect 1-factorization conjecture that every complete graph on an even number of vertices admits a perfect 1-factorization. — Graph theory · Graph coloring and labeling
• Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs — Graph theory · Graph coloring and labeling
• The Earth–Moon problem — Graph theory · Graph coloring and labeling
• The Erdős–Faber–Lovász conjecture on coloring unions of cliques — Graph theory · Graph coloring and labeling
• The graceful tree conjecture that every tree admits a graceful labeling — Graph theory · Graph coloring and labeling
• Rosa's conjecture that all triangular cacti are graceful or nearly-graceful — Graph theory · Graph coloring and labeling
• The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree — Graph theory · Graph coloring and labeling
• The Hadwiger conjecture relating coloring to clique minors — Graph theory · Graph coloring and labeling
• The Hadwiger–Nelson problem on the chromatic number of unit distance graphs — Graph theory · Graph coloring and labeling
• Jaeger's Petersen-coloring conjecture — Graph theory · Graph coloring and labeling
• The list coloring conjecture — Graph theory · Graph coloring and labeling
• overfull conjecture — Graph theory · Graph coloring and labeling
• The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree — Graph theory · Graph coloring and labeling
• The Albertson conjecture — Graph theory · Graph drawing and embedding
• Conway's thrackle conjecture that thrackles cannot have more edges than vertices — Graph theory · Graph drawing and embedding
• GNRS conjecture — Graph theory · Graph drawing and embedding
• Harborth's conjecture — Graph theory · Graph drawing and embedding
• Negami's conjecture on projective-plane embeddings of graphs with planar covers — Graph theory · Graph drawing and embedding
• The strong Papadimitriou–Ratajczak conjecture — Graph theory · Graph drawing and embedding
• Turán's brick factory problem — Graph theory · Graph drawing and embedding
• Guy's conjecture on the crossing number for complete graphs — Graph theory · Graph drawing and embedding
• Universal point sets of subquadratic size for planar graphs — Graph theory · Graph drawing and embedding
• conference graph — Graph theory · Restriction of graph parameters
• Conway's 99-graph problem — Graph theory · Restriction of graph parameters
• Degree diameter problem — Graph theory · Restriction of graph parameters
• apex graph — Graph theory · Restriction of graph parameters
• Moore graph — Graph theory · Restriction of graph parameters
• strongly regular — Graph theory · Restriction of graph parameters
• Barnette's conjecture — Graph theory · Subgraphs
• Gilbert–Pollak conjecture — Graph theory · Subgraphs
• Chvátal's toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian — Graph theory · Subgraphs
• The cycle double cover conjecture — Graph theory · Subgraphs
• The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs — Graph theory · Subgraphs
• The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph — Graph theory · Subgraphs
• The Grünbaum–Nash-Williams conjecture on whether every 4-vertex-connected toroidal graph has a Hamiltonian cycle. — Graph theory · Subgraphs
• The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree — Graph theory · Subgraphs
• The Lovász conjecture on Hamiltonian paths in symmetric graphs — Graph theory · Subgraphs
• Oberwolfach problem — Graph theory · Subgraphs
• pathwidth — Graph theory · Subgraphs
• reconstruction conjecture — Graph theory · Subgraphs
• The snake-in-the-box problem — Graph theory · Subgraphs
• Sumner's conjecture — Graph theory · Subgraphs
• Szymanski's conjecture — Graph theory · Subgraphs
• Tuza's conjecture — Graph theory · Subgraphs
• The unfriendly partition conjecture — Graph theory · Subgraphs
• Vizing's conjecture on the domination number of cartesian products of graphs — Graph theory · Subgraphs
• Walescki's theorem for hypergraphs — Graph theory · Subgraphs
• Zarankiewicz problem — Graph theory · Subgraphs
• representation — Graph theory · Word-representation of graphs
• Characterise (non-)word-representable planar graphs — Graph theory · Word-representation of graphs
• Characterise word-representable graphs in terms of (induced) forbidden subgraphs. — Graph theory · Word-representation of graphs
• word-representable — Graph theory · Word-representation of graphs
• represented — Graph theory · Word-representation of graphs
• bipartite graphs — Graph theory · Word-representation of graphs
• line graph — Graph theory · Word-representation of graphs
• representing — Graph theory · Word-representation of graphs
• Delta-conjecture (1978) — Graph theory · Miscellaneous graph theory
• The imbalance conjecture — Graph theory · Miscellaneous graph theory
• The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs — Graph theory · Miscellaneous graph theory
• Ryser's conjecture relating the maximum matching size and minimum transversal size in hypergraphs — Graph theory · Miscellaneous graph theory
• The second neighborhood problem — Graph theory · Miscellaneous graph theory
• Sidorenko's conjecture on homomorphism densities of graphs in graphons — Graph theory · Miscellaneous graph theory
• Teschner's bondage number conjecture — Graph theory · Miscellaneous graph theory
• every bridgeless graph has a nowhere-zero 5-flow — Graph theory · Miscellaneous graph theory
• every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow — Graph theory · Miscellaneous graph theory
• Woodall's conjecture — Graph theory · Miscellaneous graph theory
• The Cherlin–Zilber conjecture — Model theory and formal languages
• Generalized star height problem — Model theory and formal languages
• Hilbert's tenth problem — Model theory and formal languages
• first order theories — Model theory and formal languages
• Shelah's categoricity conjecture for L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} {\displaystyle L_{\omega _{1},\omega }} — Model theory and formal languages
• Shelah's eventual categoricity conjecture — Model theory and formal languages
• The stable field conjecture — Model theory and formal languages
• The stable forking conjecture for simple theories — Model theory and formal languages
• Tarski's exponential function problem — Model theory and formal languages
• The universality problem for C-free graphs — Model theory and formal languages
• The universality spectrum problem — Model theory and formal languages
• Vaught conjecture — Model theory and formal languages
• types — Model theory and formal languages
• Henson graphs — Model theory and formal languages
• reducts — Model theory and formal languages
• o-minimal — Model theory and formal languages
• categorical — Model theory and formal languages
• algebraically closed — Model theory and formal languages
• Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidab — Model theory and formal languages
• decidable — Model theory and formal languages
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation p — Model theory and formal languages
• Determine the structure of Keisler's order. — Model theory and formal languages
• proof-theoretic ordinal — Model theory and formal languages
• Ibragimov–Iosifescu conjecture for φ-mixing sequences — Probability theory
• Büchi's problem on sufficiently large sequences of square numbers with constant second difference. — Number theory · General
• Carmichael's totient function conjecture — Number theory · General
• Catalan–Dickson conjecture on aliquot sequences — Number theory · General
• Exponent pair conjecture — Number theory · General
• The Gauss circle problem — Number theory · General
• Grimm's conjecture — Number theory · General
• Hall's conjecture — Number theory · General
• Lehmer's totient problem — Number theory · General
• Magic square of squares — Number theory · General
• Mahler's 3/2 problem — Number theory · General
• Newman's conjecture — Number theory · General
• Scholz conjecture — Number theory · General
• Singmaster's conjecture — Number theory · General
• perfect numbers — Number theory · General
• almost perfect numbers — Number theory · General
• idoneal numbers — Number theory · General
• amicable numbers — Number theory · General
• betrothed numbers — Number theory · General
• relatively prime — Number theory · General
• Giuga numbers — Number theory · General
• Lychrel numbers — Number theory · General
• Taxicab(5, 2, n) — Number theory · General
• covering system — Number theory · General
• normal number — Number theory · General
• irrational — Number theory · General
• Erdős conjecture on arithmetic progressions — Number theory · Additive number theory
• Erdős–Turán conjecture on additive bases — Number theory · Additive number theory
• Gilbreath's conjecture on consecutive applications of the unsigned forward difference operator to the sequence of prime numbers. — Number theory · Additive number theory
• Goldbach's conjecture — Number theory · Additive number theory
• Lander, Parkin, and Selfridge conjecture — Number theory · Additive number theory
• Lemoine's conjecture — Number theory · Additive number theory
• Minimum overlap problem — Number theory · Additive number theory
• Recamán's sequence — Number theory · Additive number theory
• Skolem problem — Number theory · Additive number theory
• The values of g(k) and G(k) in Waring's problem — Number theory · Additive number theory
• Ulam numbers — Number theory · Additive number theory
• Determine growth rate of rk(N) (see Szemerédi's theorem) — Number theory · Additive number theory
• Class number problem — Number theory · Algebraic number theory
• Fontaine–Mazur conjecture — Number theory · Algebraic number theory
• Gan–Gross–Prasad conjecture — Number theory · Algebraic number theory
• Hermite's problem — Number theory · Algebraic number theory
• Hilbert's 9th problem — Number theory · Algebraic number theory
• Hilbert's 11th problem — Number theory · Algebraic number theory
• Hilbert's 12th problem — Number theory · Algebraic number theory
• Kummer–Vandiver conjecture — Number theory · Algebraic number theory
• supersingular primes — Number theory · Algebraic number theory
• Leopoldt's conjecture — Number theory · Algebraic number theory
• Stark conjectures (including Brumer–Stark conjecture) — Number theory · Algebraic number theory
• Characterize all algebraic number fields that have some power basis. — Number theory · Algebraic number theory
• Beilinson's conjectures about special values of L-functions. — Number theory · Analytic number theory
• Find the value of the De Bruijn–Newman constant. — Number theory · Analytic number theory
• Selberg class — Number theory · Analytic number theory
• Hardy–Littlewood zeta function conjectures — Number theory · Analytic number theory
• Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function — Number theory · Analytic number theory
• Hilbert–Pólya conjecture — Number theory · Analytic number theory
• Lindelöf hypothesis — Number theory · Analytic number theory
• The density hypothesis for zeroes of the Riemann zeta function. — Number theory · Analytic number theory
• Location of nontrivial zeros of L-functions, especially conjectures concerning their critical lines: — Number theory · Analytic number theory
• Grand Riemann hypothesis — Number theory · Analytic number theory
• Generalized Riemann hypothesis for Selberg class — Number theory · Analytic number theory
• Extended Riemann Hypothesis — Number theory · Analytic number theory
• Generalized Riemann hypothesis — Number theory · Analytic number theory
• Riemann hypothesis — Number theory · Analytic number theory
• Montgomery's pair correlation conjecture — Number theory · Analytic number theory
• Piltz divisor problem — Number theory · Analytic number theory
• Dirichlet's divisor problem — Number theory · Analytic number theory
• Ramanujan–Petersson conjecture — Number theory · Analytic number theory
• Selberg's 1/4 conjecture — Number theory · Analytic number theory
• Selberg's orthogonality conjecture — Number theory · Analytic number theory
• Bombieri–Lang conjecture — Number theory · Arithmetic geometry
• Erdős–Ulam problem — Number theory · Arithmetic geometry
• Manin conjecture — Number theory · Arithmetic geometry
• Sato–Tate conjecture — Number theory · Arithmetic geometry
• Rational dense problem — Number theory · Arithmetic geometry
• Vojta's conjecture — Number theory · Arithmetic geometry
• n conjecture — Number theory · Arithmetic geometry
• abc conjecture — Number theory · Arithmetic geometry
• Szpiro's conjecture — Number theory · Arithmetic geometry
• Zilber–Pink conjecture — Number theory · Arithmetic geometry
• integer factorization — Number theory · Computational number theory
• discrete logarithm — Number theory · Computational number theory
• rational number — Number theory · Computational number theory
• Littlewood conjecture — Number theory · Diophantine approximation and transcendental number theory
• Schanuel's conjecture on the transcendence degree of certain field extensions of the rational numbers. In particular — Number theory · Diophantine approximation and transcendental number theory
• The four exponentials conjecture — Number theory · Diophantine approximation and transcendental number theory
• Euler's constant — Number theory · Diophantine approximation and transcendental number theory
• (exponential) periods — Number theory · Diophantine approximation and transcendental number theory
• non-quadratic — Number theory · Diophantine approximation and transcendental number theory
• simple continued fraction — Number theory · Diophantine approximation and transcendental number theory
• Beal's conjecture — Number theory · Diophantine equations
• Brocard's problem — Number theory · Diophantine equations
• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem) — Number theory · Diophantine equations
• Erdős–Moser problem — Number theory · Diophantine equations
• Erdős–Straus conjecture — Number theory · Diophantine equations
• Fermat–Catalan conjecture — Number theory · Diophantine equations
• Goormaghtigh conjecture — Number theory · Diophantine equations
• uniqueness conjecture for Markov numbers — Number theory · Diophantine equations
• Pillai's conjecture — Number theory · Diophantine equations
• sum of three perfect cubes — Number theory · Diophantine equations
• Can every integer be written as a sum of four perfect cubes? — Number theory · Diophantine equations
• Agoh–Giuga conjecture — Number theory · Prime numbers
• Agrawal's conjecture — Number theory · Prime numbers
• Artin's conjecture on primitive roots — Number theory · Prime numbers
• Brocard's conjecture — Number theory · Prime numbers
• Bunyakovsky conjecture — Number theory · Prime numbers
• Catalan's Mersenne conjecture — Number theory · Prime numbers
• Dickson's conjecture — Number theory · Prime numbers
• Dubner's conjecture — Number theory · Prime numbers
• Elliott–Halberstam conjecture on the distribution of prime numbers in arithmetic progressions. — Number theory · Prime numbers
• Erdős–Mollin–Walsh conjecture — Number theory · Prime numbers
• Feit–Thompson conjecture — Number theory · Prime numbers
• Fortune's conjecture that no Fortunate number is composite. — Number theory · Prime numbers
• The Gaussian moat problem — Number theory · Prime numbers
• Gillies' conjecture on the distribution of prime divisors of Mersenne numbers. — Number theory · Prime numbers
• Goldbach conjecture — Number theory · Prime numbers
• Legendre's conjecture — Number theory · Prime numbers
• Twin prime conjecture — Number theory · Prime numbers
• Are there infinitely many primes of the form n 2 + 1 {\displaystyle n^{2}+1} {\displaystyle n^{2}+1}? — Number theory · Prime numbers
• New Mersenne conjecture — Number theory · Prime numbers
• Polignac's conjecture — Number theory · Prime numbers
• Schinzel's hypothesis H — Number theory · Prime numbers
• Selfridge's conjecture — Number theory · Prime numbers
• converse of Wolstenholme's theorem — Number theory · Prime numbers
• Mersenne numbers — Number theory · Prime numbers
• Are there any composite c satisfying 2 ≡ 1 (mod c)? — Number theory · Prime numbers
• Wieferich primes — Number theory · Prime numbers
• balanced primes — Number theory · Prime numbers
• cluster primes — Number theory · Prime numbers
• cousin primes — Number theory · Prime numbers
• Cullen primes — Number theory · Prime numbers
• Euclid primes — Number theory · Prime numbers
• Fibonacci primes — Number theory · Prime numbers
• Kummer primes — Number theory · Prime numbers
• Mersenne primes — Number theory · Prime numbers
• Newman–Shanks–Williams primes — Number theory · Prime numbers
• palindromic primes — Number theory · Prime numbers
• Pierpont primes — Number theory · Prime numbers
• prime quadruplets — Number theory · Prime numbers
• prime triplets — Number theory · Prime numbers
• Siegel's conjecture — Number theory · Prime numbers
• safe and Sophie Germain primes — Number theory · Prime numbers
• Wagstaff primes — Number theory · Prime numbers
• Wilson primes — Number theory · Prime numbers
• Wolstenholme primes — Number theory · Prime numbers
• Woodall primes — Number theory · Prime numbers
• Can a prime p satisfy 2 p − 1 ≡ 1 ( mod p 2 ) {\displaystyle 2^{p-1}\equiv 1{\pmod {p^{2}}}} {\displaystyle 2^{p-1}\equiv 1{\pmod — Number theory · Prime numbers
• Euclid–Mullin sequence — Number theory · Prime numbers
• Lucas–Wieferich primes — Number theory · Prime numbers
• For any given integer a > 0, are there infinitely many primes p such that a ≡ 1 (mod p)? — Number theory · Prime numbers
• repunit — Number theory · Prime numbers
• For any given integers k ≥ 1 , b ≥ 2 , c ≠ 0 {\displaystyle k\geq 1,b\geq 2,c\neq 0} {\displaystyle k\geq 1,b\geq 2,c\neq 0}, with — Number theory · Prime numbers
• Fermat number — Number theory · Prime numbers
• partition principle — Set theory
• Woodin — Set theory
• generalized continuum hypothesis — Set theory
• Singular cardinals hypothesis — Set theory
• The problem of finding the ultimate core model, one that contains all large cardinals. — Set theory
• Woodin's Ω-conjecture — Set theory
• consistency — Set theory
• Jónsson algebra — Set theory
• open coloring axiom — Set theory
• Reinhardt cardinals — Set theory
• Baum–Connes conjecture — Topology
• Berge conjecture that the only knots in the 3-sphere which admit lens space surgeries are Berge knots. — Topology
• Bing–Borsuk conjecture — Topology
• Borel conjecture — Topology
• Halperin conjecture on rational Serre spectral sequences of certain fibrations. — Topology
• Hilbert–Smith conjecture — Topology
• Novikov conjecture — Topology
• Quadrisecants of wild knots — Topology
• Telescope conjecture — Topology
• Unknotting problem — Topology
• Volume conjecture relating quantum invariants of knots to the hyperbolic geometry of their knot complements. — Topology
• Whitehead conjecture — Topology
• Zeeman conjecture — Topology
• Nearby Lagrangian conjecture — Topology
• Dixmier conjecture — Algebra
• Mazur's conjecture B (Vessilin Dimitrov, Ziyang Gao, and Philipp Habegger, 2020) — Algebra
• Suita conjecture (Qi'an Guan and Xiangyu Zhou, 2015) — Algebra
• Carlitz–Wan conjecture (Hendrik Lenstra, 1995) — Algebra
• Serre's nonnegativity conjecture (Ofer Gabber, 1995) — Algebra
• Mizohata–Takeuchi conjecture (Hannah Cairo, 2025). — Analysis
• Kadison–Singer problem — Analysis
• Ahlfors measure conjecture (Ian Agol, 2004) — Analysis
• Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999) — Analysis
• Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018) — Combinatorics
• McMullen's g-conjecture — Combinatorics
• Hirsch conjecture (Francisco Santos Leal, 2010) — Combinatorics
• Gessel's lattice path conjecture (Manuel Kauers, Christoph Koutschan, and Doron Zeilberger, 2009) — Combinatorics
• Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004) (and also the Alon–Friedgut conjecture) — Combinatorics
• Kemnitz's conjecture (Christian Reiher, 2003, Carlos di Fiore, 2003) — Combinatorics
• Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003) — Combinatorics
• Eremenko's conjecture — Dynamical systems
• Zimmer's conjecture (Aaron Brown, David Fisher, and Sebastián Hurtado-Salazar, 2017) — Dynamical systems
• Existence of a non-terminating game of beggar-my-neighbour (Brayden Casella, 2024) — Game theory
• The angel problem (Various independent proofs, 2006) — Game theory
• Einstein problem (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2024) — Geometry · 21st century
• Maximal rank conjecture (Eric Larson, 2018) — Geometry · 21st century
• Weibel's conjecture (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018) — Geometry · 21st century
• Willmore conjecture (Fernando Codá Marques and André Neves, 2012) — Geometry · 21st century
• Erdős distinct distances problem (Larry Guth, Nets Hawk Katz, 2011) — Geometry · 21st century
• Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008) — Geometry · 21st century
• Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004) — Geometry · 21st century
• Carpenter's rule problem (Robert Connelly, Erik Demaine, Günter Rote, 2003) — Geometry · 21st century
• Lambda g conjecture (Carel Faber and Rahul Pandharipande, 2003) — Geometry · 21st century
• Nagata's conjecture (Ivan Shestakov, Ualbai Umirbaev, 2003) — Geometry · 21st century
• Double bubble conjecture (Michael Hutchings, Frank Morgan, Manuel Ritoré, Antonio Ros, 2002) — Geometry · 21st century
• Honeycomb theorem (Thomas Callister Hales, 1999) — Geometry · 20th century
• Lange's conjecture (Montserrat Teixidor i Bigas and Barbara Russo, 1999) — Geometry · 20th century
• Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998) — Geometry · 20th century
• Kepler conjecture (Samuel Ferguson, Thomas Callister Hales, 1998) — Geometry · 20th century
• Dodecahedral conjecture (Thomas Callister Hales, Sean McLaughlin, 1998) — Geometry · 20th century
• Kahn–Kalai conjecture (Jinyoung Park and Huy Tuan Pham, 2022) — Graph theory
• Blankenship–Oporowski conjecture — Graph theory
• Ringel's conjecture — Graph theory
• Disproof of Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019) — Graph theory
• Kelmans–Seymour conjecture (Dawei He, Yan Wang, and Xingxing Yu, 2020) — Graph theory
• Goldberg–Seymour conjecture (Guantao Chen, Guangming Jing, and Wenan Zang, 2019) — Graph theory
• Babai's problem (Alireza Abdollahi, Maysam Zallaghi, 2015) — Graph theory
• Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014) — Graph theory
• Alon–Saks–Seymour conjecture (Hao Huang, Benny Sudakov, 2012) — Graph theory
• Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009) — Graph theory
• Erdős–Menger conjecture (Ron Aharoni, Eli Berger 2007) — Graph theory
• Road coloring conjecture (Avraham Trahtman, 2007) — Graph theory
• Robertson–Seymour theorem (Neil Robertson, Paul Seymour, 2004) — Graph theory
• Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002) — Graph theory
• Toida's conjecture (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001) — Graph theory
• Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen, 1996) — Graph theory
• Hanna Neumann conjecture (Joel Friedman, 2011, Igor Mineyev, 2011) — Group theory
• Density theorem (Hossein Namazi, Juan Souto, 2010) — Group theory
• Full classification of finite simple groups (Koichiro Harada, Ronald Solomon, 2008) — Group theory
• André–Oort conjecture (Jonathan Pila, Ananth Shankar, Jacob Tsimerman, 2021) — Number theory · 21st century
• Duffin–Schaeffer theorem (Dimitris Koukoulopoulos, James Maynard, 2019) — Number theory · 21st century
• Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015) — Number theory · 21st century
• Goldbach's weak conjecture (Harald Helfgott, 2013) — Number theory · 21st century
• Existence of bounded gaps between arbitrarily large primes (Yitang Zhang, Polymath8, James Maynard, 2013) — Number theory · 21st century
• Sidon set problem (Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa, 2010) — Number theory · 21st century
• Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008) — Number theory · 21st century
• Green–Tao theorem (Ben J. Green and Terence Tao, 2004) — Number theory · 21st century
• Catalan's conjecture (Preda Mihăilescu, 2002) — Number theory · 21st century
• Erdős–Graham problem (Ernest S. Croot III, 2000) — Number theory · 21st century
• Lafforgue's theorem (Laurent Lafforgue, 1998) — Number theory · 20th century
• Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995) — Number theory · 20th century
• Burr–Erdős conjecture (Choongbum Lee, 2017) — Ramsey theory
• Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor W. Marek, 2016) — Ramsey theory
• Sensitivity conjecture for Boolean functions (Hao Huang, 2019) — Theoretical computer science
• Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020) — Topology
• Virtual Haken conjecture — Topology
• Hsiang–Lawson's conjecture (Simon Brendle, 2012) — Topology
• Ehrenpreis conjecture (Jeremy Kahn, Vladimir Markovic, 2011) — Topology
• Atiyah conjecture for groups with finite subgroups of unbounded order (Austin, 2009) — Topology
• Cobordism hypothesis (Jacob Lurie, 2008) — Topology
• Spherical space form conjecture (Grigori Perelman, 2006) — Topology
• Poincaré conjecture (Grigori Perelman, 2002) — Topology
• Geometrization conjecture (Grigori Perelman, series of preprints in 2002–2003) — Topology
• Nikiel's conjecture (Mary Ellen Rudin, 1999) — Topology
• Disproof of the Ganea conjecture (Iwase, 1997) — Topology
• Erdős discrepancy problem (Terence Tao, 2015) — Uncategorised · 2010s
• Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015) — Uncategorised · 2010s
• Jeff Cheeger — Uncategorised · 2010s
• Gaussian correlation inequality (Thomas Royen, 2014) — Uncategorised · 2010s
• Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman, Aleksandar Nikolov, 2011) — Uncategorised · 2010s
• Bloch–Kato conjecture — Uncategorised · 2010s
• Kauffman–Harary conjecture (Thomas Mattman, Pablo Solis, 2009) — Uncategorised · 2000s
• Surface subgroup conjecture (Jeremy Kahn, Vladimir Markovic, 2009) — Uncategorised · 2000s
• Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Zhiqin Lu, 2007) — Uncategorised · 2000s
• Nirenberg–Treves conjecture (Nils Dencker, 2005) — Uncategorised · 2000s
• Lax conjecture (Adrian Lewis, Pablo Parrilo, Motakuri Ramana, 2005) — Uncategorised · 2000s
• The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004) — Uncategorised · 2000s
• Milnor conjecture (Vladimir Voevodsky, 2003) — Uncategorised · 2000s
• Kirillov's conjecture (Ehud Baruch, 2003) — Uncategorised · 2000s
• Kouchnirenko's conjecture (Bertrand Haas, 2002) — Uncategorised · 2000s
• n! conjecture (Mark Haiman, 2001) (and also Macdonald positivity conjecture) — Uncategorised · 2000s
• Kato's conjecture (Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philipp Tchamitchian, 2001) — Uncategorised · 2000s
• Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001) — Uncategorised · 2000s
• Modularity theorem (Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, 2001) — Uncategorised · 2000s
• Erdős–Stewart conjecture (Florian Luca, 2001) — Uncategorised · 2000s
• Berry–Robbins problem (Michael Atiyah, 2000) — Uncategorised · 2000s