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List of unsolved problems in mathematics

Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention.

This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative. Although this list may never be comprehensive, the problems listed here vary widely in both difficulty and importance.

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

Millennium Prize Problems
Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:

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

The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman in 2003. However, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is unsolved.

Notebooks

 * The Kourovka Notebook (Коуровская тетрадь) is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.
 * The Sverdlovsk Notebook (Свердловская тетрадь) is a collection of unsolved problems in semigroup theory, first published in 1969 and updated many times since.
 * The Dniester Notebook (Днестровская тетрадь) lists several hundred unsolved problems in algebra, particularly ring theory and modulus theory.
 * The Erlagol Notebook (Эрлагольская тетрадь) lists unsolved problems in algebra and model theory.

Algebra

 * 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.
 * Bombieri–Lang conjectures on densities of rational points of algebraic surfaces and algebraic varieties defined on number fields and their field extensions.
 * 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?
 * Zilber–Pink conjecture that if $$X$$ is a mixed Shimura variety or semiabelian variety defined over $$\mathbb{C}$$, and $$V \subseteq X$$ is a subvariety, then $$V$$ contains only finitely many atypical subvarieties.

Group theory

 * 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
 * Burnside problem: for which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) 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?
 * 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 of 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 multivalent 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

Transcendental numbers and diophantine approximation

 * The four exponentials conjecture: the transcendence of at least one of four exponentials of combinations of irrationals
 * Schanuel's conjecture on the transcendence degree of exponentials of linearly independent irrationals
 * Are $$\gamma$$ (the Euler–Mascheroni constant),$$\pi + e, \pi - e, \pi e, \pi/e, \pi^e, \pi^{\sqrt{2}}, \pi^{\pi}, e^{\pi^2}, \ln\pi, 2^e, e^e$$, Catalan's constant, or Khinchin's constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?

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?
 * Map folding – various problems in map folding and stamp folding.
 * No-three-in-line problem – how many points can be placed in the $$n \times n$$ grid so that no three of them lie on a line?
 * Rudin's conjecture on the number of squares in finite arithmetic progressions
 * 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 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

 * 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 (aka 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.
 * Eremenko's conjecture: every component of the escaping set of an entire transcendental function is unbounded.
 * 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.
 * 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?
 * 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?

Combinatorial games

 * Sudoku:
 * How many puzzles have exactly one solution?
 * How many puzzles with exactly one solution are minimal?
 * What is the maximum number of givens for a minimal puzzle?
 * 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?
 * 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 $$A[t_{1}, \ldots, t_{n}]$$.
 * 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
 * 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
 * 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 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

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
 * 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

 * 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 Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n 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
 * How many unit distances can be determined by a set of $n$ points in the Euclidean plane?
 * Finding matching upper and lower bounds for k-sets and halving lines
 * 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?
 * 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 $$(n+1)^{n}/n!$$
 * The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?
 * 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
 * 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.
 * 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?
 * 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

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

 * Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs
 * Meyniel's conjecture that cop number is $$O(\sqrt n)$$

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?

 * Universal point sets of subquadratic size for planar graphs

Restriction of graph parameters

 * 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 K6-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 $t$ such that every $t$-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 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 $n$-vertex cubic graph?
 * The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.
 * The 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?
 * Vizing's conjecture on the domination number of cartesian products of graphs
 * 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
 * Characterise word-representable graphs in terms of (induced) forbidden subgraphs.
 * Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs )
 * Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter
 * Is 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?
 * Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?

Miscellaneous graph theory

 * 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
 * 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.
 * Shelah's eventual categoricity conjecture: For every cardinal $$\lambda$$ there exists a cardinal $$\mu(\lambda)$$ such that if an AEC K with LS(K)<= $$\lambda$$ is categorical in a cardinal above $$\mu(\lambda)$$ then it is categorical in all cardinals above $$\mu(\lambda)$$.
 * 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.

Probability theory

 * Ibragimov–Iosifescu conjecture for φ-mixing sequences

General

 * Beilinson's conjectures
 * Brocard's problem: are there any integer solutions to $$n! + 1 = m^{2}$$ other than $$n = 4, 5, 7$$?
 * 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 $$1$$?
 * 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?
 * Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating.
 * 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 $$\epsilon > 0$$, is the pair $$(\epsilon, 1/2 + \epsilon)$$ 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?
 * 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: 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 $$t$$?
 * 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 $$\epsilon > 0$$, there is some constant $$c(\epsilon)$$ such that either $$y^{2} = x^{3}$$ or $$|y^{2} - x^{3}| > c(\epsilon)x^{1/2 - \epsilon}$$.
 * 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 $$k$$-th order in a general algebraic number field, where $$k$$ is a power of a prime.
 * Hilbert's twelfth problem: extend the Kronecker–Weber theorem on Abelian extensions of $$\mathbb{Q}$$ to any base number field.
 * Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function
 * Lehmer's totient problem: if $$\phi(n)$$ divides $$n - 1$$, must $$n$$ 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 $$\epsilon > 0$$, $$\zeta(1/2 + it) = o(t^{\epsilon})$$
 * The density hypothesis for zeroes of the Riemann zeta function
 * Littlewood conjecture: for any two real numbers $$\alpha, \beta$$, $$\liminf_{n \rightarrow \infty} n\,\Vert n\alpha\Vert\,\Vert n\beta\Vert = 0$$, where $$\Vert x\Vert$$ is the distance from $$x$$ to the nearest integer.
 * 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 $$n$$.
 * 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 $$\epsilon > 0$$, $$\text{rad}(abc)^{1+\epsilon} < c$$ is true for only finitely many positive $$a, b, c$$ such that $$a + b = c$$.
 * Szpiro's conjecture: for any $$\epsilon > 0$$, there is some constant $$C(\epsilon)$$ such that, for any elliptic curve $$E$$ defined over $$\mathbb{Q}$$ with minimal discriminant $$\Delta$$ and conductor $$f$$, we have $$|\Delta| \leq C(\epsilon) \cdot f^{6+\epsilon}$$.
 * Newman's conjecture: the partition function satisfies any arbitrary congruence infinitely often.
 * 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 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 $$2^{n} - 1$$ is at most $$n - 1$$ plus the length of the shortest addition chain producing $$n$$.
 * 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?
 * 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?
 * 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?
 * Can a 3×3 magic square be constructed from 9 distinct perfect square numbers?
 * Find the value of the De Bruijn–Newman constant.

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 $$n$$?
 * 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 $$\sqrt{X}/\ln{X}$$
 * Selberg's 1/4 conjecture: the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least $$1/4$$.
 * 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?

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.
 * 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 $$1/m + 1/n + 1/k < 1$$.
 * 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

 * 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 2c − 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 $$e^{-1/2}$$?
 * 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 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 $$k\geq 1, b\geq 2, c\neq 0$$, with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form $$(k\times b^n+c)/\text{gcd}(k+c,b-1)$$ 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: These conjectures are about models of Zermelo-Frankel set theory with choice, and may not be able to be expressed in models of other set theories such as the various constructive set theories or non-wellfounded set theory.
 * (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 $$2^{\aleph_\omega} < \aleph_{\omega_1}$$ (see Singular cardinals hypothesis)? The best bound, ℵω 4, 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

 * 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?

Algebra

 * Mazur's conjecture B (Vessilin Dimitrov, Ziyang Gao, and Philipp Habegger, 2020)
 * Suita conjecture (Qi'an Guan and Xiangyu Zhou, 2015)
 * Torsion conjecture (Loïc Merel, 1996)
 * Carlitz–Wan conjecture (Hendrik Lenstra, 1995)
 * Serre's nonnegativity conjecture (Ofer Gabber, 1995)

Analysis

 * Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013) (and the Feichtinger's conjecture, Anderson's paving conjectures, Weaver's discrepancy theoretic $$KS_r$$ and $$KS'_r$$ conjectures, Bourgain-Tzafriri conjecture and $$R_\epsilon$$-conjecture)
 * Ahlfors measure conjecture (Ian Agol, 2004)
 * Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)

Combinatorics

 * Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)
 * McMullen's g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)
 * Hirsch conjecture (Francisco Santos Leal, 2010)
 * Gessel's lattice path conjecture (Manuel Kauers, Christoph Koutschan, and Doron Zeilberger, 2009)
 * Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004) (and also the Alon–Friedgut conjecture)
 * Kemnitz's conjecture (Christian Reiher, 2003, Carlos di Fiore, 2003)
 * Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)

Dynamical systems

 * Zimmer's conjecture (Aaron Brown, David Fisher, and Sebastián Hurtado-Salazar, 2017)
 * Painlevé conjecture (Jinxin Xue, 2014)

Game theory

 * Existence of a non-terminating game of beggar-my-neighbour (Brayden Casella, 2024)
 * The angel problem (Various independent proofs, 2006)

21st century

 * Einstein problem (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2023, still in peer review)
 * Maximal rank conjecture (Eric Larson, 2018)
 * Weibel's conjecture (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018)
 * Yau's conjecture (Antoine Song, 2018)
 * Pentagonal tiling (Michaël Rao, 2017)
 * Willmore conjecture (Fernando Codá Marques and André Neves, 2012)
 * Erdős distinct distances problem (Larry Guth, Nets Hawk Katz, 2011)
 * Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)
 * Tameness conjecture (Ian Agol, 2004)
 * Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)
 * Carpenter's rule problem (Robert Connelly, Erik Demaine, Günter Rote, 2003)
 * Lambda g conjecture (Carel Faber and Rahul Pandharipande, 2003)
 * Nagata's conjecture (Ivan Shestakov, Ualbai Umirbaev, 2003)
 * Double bubble conjecture (Michael Hutchings, Frank Morgan, Manuel Ritoré, Antonio Ros, 2002)

20th century

 * Honeycomb conjecture (Thomas Callister Hales, 1999)
 * Lange's conjecture (Montserrat Teixidor i Bigas and Barbara Russo, 1999)
 * Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)
 * Kepler conjecture (Samuel Ferguson, Thomas Callister Hales, 1998)
 * Dodecahedral conjecture (Thomas Callister Hales, Sean McLaughlin, 1998)

Graph theory

 * Kahn–Kalai conjecture (Jinyoung Park and Huy Tuan Pham, 2022)
 * Blankenship–Oporowski conjecture on the book thickness of subdivisions (Vida Dujmović, David Eppstein, Robert Hickingbotham, Pat Morin, and David Wood, 2021)
 * Ringel's conjecture that the complete graph $$K_{2n+1}$$ can be decomposed into $$2n+1$$ copies of any tree with $$n$$ edges (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)
 * Disproof of Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)
 * Kelmans–Seymour conjecture (Dawei He, Yan Wang, and Xingxing Yu, 2020)
 * Goldberg–Seymour conjecture (Guantao Chen, Guangming Jing, and Wenan Zang, 2019)
 * Babai's problem (Alireza Abdollahi, Maysam Zallaghi, 2015)
 * Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
 * Alon–Saks–Seymour conjecture (Hao Huang, Benny Sudakov, 2012)
 * Read–Hoggar conjecture (June Huh, 2009)
 * Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)
 * Erdős–Menger conjecture (Ron Aharoni, Eli Berger 2007)
 * Road coloring conjecture (Avraham Trahtman, 2007)
 * Robertson–Seymour theorem (Neil Robertson, Paul Seymour, 2004)
 * Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)
 * Toida's conjecture (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001)
 * Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen, 1996)

Group theory

 * Hanna Neumann conjecture (Joel Friedman, 2011, Igor Mineyev, 2011)
 * Density theorem (Hossein Namazi, Juan Souto, 2010)
 * Full classification of finite simple groups (Koichiro Harada, Ronald Solomon, 2008)

21st century

 * André–Oort conjecture (Jonathan Pila, Ananth Shankar, Jacob Tsimerman, 2021)
 * Duffin-Schaeffer conjecture (Dimitris Koukoulopoulos, James Maynard, 2019)
 * Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)
 * Goldbach's weak conjecture (Harald Helfgott, 2013)
 * Existence of bounded gaps between primes (Yitang Zhang, Polymath8, James Maynard, 2013)
 * Sidon set problem (Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa, 2010)
 * Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)
 * Green–Tao theorem (Ben J. Green and Terence Tao, 2004)
 * Catalan's conjecture (Preda Mihăilescu, 2002)
 * Erdős–Graham problem (Ernest S. Croot III, 2000)

20th century

 * Lafforgue's theorem (Laurent Lafforgue, 1998)
 * Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)

Ramsey theory

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

Theoretical computer science

 * Sensitivity conjecture for Boolean functions (Hao Huang, 2019)

Topology

 * Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020)
 * Virtual Haken conjecture (Ian Agol, Daniel Groves, Jason Manning, 2012) (and by work of Daniel Wise also virtually fibered conjecture)
 * Hsiang–Lawson's conjecture (Simon Brendle, 2012)
 * Ehrenpreis conjecture (Jeremy Kahn, Vladimir Markovic, 2011)
 * Atiyah conjecture for groups with finite subgroups of unbounded order (Austin, 2009)
 * Cobordism hypothesis (Jacob Lurie, 2008)
 * Spherical space form conjecture (Grigori Perelman, 2006)
 * Poincaré conjecture (Grigori Perelman, 2002)
 * Geometrization conjecture, (Grigori Perelman, series of preprints in 2002–2003)
 * Nikiel's conjecture (Mary Ellen Rudin, 1999)
 * Disproof of the Ganea conjecture (Iwase, 1997)

2010s

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

2000s

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


text:
Talk:1360

[Untitled]
King Valdemar Atterdag of Denmark seizes the Swedish province of Scania (Skåne).

I suppose what is meant is: Scania, which today is a Swedish province.

S.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:1360


text:
Gavrila Derzhavin

Gavriil (Gavrila) Romanovich Derzhavin (Гаврии́л (Гаври́ла) Рома́нович Держа́вин; 14 July 1743 – 20 July 1816) was one of the most highly esteemed Russian poets before Alexander Pushkin, as well as a statesman. Although his works are traditionally considered literary classicism, his best verse is rich with antitheses and conflicting sounds in a way reminiscent of John Donne and other metaphysical poets.

Early life and family
Derzhavin was born in the Kazan Governorate into a landed family of impoverished Russian nobility. His 15th-century Tatar ancestor Morza Bagrim, converted to Christianity and became a vassal of Grand Prince Vasily II. Bagrim was rewarded with lands for his service to the prince, and from him descended noble families of Narbekov, Akinfov and Keglev (or Teglev).

A member of the Narbekov family, who received the nickname Derzhava (Держа́ва; Дрьжа́ва; Russian for "orb" or "power"), was the patriarch of the Derzhavin family. The Derzhavins once held profitable estates along the Myosha River, about 25 mi from the capital city of Kazan, but over time they were divided, sold or mortgaged. By the time Gavrila Derzhavin's father, Roman Nikolayevich Derzhavin, was born in 1706, he stood to inherit only a few parcels of land, occupied by few peasants. Roman joined the military and in 1742, at age 36, he married a widowed distant relative Fyokla Andreyevna Gorina (née Kozlova). She was from a similar background and also possessed a few scattered estates. The estates were the source of constant lawsuits, fights and feuds with neighbors, sometimes extending into violence.

Derzhavin was born nearly nine months after his parents were wed, but the location of his birth remains a point of dispute. Derzhavin considered himself a native of Kazan — which proudly proclaims itself as the city of his birth — but he was possibly born at one of his family's estates in Sokury or Karmachi, in Laishevsky County. The Laishevsky District is informally known as the Derzhavinsky District because of its association with Derzhavin. He was named Gavriil (Russian for Gabriel), as his birth was 10 days before the Synaxis of the Archangel Gabriel, celebrated on 13 July in Slavic Orthodoxy. He was a sickly child, and his parents followed the traditional practice of the era and "baked the baby" («перепекание ребенка») — an ancient ceremony in which sickly or premature babies are placed on a bread peel and put in and out of the oven three times.

Derzhavin's father was transferred to Yaransk and then Stavropol. Two more children were born, a boy and a girl, although the latter died young.

Education
As members of the nobility, albeit minor, the Derzhavins were required to educate their children, but options were limited given their poverty and the few educational institutions in Russia at the time. Male members of the nobility were expected to enter government roles as civil servants or military officers at age 20. Nobility unable to send their children to one of the three educational institutions were given a waiver to educate their children at home, but the children were given examinations at 7, 12 and 16 to inspect their progress. Known as Ganyushka, Gavrilo's education began at age 3 when he was taught to read and write by local churchmen (as his mother was essentially illiterate). When he was 8, the family was sent to Orenburg near present-day Kazakhstan. The Russian Empire, eager to extend its reach, sent convicts to Orenburg to construct the city. A German named Joseph Rose opened a co-ed school to instruct the children of the nobility. Rose, in addition to being a criminal, had no formal education and was only able to instruct the children in the German language, which was then the most desirable language among the enlightened class in Russia.

When Gavrila was 10, the Derzhavins moved back to their estates in Kazan after two years in Orenburg. In the fall of 1753, he made his first trip to Moscow. Roman Derzhavin, who was suffering from consumption, needed to formally apply for retirement in Moscow, and then planned to continue to Saint Petersburg to register his son for future enlistment as required. However, he was delayed in Moscow until early January; by the time he received his discharge, he had no money to continue the journey to Saint Petersburg. They were forced to return to Kazan, where his father died later that year. His father owned half the land in Sokury, which Gavrila inherited along with other estates in Laishevsky. However, they provided very little income and the neighbors continued to encroach on their lands, flooding their estates or simply seizing land for themselves. His mother, a penniless widow with no powerful relatives, was unable to get any redress in the courts and was snubbed by judges. Derzhavin later wrote that his "mother's suffering from injustice remained eternally etched on his heart." Nevertheless, his mother was able to hire two tutors to teach her sons geometry and arithmetic.

In 1758, a new school opened in Kazan, saving his mother the difficulty of sending him to Saint Petersburg. The grammar school offered instruction in Latin, French, German, and arithmetic, as well as dancing, fencing and music. The instruction quality was still poor overall, with no textbooks. The school also offered opportunities for the students to perform tragedies by Molière and Alexander Sumarokov. Derzhavin eventually excelled in geometry and was informed he would be joining the corps of engineers in Saint Petersburg. However, a bureaucratic mistake led to him being made a private in the Preobrazhensky Regiment, the bodyguards of the royal family.

Career
In Saint Petersburg, Derzhavin rose from the ranks as a common soldier to the highest offices of state under Catherine the Great. He first impressed his commanders during Pugachev's Rebellion. Politically astute, his career advanced when he left the military service for civil service. He rose to the position of governor of Olonets (1784) and Tambov (1785), personal secretary to the Empress (1791), President of the College of Commerce (1794), and finally the Minister of Justice (1802).

He was dismissed from his post in 1803 and spent much of the rest of his life in the country estate at Zvanka near Novgorod, writing idylls and anacreontic verse. At his Saint Petersburg house, he held monthly meetings of the conservative Lovers of the Russian Word society. He died in 1816 and was buried in the Khutyn Monastery near Zvanka, reburied by the Soviets in the Novgorod Kremlin, and then reinterred at Khutyn.

Works
Derzhavin is best remembered for his odes, dedicated to the Empress and other courtiers. He paid little attention to the prevailing system of genres, and many a time would fill an ode with elegiac, humorous, or satiric contents. In his grand ode to the Empress, for instance, he mentions searching for fleas in his wife's hair and compares his own poetry with lemonade.

Unlike other Classicist poets, Derzhavin found delight in carefully chosen details, such as a colour of wallpaper in his bedroom or a poetic inventory of his daily meal. He believed that French was a language of harmony but that Russian was a language of conflict. Although he relished harmonious alliterations, sometimes he deliberately instrumented his verse with cacophonous effect.

Derzhavin's major odes were the impeccable "On the Death of Prince Meschersky" (1779); the playful "Ode to Felica" (1782); the lofty "God" (1785), which was translated into many European languages; "Waterfall" (1794), occasioned by the death of Prince Potemkin; and "Bullfinch"  (1800), a poignant elegy on the death of his friend Suvorov. He also provided lyrics for the first Russian national anthem, Let the thunder of victory sound!

In 1800, Derzhavin wrote the political work Opinion in response to a request by Emperor Paul I to investigate recent famines in Mogilev Governorate. In the Opinion, Derzhavin blamed Belarusian famines on the "mercenary trades" of Jews, who exploited peasants through leaseholding of estates and distilling of alcohol, as well as the indifference of the local magnates who allowed this exploitation to occur. In response to these issues, Derzhavin proposed a series of reforms to substantially restrict the freedoms of the magnates, abolish the Jewish Qahal, end the autonomy of the Russian Jewish community, and resettle Russian Jews in colonies along the Black Sea. The Opinion became an influential source of information during the early reign of Alexander I, who eventually implemented several of Derzhavin's suggested reforms in the 1804 Statute Concerning the Organization of the Jews.

Aleksandr Sozhenitsyn in his book - Two Hundred Years Together: history of the Jews in Russia, disputed the allegation of antisemitism taken from the Jewish Encyclopedia responding:

Influence
According to D.S. Mirsky, "Derzhavin's poetry is a universe of amazing richness; its only drawback was that the great poet was of no use either as a master or as an example. He did nothing to raise the level of literary taste or to improve the literary language, and as for his poetical flights, it was obviously impossible to follow him into those giddy spheres." Nevertheless, Nikolai Nekrasov professed to follow Derzhavin rather than Pushkin, and Derzhavin's line of broken rhythms was continued by Marina Tsvetaeva in the 20th century. The name of Gavrila Derzhavin was assigned to Tambov State University.

Memorable lines

 * Where used to be a table full of viands, a coffin now stands (Где стол был яств, там гроб стоит. Gde stol byl yastv, tam grob stoit)
 * I'm a czar - I'm a slave - I'm a worm - I'm a god (Я - царь, я - раб, я - червь, я - бог. Ya tsar, - ya rab, - ya cherv, - ya bog)
 * Heart of a lion, wings of an eagle are no longer with us! – How can we fight? (Львиного сердца, крыльев орлиных нет теперь с нами. Что воевать? Lvinogo serdtsa, krylyev orlinykh net teper s nami. Chto voyevat?)

Lines found at Derzhavin's table after his death

 * The current of Time's river
 * Will carry off all human deeds
 * And sink into oblivion
 * All peoples, kingdoms and their kings.

 * And if there's something that remains
 * Through sounds of horn and lyre,
 * It too will disappear into the maw of time
 * And not avoid the common pyre...

title: Gavrila Derzhavin


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Okayama Prefecture

Okayama Prefecture (岡山県) is a prefecture of Japan located in the Chūgoku region of Honshu. Okayama Prefecture has a population of 1,906,464 (1 February 2018) and has a geographic area of 7,114 km2 (2,746 sq mi). Okayama Prefecture borders Tottori Prefecture to the north, Hyōgo Prefecture to the east, and Hiroshima Prefecture to the west.

Okayama is the capital and largest city of Okayama Prefecture, with other major cities including Kurashiki, Tsuyama, and Sōja. Okayama Prefecture's south is located on the Seto Inland Sea coast across from Kagawa Prefecture on the island of Shikoku, which are connected by the Great Seto Bridge, while the north is characterized by the Chūgoku Mountains.

History
Prior to the Meiji Restoration of 1868, the area of present-day Okayama Prefecture was divided between Bitchū, Bizen and Mimasaka Provinces. Okayama Prefecture was formed and named in 1871 as part of the large-scale administrative reforms of the early Meiji period (1868–1912), and the borders of the prefecture were set in 1876.

Geography
Okayama Prefecture borders Hyōgo Prefecture, Tottori Prefecture, and Hiroshima Prefecture. It faces Kagawa Prefecture in Shikoku across the Seto Inland Sea and includes 90 islands in the sea.

Okayama Prefecture is home to the historic town of Kurashiki. Most of the population is concentrated around Kurashiki and Okayama. The small villages in the northern mountain region are aging and declining in population - more than half of the prefecture's municipalities are officially designated as depopulated.

As of 1 April 2014, 11% of the total land area of the prefecture was designated as Natural Parks, namely the Daisen-Oki and Setonaikai National Parks; the Hyōnosen-Ushiroyama-Nagisan Quasi-National Park; and seven Prefectural Natural Parks.

Cities
Fifteen cities are located in Okayama Prefecture:

Towns and villages
These are the towns and villages in each district:

Demographics
Per Japanese census data, and, Okayama prefecture has had continual negative population growth since 2005

Universities

 * Okayama
 * Okayama University
 * Notre Dame Seishin University
 * Okayama University of Science
 * Okayama Shoka University
 * Sanyo Gakuen University
 * Shujitsu University
 * Kurashiki
 * Okayama Gakuin University
 * Kurashiki Sakuyo University
 * Kawasaki University of Medical Welfare
 * Soja
 * Okayama Prefectural University
 * Tsuyama
 * Mimasaka University
 * Niimi
 * Niimi Public University

High schools

 * Okayama
 * Okayama Ichinomiya Senior High School
 * Okayama Asahi Senior High School
 * Okayama Sozan Senior High School
 * Okayama Hosen Senior High School
 * Okayama Joto Senior High School
 * Okayama Sakuyo High School
 * Kurashiki High School

Rail

 * JR West
 * Sanyo Shinkansen
 * Sanyo Line
 * Hakubi Line
 * Tsuyama Line
 * Kibi Line
 * Ako Line
 * Uno Line
 * Kishin Line
 * Geibi Line
 * Inbi Line
 * JR West and JR Shikoku
 * Seto-Ōhashi Line
 * Honshi-bisan Line
 * Chizu Express
 * Ibara Railway
 * Mizushima Rinkai Railway

Tramways

 * Okayama Electric Tramway

Expressways

 * Sanyo Expressway
 * Chugoku Expressway
 * Seto Central Expressway
 * Yonago Expressway
 * Okayama Expressway
 * Tottori Expressway

National highways

 * Route 2 (Osaka-Kobe-Himeji-Bizen-Okayama-Kurashiki-Asakuchi-Onomichi-Hiroshima-Shūnan-Shimonoseki-Kitakyushu)
 * Route 30 (Okayama-Uno-Takamatsu
 * Route 53 (Okayama-Tsuyama-Tottori)
 * Route 179
 * Route 180 (Okayama-Takahashi-Niimi)
 * Route 181 (Tsuyama-Maniwa-Yonago-Yasugi-Matsue)
 * Route 182
 * Route 183
 * Route 250 (Okayama-Setouchi-Ako-Aioi-Takasago-Akashi)
 * Route 313 (Fukuyama-Takahashi-Maniwa-Kurayoshi)
 * Route 373
 * Route 374
 * Route 429
 * Route 430
 * Route 482 (Kyotango-Toyooka-Wakasa-Kagamino-Maniwa-Kōfu of Tottori
 * Route 484

Airport

 * Okayama Airport

Culture

 * Bizen-yaki (Bizen pottery)
 * Bizen Osafune/Bitchu Aoe swords

Association with Momotarō legend
Okayama Prefecture is closely associated with the folklore hero, Momotarō. This tale is said to have roots in the legendary story of Kibitsuhiko-no-mikoto and Ura which explains that the Prince Ura of Kudara used to live in Kinojo (castle of the devil) and was a cause of trouble for the people living in the village. The emperor's government sent Kibitsuhiko-no-mikoto (Momotarō) to defeat Ura. The city of Okayama holds an annual Momotarō-matsuri, or Momotarō Festival.

Arts

 * Okayama Prefectural Museum
 * Okayama Orient Museum
 * Okayama Prefectural Museum of Art
 * Hayashibara Museum of Art
 * Yumeji Art Museum
 * Okayama Symphony Hall
 * Inryoji Temple

Sports
The sports teams listed below are based in Okayama.

Football

 * Fagiano Okayama FC (Okayama city)
 * Mitsubishi Motors Mizushima FC (Kurashiki)

Volleyball

 * Okayama Seagulls (Okayama city)

Basketball

 * Tryhoop Okayama (B3 League, Okayama city)

Tourism
Some tourist attractions are:
 * Koraku-en Japanese garden in Okayama
 * Okayama Castle, Okayama
 * Ki Castle, Sōja
 * Shizutani School, Bizen
 * Bikan Historical Area (倉敷美観地区), Kurashiki
 * Ohara Museum of Art, Kurashiki
 * Brazilian Park Washuzan Highland, Kurashiki
 * Bitchu Matsuyama Castle, Takahashi
 * Kakuzan Park, Tsuyama
 * Bisei Astronomical Observatory (美星天文台), Ibara Town (following dissolution of Bisei Town)
 * Maki-do Cave, in Niimi

Notable people

 * Yuko Arimori (born 1966), marathon runner
 * Kenji Doihara (1883–1948), army officer
 * Tesshō Genda (born 1948), voice actor
 * Morihiro Hashimoto (1977–2017), darts player
 * Naoki Hoshino (1892–1978), politician
 * Koshi Inaba (born 1964), singer
 * Masaki Kajishima (born 1962), creator of Tenchi Muyo!
 * Shiro Kawase (1889–1946), admiral
 * Sadahiko Miyake (1891–1956), general
 * Chiura Obata (1885–1975), artist
 * Mori Takashi, former member of Gentouki
 * Yōji Takikawa (born 1949), pedagogist
 * Mutsuo Toi (1917–1938), perpetrator of the Tsuyama massacre
 * Inukai Tsuyoshi (1855–1932), former Prime Minister of Japan
 * Jiro Watanabe (born 1955), boxer
 * Takeo Yasuda (1889–1964), lieutenant general
 * Eisuke Yoshiyuki (1906–1940), author
 * Fujii Kaze (born 1997), singer
 * Nishimura Riki, also known as Ni-ki of ENHYPEN (born 2005), K-pop idol, dancer, singer, rapper
 * Issei Mamehara (born 2002), singer, dancer, member of JO1

title: Okayama Prefecture


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Sausage roll

A sausage roll is a savoury dish, popular in current and former Commonwealth nations, consisting of sausage meat wrapped in puff pastry. Although variations are known throughout Europe and in other regions, the sausage roll is most closely associated with British cuisine.

Composition
The basic composition of a sausage roll is sheets of puff pastry formed into tubes around sausage meat and glazed with egg or milk before being baked. They can be served either hot or cold. In the 19th century, they were made using shortcrust pastry instead of puff pastry.

A vegetarian or vegan approximation of a sausage roll can be made in the same manner, using a meat substitute.

Sales
In the UK, the bakery chain Greggs sells around 2.5 million sausage rolls per week, or around 140 million per year.

History
The wrapping of meat or other foodstuffs into dough can be traced back to the Classical Greek or Roman eras. Early versions of the roll with pork as a filling proved popular in London during the Napoleonic Wars and it became identified as an English dish.

On 20 September 1809, the Bury and Norwich Post mentions T. Ling, aged 75, (an industrious vendor of saloop, buns, and sausage rolls). The Times first mentions the food item in 1864 when William Johnstone, "wholesale pork pie manufacturer and sausage roll maker", was fined £15 (£0 in 2021), under the Nuisances Removal Act (Amendment) Act 1863, for having on his premises a large quantity of meat unsound, unwholesome and unfit for food. In 1894, a theft case provided further insights into the Victorian sausage roll production whereby the accused apprentice was taught to soak brown bread in red ochre, salt, and pepper to give the appearance of beef sausage for the filling.

National variants
Similar meat and pastry recipes include the Czech klobásník, the Belgian worstenbroodje, the Dutch saucijzenbroodje, the German Münsterländer Wurstbrötchen and sausage bread in the United States.

Hong Kong has developed its own style of sausage roll. Instead of having sausage meat wrapped in puff pastry like the traditional western style, the Hong Kong style "sausage bun" (Chinese: 腸仔包) consists of a sausage wrapped inside a soft milk bread style bun.

In popular culture

 * The 1896 Gilbert and Sullivan operetta The Grand Duke features sausage rolls as a plot device, where conspirators recognise one another by eating sausage rolls.
 * From 2018 to 2022, English YouTuber LadBaby had five consecutive Christmas number ones on the UK Singles Chart with cover versions of songs where the lyrics were rewritten to reference sausage rolls, including "I Love Sausage Rolls", "Don't Stop Me Eatin'", and "Sausage Rolls for Everyone".

title: Sausage roll


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Red Rooster

Red Rooster is an Australian fast food chain. It sells roast chicken alongside common fast food items, such as burgers, chips, salads, and beverages. Since 2021, it has offered fried chicken.

In 1972, Peter and Theo Kailis opened the first Red Rooster in Kelmscott, a suburb of Perth. They sought to compete with American fast food chains such as Kentucky Fried Chicken, which came to Australia in 1968, by emulating their standardised model. Red Rooster proved successful, growing to 45 stores in Western Australia and Victoria before Myer bought it in 1981. Myer's 1986 purchase and merger of another chain, Big Rooster, into Red Rooster expanded it into the eastern states.

In 2002, Chicken Treat owner Australian Fast Foods (AFF) acquired Red Rooster. It changed hands between private equity firms several times and is currently owned by Craveable Brands, a holding company of PAG Asia Capital which also owns Chicken Treat and Oporto. In 2010, most Red Roosters were converted into franchises. While it has experienced a decline in popularity in recent years, as of 2021, Red Rooster is the sixth-most popular fast food restaurant in Australia.

Beginnings (1972–1981)
In 1972, Peter and Theo Kailis opened the first Red Rooster in Kelmscott, a suburb of Perth. The brothers were inspired by a chicken shop on Wanneroo Road. It marked a departure from their family background in fishing, pearling, and seafood. Unlike local restaurants, Red Rooster sought to compete with American fast food chains by emulating their standardised menu, branding, and marketing strategies.

At the time, American franchises such as McDonald's were yet to expand to Western Australia. They entered the Australian market in 1968, when Kentucky Fried Chicken (KFC) opened in Sydney, proving popular with the country's fledgling restaurant scene.

Red Rooster quickly proved successful. In July 1981, when the Myer Emporium bought the chain for $8.97 million, it was the fourth-largest fast food group in Australia. It had 28 locations in Western Australia and 12 in Victoria, alongside five Red Bull hamburger outlets, which were separate buildings on the same site as Red Roosters. After the deal, Peter Kailis retained his positions as chairman and general manager.

Myer (1982–2002)
In 1986, Coles Myer bought the Big Rooster chain to expand into the eastern states (except non-Steggles' Queensland stores, formerly known as "Big Rooster", which were purchased in 1992), and renamed the stores "Red Rooster". Big Rooster remains operational in Papua New Guinea.

Modern era (2003–)
In 2002, Red Rooster was purchased by Western Australian company Australian Fast Foods, which owned the competing Chicken Treat fast food chain. In 2007, both chains were sold for $180 million to a consortium formed by the management and Quadrant Private Equity.

In 2009, the Red Rooster chain in New Zealand closed its stores. The first New Zealand outlet, in Takanini, had opened in December 2004.

In 2010, Red Rooster changed company-owned stores to franchises. In 2011, Quadrant Private Equity sold parent company Quick Service Restaurant Holdings (later renamed Craveable Brands) to Archer Capital.

In 2019, ownership switched to PAG Asia Capital, a private equity group based in Hong Kong, who bought Craveable Brands for about $500 million. In Queensland later that year, seven Red Rooster stores on the Sunshine Coast closed when the franchisee went into voluntary administration.

The demographer Bernard Salt has noted that Red Rooster restaurants in Sydney are almost all in Greater Western Sydney, with the 'Red Rooster Line' dividing the city between the richer east and north, and the poorer west and south.

Marketing and promotions
In 2009, Red Rooster ran an advertising campaign called "They don't get it in America" featuring comedian Tom Gleeson in the United States asking people about Red Rooster.

In 2010, Red Rooster was a sponsor of Supercars Championship team Holden Racing Team. In 2016, the team returned as the title sponsor of the Sydney SuperNight 300.

In 2011, Red Rooster changed to promoting its restaurants as healthy, fresh and quick.

Red Rooster launched its trial delivery service through Menulog in September 2014 from the Baulkham Hills, New South Wales restaurant, in partnership with Menulog. As well as delivery to homes, it was announced delivery options to businesses, sporting clubs and local organisations would be available.

title: Red Rooster


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Wikipedia talk:How to use the Recent Changes page

Is there a URL for changes from, say, 8:45 to 9:45? For example, I checked recent changes at 11:00 and there were so many that I got them from 9:45, instead of the desired 8:45. It is a bit wasteful on server resources to ask now for all those from 8:45 (using a higher limit). - Patrick 10:00 Feb 25, 2003 (UTC)


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Talk:List of comedy television series

Organisation of Page
Shouldn't the USA and UK have their own seperate pages, othewise it could become extremely long.--RMHED 01:42, 24 July 2006 (UTC)

 * I arrived here from a link to "list of comedies" and didn't realize I was looking at a page that was devoted only to television comedies. The organization of the page didn't help. When I got down to the list of stage comedies, there was no "See also" to explain why there were so few other types of comedies listed.

 * I'm wondering why it makes sense at all to have such an extensive list of television comedies if no comparable list of other comedies is offered. The implication is that Wikipedia values television comedies more than other types. (Toddmg)

 * Considering the actual page TITLE is "List of comedy television series", and then at the end there are two more lists: Theatrical and Radio Comedies, this is crazy. We need to break the two other lists off as separate pages. There are quite a few other WP comedy lists: Films (broken down into individual decades) Sitcoms (should be deleted, probably) Situation Comedies (hmmm!!) Comedy-Drama Films, etc, etc, take a look yourself...   I propose I (or someone else if they beat me to it) break off the two Theatrical and Radio lists into new pages.  Does anyone object to that?  It's now end July 2012, if no-one has objected by end August, I'll go ahead and do it then. TheBustopher (talk) 16:07, 29 July 2012 (UTC)

Fall and Rise of Reginald Perrin
The Fall and Rise of Reginald Perrin was Fall and Rise in that order. For those of you who may not know, the central character Reggie Perrin faked his own death and returned, hence fall, then rise. --rbrwr

Dinosaur Pictures?
Anyone else think it's kind of weird how the only picture on the page is of Dinosaurs?
 * Because when I think of comedy, I think of Dinosaurs. Mole 20:17, 26 Jul 2004 (UTC)
 * Well, the pictures are all gone now, so the point is moot TheBustopher (talk) 16:13, 29 July 2012 (UTC)

Red Dwarf UK
Hey i was just wondering that the Red Dwarf on the UK part says 1988 - present. However, the series ended a while ago now so i thought somebody could find the year it ended? Thanks --Lewisjg 20:13, 21 December 2006 (UTC)
 * Technically this is true, there is a tenth series of Red Dwarf planned for 2012 (let's keep our fingers crossed). Series 9 was "Back to Earth" - a special 3 part series made in 2008 (broadcast in 2009).  Series 8 was sort of the last full-on shows in 1999 which I guess is sort of the answer you seek. TheBustopher (talk) 13:16, 9 December 2011 (UTC)

Friends.
Friends appears to be listed under both the UK and the US. Since the shows are listed under their country of origin, should Friends not be removed from the UK section? (Unless I'm missing a subtlety.) Paulbuckingham (talk) 03:06, 16 April 2010 (UTC)

WorldCat Genres
Maximilianklein (talk) 23:06, 5 December 2012 (UTC)

Children's stuff
Should children's TV be listed? Things like cartoons? I can understand putting in things that are made specifically as comedy, but if cartoons are added then almost every cartoon ever made will qualify and the quality of the page will decrease significantly. At the very least, children's comedy should have its own section or - more likely - its own page.Yb2 (talk) 21:12, 15 April 2015 (UTC)

Perhaps add this to list of US comedies
https://en.wikipedia.org/wiki/Kevin_Can_F**k_Himself Flakey712 (talk) 19:49, 23 September 2023 (UTC)

title: Talk:List of comedy television series


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Yamagata Prefecture

Yamagata Prefecture (山形県) is a prefecture of Japan located in the Tōhoku region of Honshu. Yamagata Prefecture has a population of 1,028,055 (1 August 2023) and has a geographic area of 9,325 km2 (3,600 sq mi). Yamagata Prefecture borders Akita Prefecture to the north, Miyagi Prefecture to the east, Fukushima Prefecture to the south, and Niigata Prefecture to the southwest.

Yamagata is the capital and largest city of Yamagata Prefecture, with other major cities including Tsuruoka, Sakata, and Yonezawa. Yamagata Prefecture is located on Japan's western Sea of Japan coast and its borders with neighboring prefectures are formed by various mountain ranges, with 17% of its total land area being designated as Natural Parks. Yamagata Prefecture formed the southern half of the historic Dewa Province with Akita Prefecture and is home to the Three Mountains of Dewa, which includes the Haguro Five-story Pagoda, a recognised National Treasure of Japan.

History
The aboriginal Ezo (蝦夷) people once inhabited the area now known as Yamagata. Yamagata and Akita Prefecture were known as Dewa Province until the Meiji Restoration.

During the Heian period (794–1185), the Fujiwara (藤原) family ruled the area. Yamagata City flourished during the Edo period (1603–1867) due to its status as a castle town and post station, famous for beni (red safflower dye used in the production of handspun silk). In 1689, the famous haiku poet, Matsuo Bashō visited Yamagata during his five-month trip to the northern regions of Japan.

Geography

Yamagata Prefecture is located in the southwest corner of Tōhoku, facing the Sea of Japan. It borders Niigata Prefecture and Fukushima Prefecture on the south, Miyagi Prefecture on the east, and Akita Prefecture on the north. All of these boundaries are marked by mountains, with most of the population residing in a limited central plain.

As of 31 March 2020, 17 percent of the total land area of the prefecture was designated as Natural Parks, namely the Bandai-Asahi National Park; Chōkai, Kurikoma, and Zaō Quasi-National Parks; and Goshōzan, Kabusan, Kennan, Mogamigawa, Shōnai Kaihin, and Tendō Kōgen Prefectural Natural Parks.

Cities
Thirteen cities are located in Yamagata Prefecture:

Towns and villages
These are the towns and villages in each district:

Climate
The climate of Yamagata Prefecture is characterized by long, hot, and humid summers and long, snowy winters. Both spring and autumn are short, the former often cold, the latter often warm, but both quite dry and sunny. Yamagata Prefecture, along with northern parts of Miyagi and Iwate are the transition areas from humid subtropical climate (Koppen Cfa/Cwa) to humid continental within the Japan mainland. Winter temperatures rarely fall below -10 °C in populated areas; they frequently rise above 30 °C in July and August. Precipitation falls all year round and the remnants of one or perhaps two typhoons usually pass through between August and October. The winters see heavy snowfall especially at higher elevations, though the Japan Sea coast (Sakata) is milder and has more rain. Snowfall for Shinjō is typical of populated mountainous areas, snowfall for Yamagata City typical of the valleys. The central mountains around Gassan may see as much as 3000. cm of snow in a season with depths up to 8 m at higher elevations.

Fruit

Yamagata Prefecture is the largest producer of cherries and pears in Japan. A large quantity of other kinds of fruits such as grapes, apples, peaches, melons, persimmons and watermelons are also produced.

Demographics
As of October 2020, Yamagata Prefecture had a population of 1,068,027. As of October 2019, 33.4% of the population was over 65 with 15.5% aged between 65 and 74 and 17.9% over 75. This is an increase of 8.5% over census data from 2004 where 12.8% of the population was aged between 65 and 74, and 12.1% was over 75.

Yamagata prefecture experienced its greatest growth period following the end of World War II, but then quickly began to slow down and eventually decline steadily. Today, the population is nearly the same as it was in 1930.

City life
Yamagata City is the central hub for Yamagata Prefecture and has many shops and restaurants around its main station. It is also the location with the most western hotels and the primary place of lodging for visitors visiting this region. This city serves as a good hub to visit the surrounding cities and towns around this prefecture with bus lines and train lines linking almost every part of Yamagata from the station.

There are also many bars near the station giving this city a good night life for visitors to enjoy at the front of the station as well as the nanukamachi district in the downtown area of the city.

Festivals and events
Yamagata Prefecture has a number of annual festivals and events.

The largest is the Yamagata Hanagasa Festival (花笠祭り) which takes place in Yamagata City on the first weekend in August, when thousands of people perform the hanagasa dance in the city centre and attracts up to 300,000 spectators. Yamagata City is the home of the bi-annual Yamagata International Documentary Film Festival in October.

In February, a snow lantern festival is held in Yonezawa at the Uesugi Shrine. Hundreds of candle-lit lanterns light pathways dug into the snow around the shrine. Yonezawa is also the site of the Uesugi Festival (上杉祭り, uesugi matsuri) in mid-spring. The festival's highlight is a re-enactment of the Battle of Kawanakajima on the banks of the Matsukawa River.

In September, Yamagata Prefecture is famous for its imoni, a taro-root stew popular in Northern Japan during the autumn. Imonikai, taro-root soup parties, are very popular during this season, and many tourists come to Yamagata Prefecture specifically for its particular style of imoni.

Art
Beginning in 2003, Yamagata city officials with the aid of Tōhoku University of Art and Design began a three-year project in which the Buddhist art of the city's temples would be catalogued and compared to a set of guidelines in order to identify "cultural assets". One hundred and ninety temples have had their works of art examined and several significant examples of Buddhist sculpture have been discovered. At Heisenji Temple, in the Hirashimizu district, a particularly rare statue, a seated Vairocana Buddha made from zelkova wood, was found. Other significant works include sculptures from the Heian period (794–1185) and Kamakura period (1192–1333).

The Yamagata Museum of Art, located in Yamagata City, was opened in 1964 through the efforts of a foundation led by Yoshio Hattori, the president of Yamagata Shimbun and Yamagata Broadcasting Co, Ltd. The permanent collection consists of three types of art: Japanese and Asian, regional, and French. Special exhibitions are held periodically.

Language
Yamagata Prefecture is known for its local dialect Yamagata-ben, sometimes thought of as backward sounding in other parts of Japan. The 2004 movie Swing Girls (スウィングガールズ), co-written and directed by Shinobu Yaguchi, is set in Yamagata and makes use of Yamagata-ben for comedic purposes.

Sports
The sports teams listed below are based in Yamagata.

Football
 * Montedio Yamagata

Volleyball
 * Pioneer Red Wings (Tendō)

Basketball
 * Passlab Yamagata Wyverns

Airports

 * Yamagata Airport (Tokyo, Osaka, Nagoya, Sapporo by Japan Airlines)
 * Shonai Airport (Tokyo, Osaka, Sapporo by All Nippon Airways)

Railways

 * Yamagata Shinkansen
 * Senzan Line
 * Aterazawa Line
 * Yonesaka Line
 * Rikuu West Line
 * Rikuu East Line
 * Flower Nagai Line
 * Ōu Main Line
 * Uetsu Main Line

Tourism
The temple of Yama-dera, carved into the mountainside near the city of Yamagata, is a major attraction.

The Dewa Sanzan are three holy mountains that form a traditional pilgrimage for followers of the Shugendō branch of Shintō. The famous Gojudo (five-story pagoda) is at the base of Mount Haguro, the lowest of the three mountains.

Mount Zaō is a famous winter ski resort, also known for its snow monsters (frozen snow-covered trees) in the winter, and the Okama crater lake, also known as the Goshiki Numa (Five Color Lake) because its colour changes according to the weather.

Ginzan Onsen (銀山温泉) is a Silver Mountain hot spring town located in the mountains of Obanazawa City, Yamagata Prefecture. The area originally developed around a silver mine.

Newspapers

 * Yamagata Shimbun
 * Yonezawa Shimbun
 * Shonai Nippo
 * Okitama Times

TV and radio

 * YBC TV and Radio (NNN, JRN, and NRN)
 * YTS TV (ANN)
 * TUY TV (JNN)
 * SAY TV (FNN)
 * Rhythm Station (JFN)

Film
Yamagata is pictured in the 1991 Studio Ghibli film Only Yesterday directed by Isao Takahata.
 * Yamagata International Documentary Film Festival

title: Yamagata Prefecture