List of unsolved problems in mathematics Information
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 longstanding 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.
List  Number of problems 
Number unsolved or incompletely solved 
Proposed by  Proposed in 

Hilbert's problems^{ [1]}  23  15  David Hilbert  1900 
Landau's problems^{ [2]}  4  4  Edmund Landau  1912 
Taniyama's problems^{ [3]}  36    Yutaka Taniyama  1955 
Thurston's 24 questions^{ [4]}^{ [5]}  24    William Thurston  1982 
Smale's problems  18  14  Stephen Smale  1998 
Millennium Prize Problems  7  6^{ [6]}  Clay Mathematics Institute  2000 
Simon problems  15  <12^{ [7]}^{ [8]}  Barry Simon  2000 
Unsolved Problems on Mathematics for the 21st Century^{ [9]}  22    Jair Minoro Abe, Shotaro Tanaka  2001 
DARPA's math challenges^{ [10]}^{ [11]}  23    DARPA  2007 
Millennium Prize Problems
Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:^{ [6]}
 Birch and SwinnertonDyer 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.^{ [12]} However, a generalization called the smooth fourdimensional Poincaré conjecture—that is, whether a fourdimensional topological sphere can have two or more inequivalent smooth structures—is unsolved.^{ [13]}
Unsolved problems
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 applied to a complex matrix is at most twice the supremum of over the field of values of .
 Demazure conjecture on representations of algebraic groups over the integers.
 Eilenberg–Ganea conjecture: a group with cohomological dimension 2 also has a 2dimensional Eilenberg–MacLane space .

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?^{ [14]}
 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 pcurvature conjecture: a conjectured local–global principle for linear ordinary differential equations.
 Hadamard conjecture: for every positive integer , a Hadamard matrix of order exists.
 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 Mcurves?
 Homological conjectures in commutative algebra
 Jacobson's conjecture: the intersection of all powers of the Jacobson radical of a leftandright Noetherian ring is precisely 0.
 Kaplansky's conjectures
 Köthe conjecture: if a ring has no nil ideal other than , then it has no nil onesided ideal other than .
 Existence of perfect cuboids and associated cuboid conjectures
 Pierce–Birkhoff conjecture: every piecewisepolynomial is the maximum of a finite set of minimums of finite collections of polynomials.
 Rota's basis conjecture: for matroids of rank with disjoint bases , it is possible to create an matrix whose rows are and whose columns are also bases.
 Sendov's conjecture: if a complex polynomial with degree at least has all roots in the closed unit disk, then each root is within distance from some critical point.
 Serre's conjecture II: if is a simply connected semisimple algebraic group over a perfect field of cohomological dimension at most , then the Galois cohomology set is zero.
 Serre's multiplicity conjectures
 Uniform boundedness conjecture for rational points: do algebraic curves of genus over number fields have at most some bounded number of  rational points?
 Wild problems: problems involving classification of pairs of matrices under simultaneous conjugation.
 Zariski–Lipman conjecture: for a complex algebraic variety with coordinate ring , if the derivations of are a free module over , then is smooth.
 Zauner's conjecture: do SICPOVMs exist in all dimensions?
Notebook problems
 The Dniester Notebook ( Russian: Днестровская тетрадъ) lists several hundred unsolved problems in algebra, particularly ring theory and modulus theory.^{ [15]}
 The Erlagol Notebook ( Russian: Эрлаголъская тетрадъ) lists unsolved problems in algebra and model theory.^{ [16]}
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
 The four exponentials conjecture: the transcendence of at least one of four exponentials of combinations of irrationals^{ [17]}
 Goodman's conjecture on the coefficients of multivalent functions
 Invariant subspace problem – does every bounded operator on a complex Banach space send some nontrivial closed subspace to itself?
 Kung–Traub conjecture on the optimal order of a multipoint iteration without memory^{ [18]}
 Lehmer's conjecture on the Mahler measure of noncyclotomic polynomials^{ [19]}
 The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy^{ [20]}
 Schanuel's conjecture on the transcendence degree of exponentials of linearly independent irrationals^{ [17]}
 Vitushkin's conjecture on compact subsets of with analytic capacity
 Are (the Euler–Mascheroni constant),, Catalan's constant, or Khinchin's constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?^{ [21]}^{ [22]}^{ [23]}
 What is the exact value of Landau's constants, including Bloch's constant?
 How are suspended infiniteinfinitesimals paradoxes justified?
 Regularity of solutions of Euler equations
 Convergence of Flint Hills series
 Regularity of solutions of Vlasov–Maxwell equations
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?^{ [24]}
 Problems in Latin squares – open questions concerning Latin squares
 The lonely runner conjecture – if runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance from each other runner) at some time?^{ [25]}
 The sunflower conjecture: can the number of size sets required for the existence of a sunflower of sets be bounded by an exponential function in for every fixed ?
 Nothreeinline problem – how many points can be placed in the grid so that no three of them lie on a line?
 Frankl's unionclosed 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^{ [26]}
 Give a combinatorial interpretation of the Kronecker coefficients^{ [27]}
 The values of the Dedekind numbers for ^{ [28]}
 The values of the Ramsey numbers, particularly
 The values of the Van der Waerden numbers
 Finding a function to model nstep selfavoiding walks^{ [29]}
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?^{ [30]}
 Birkhoff conjecture – if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse?^{ [31]}
 Collatz conjecture (aka the conjecture)
 Eremenko's conjecture: every component of the escaping set of an entire transcendental function is unbounded.
 Furstenberg conjecture – is every invariant and ergodic measure for the 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 higherrank 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 largefrequency eigenfunctions of the Laplacian on a negativelycurved manifold^{ [32]}
 Rokhlin's multiple mixing problem – are all strongly mixing systems also strongly 3mixing?^{ [33]}
 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 the Lyapunov’s second method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion?
 Is every reversible cellular automaton in three or more dimensions locally reversible?^{ [34]}
Games and puzzles
Combinatorial games
 Is there a nonterminating game of beggarmyneighbour?

Sudoku:
 How many puzzles have exactly one solution?^{
[35]}
 How many puzzles with exactly one solution are minimal?^{ [35]}
 What is the maximum number of givens for a minimal puzzle?^{ [35]}
 How many puzzles have exactly one solution?^{
[35]}

Tictactoe variants:
 Given a width of tictactoe board, what is the smallest dimension such that X is guaranteed a winning strategy?^{ [36]}
 What is the Turing completeness status of all unique elementary cellular automata?
Games with imperfect information
Geometry
Algebraic geometry
 Abundance conjecture: if the canonical bundle of a projective variety with Kawamata log terminal singularities is nef, then it is semiample.
 Bass conjecture on the finite generation of certain algebraic Kgroups.
 Deligne conjecture: any one of numerous named for Pierre Deligne.
 Dixmier conjecture: any endomorphism of a Weyl algebra is an automorphism.
 Fröberg conjecture on the Hilbert functions of a set of forms.
 Fujita conjecture regarding the line bundle constructed from a positive holomorphic line bundle on a compact complex manifold and the canonical line bundle of
 Hartshorne's conjectures^{ [37]}
 Jacobian conjecture: if a polynomial mapping over a characteristic0 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^{ [38]}
 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 Kgroups 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 finitelygenerated fields to the Galois group of .
 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^{ [39]}
 Are infinite sequences of flips possible in dimensions greater than 3?
 Resolution of singularities in characteristic
Covering and packing
 Borsuk's problem on upper and lower bounds for the number of smallerdiameter subsets needed to cover a bounded ndimensional set.
 The covering problem of Rado: if the union of finitely many axisparallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?^{ [40]}
 The Erdős–Oler conjecture: when is a triangular number, packing circles in an equilateral triangle requires a triangle of the same size as packing circles^{ [41]}
 The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24^{ [42]}
 Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrallysymmetric convex plane sets^{ [43]}
 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?^{ [44]}
 Ulam's packing conjecture about the identity of the worstpacking convex solid^{ [45]}
Differential geometry
 The spherical Bernstein's problem, a generalization of Bernstein's problem
 Carathéodory conjecture: any convex, closed, and twicedifferentiable surface in threedimensional 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 closelyrelated 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.^{ [46]}
 The filling area conjecture, that a hemisphere has the minimum area among shortcutfree surfaces in Euclidean space whose boundary forms a closed curve of given length^{ [47]}
 The Hopf conjectures relating the curvature and Euler characteristic of higherdimensional Riemannian manifolds^{ [48]}
 Yau's conjecture: a closed Riemannian 3manifold has an infinite number of smooth closed immersed minimal surfaces.
 Yau's conjecture on the first eigenvalue that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurface of is .
Discrete geometry
 The Hadwiger conjecture on covering ndimensional convex bodies with at most 2^{n} smaller copies^{ [49]}
 Solving the happy ending problem for arbitrary ^{ [50]}
 Improving lower and upper bounds for the Heilbronn triangle problem.
 Kalai's 3^{d} conjecture on the least possible number of faces of centrally symmetric polytopes.^{ [51]}
 The Kobon triangle problem on triangles in line arrangements^{ [52]}
 The Kusner conjecture: at most points can be equidistant in spaces^{ [53]}
 The McMullen problem on projectively transforming sets of points into convex position^{ [54]}
 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?^{ [55]}
 Finding matching upper and lower bounds for ksets and halving lines^{ [56]}
 Tripod packing:^{ [57]} 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 by matrix depending on points in ^{ [58]}
 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^{ [59]}
 Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?^{ [60]}
 Ehrhart's volume conjecture: a convex body in dimensions containing a single lattice point in its interior as its center of mass cannot have volume greater than
 The einstein problem – does there exist a twodimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?^{ [61]}
 Falconer's conjecture: sets of Hausdorff dimension greater than in must have a distance set of nonzero Lebesgue measure^{ [62]}
 Inscribed square problem, also known as Toeplitz' conjecture and the square peg problem – does every Jordan curve have an inscribed square?^{ [63]}
 The Kakeya conjecture – do dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to ?^{ [64]}
 The Kelvin problem on minimumsurfacearea partitions of space into equalvolume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem^{ [65]}
 Lebesgue's universal covering problem on the minimumarea convex shape in the plane that can cover any shape of diameter one^{ [66]}
 Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar.^{ [67]}
 Moser's worm problem – what is the smallest area of a shape that can cover every unitlength curve in the plane?^{ [68]}
 The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unitwidth Lshaped corridor?^{ [69]}
 Does every convex polyhedron have Rupert's property?^{ [70]}^{ [71]}
 Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net, or simple edgeunfolding?^{ [72]}^{ [73]}
 Is there a nonconvex polyhedron without selfintersections with more than seven faces, all of which share an edge with each other?
 The Thomson problem – what is the minimum energy configuration of mutuallyrepelling particles on a unit sphere?^{ [74]}
 Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?^{ [75]}
 Dissection into orthoschemes – is it possible for simplices of every dimension?^{ [76]}
 Uniform 5polytopes – find and classify the complete set of these shapes^{ [77]}
Graph theory
Graph coloring and labeling
 Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs^{ [78]}
 The Erdős–Faber–Lovász conjecture on coloring unions of cliques^{ [79]}
 The Gyárfás–Sumner conjecture on χboundedness of graphs with a forbidden induced tree^{ [80]}
 The Hadwiger conjecture relating coloring to clique minors^{ [81]}
 The Hadwiger–Nelson problem on the chromatic number of unit distance graphs^{ [82]}
 Jaeger's Petersencoloring conjecture: every bridgeless cubic graph has a cyclecontinuous mapping to the Petersen graph^{ [83]}
 The list coloring conjecture:, for every graph, the list chromatic index equals the chromatic index^{ [84]}
 The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree^{ [85]}
Graph drawing
 The Albertson conjecture: the crossing number can be lowerbounded by the crossing number of a complete graph with the same chromatic number^{ [86]}
 Conway's thrackle conjecture^{ [87]} that thrackles cannot have more edges than vertices
 Harborth's conjecture: every planar graph can be drawn with integer edge lengths^{ [88]}
 Negami's conjecture on projectiveplane embeddings of graphs with planar covers^{ [89]}
 The strong Papadimitriou–Ratajczak conjecture: every polyhedral graph has a convex greedy embedding^{ [90]}
 Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?^{ [91]}
 Universal point sets of subquadratic size for planar graphs^{ [92]}
Paths and cycles in graphs
 Barnette's conjecture: every cubic bipartite threeconnected planar graph has a Hamiltonian cycle^{ [93]}
 Chvátal's toughness conjecture, that there is a number t such that every ttough graph is Hamiltonian^{ [94]}
 The cycle double cover conjecture: every bridgeless graph has a family of cycles that includes each edge twice^{ [95]}
 The Erdős–Gyárfás conjecture on cycles with poweroftwo lengths in cubic graphs^{ [96]}
 The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree^{ [97]}
 The Lovász conjecture on Hamiltonian paths in symmetric graphs^{ [98]}
 The Oberwolfach problem on which 2regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edgedisjoint copies of the given graph.^{ [99]}
 Szymanski's conjecture: every permutation on the dimensional doubly directed hypercube graph can be routed with edgedisjoint paths.
Wordrepresentation of graphs
 Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter?^{ [100]}^{ [101]}^{ [102]}^{ [103]}
 Characterise (non) wordrepresentable planar graphs^{ [100]}^{ [101]}^{ [102]}^{ [103]}
 Characterise wordrepresentable graphs in terms of (induced) forbidden subgraphs.^{ [100]}^{ [101]}^{ [102]}^{ [103]}
 Characterise wordrepresentable neartriangulations containing the complete graph K_{4} (such a characterisation is known for K_{4}free planar graphs^{ [104]})
 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^{ [105]}
 Is it true that out of all bipartite graphs, crown graphs require longest wordrepresentants?^{ [106]}
 Is the line graph of a non wordrepresentable graph always non wordrepresentable?^{ [100]}^{ [101]}^{ [102]}^{ [103]}
 Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?^{ [100]}^{ [101]}^{ [102]}^{ [103]}
Miscellaneous graph theory
 Babai's problem: which groups are Babai invariant groups?
 Brouwer's conjecture on upper bounds for sums of eigenvalues of Laplacians of graphs in terms of their number of edges.
 Conway's 99graph problem: does there exist a strongly regular graph with parameters (99,14,1,2)?^{ [107]}
 Degree diameter problem: given two positive integers , what is the largest graph of diameter such that all vertices have degrees at most ?
 The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph^{ [108]}
 The GNRS conjecture on whether minorclosed graph families have embeddings with bounded distortion^{ [109]}
 Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs^{ [110]}
 The implicit graph conjecture on the existence of implicit representations for slowlygrowing hereditary families of graphs^{ [111]}
 Jørgensen's conjecture that every 6vertexconnected K_{6}minorfree graph is an apex graph^{ [112]}
 Meyniel's conjecture that cop number is ^{ [113]}
 The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertexdeleted subgraphs.^{ [114]}^{ [115]}
 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?^{ [116]}
 Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?^{ [117]}
 Sumner's conjecture: does every vertex tournament contain as a subgraph every vertex oriented tree?^{ [118]}
 Tutte's conjectures:
 every bridgeless graph has a nowherezero 5flow^{ [119]}
 every Petersen minorfree bridgeless graph has a nowherezero 4flow^{ [120]}
 Vizing's conjecture on the domination number of cartesian products of graphs^{ [121]}
 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?
 Does a Moore graph with girth 5 and degree 57 exist?^{ [122]}
 What is the largest possible pathwidth of an nvertex cubic graph?^{ [123]}
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
 Guralnick–Thompson conjecture on the composition factors of groups in genus0 systems^{ [124]}
 Herzog–Schönheim conjecture: if a finite system of left cosets of subgroups of a group form a partition of , 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?
 Problems in loop theory and quasigroup theory consider generalizations of groups
 Are there an infinite number of Leinster groups?
 Does generalized moonshine exist?
 For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
 Is every finitely presented periodic group finite?
 Is every group surjunctive?
Notebook problems
 The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.^{ [125]}
Model theory and formal languages
 The Cherlin–Zilber conjecture: A simple group whose firstorder theory is stable in 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^{ [126]}
 The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for saturated models of a countable theory.^{ [127]}
 Shelah's categoricity conjecture for : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.^{ [127]}
 Shelah's eventual categoricity conjecture: For every cardinal there exists a cardinal such that if an AEC K with LS(K)<= is categorical in a cardinal above then it is categorical in all cardinals above .^{ [127]}^{ [128]}
 The stable field conjecture: every infinite field with a stable firstorder theory is separably closed.
 The stable forking conjecture for simple theories^{ [129]}
 Tarski's exponential function problem: is the theory of the real numbers with the exponential function decidable?
 The universality problem for Cfree graphs: For which finite sets C of graphs does the class of Cfree countable graphs have a universal member under strong embeddings?^{ [130]}
 The universality spectrum problem: Is there a firstorder theory whose universality spectrum is minimum?^{ [131]}
 Vaught conjecture: the number of countable models of a firstorder complete theory in a countable language is either finite, , or .
 Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality does it have a model of cardinality continuum?^{ [132]}
 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 ominimal first order theory with a transexponential (rapid growth) function?
 If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?^{ [133]}^{ [134]}
 Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or cofinite.)
 Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of wellordering (MTWO) consistently decidable?^{ [135]}
 Is the theory of the field of Laurent series over decidable? of the field of polynomials over ?
 Is there a logic L which satisfies both the Beth property and Δinterpolation, is compact but does not satisfy the interpolation property?^{ [136]}
 Determine the structure of Keisler's order.^{ [137]}^{ [138]}
Number theory
General

n conjecture: a generalization of the abc conjecture to more than three integers.
 abc conjecture: for any , is true for only finitely many positive such that .
 Szpiro's conjecture: for any , there is some constant such that, for any elliptic curve defined over with minimal discriminant and conductor , we have .
 Hardy–Littlewood zetafunction conjectures
 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 th order in a general algebraic number field, where is a power of a prime.
 Hilbert's twelfth problem: extend the Kronecker–Weber theorem on Abelian extensions of to any base number field.

Grand Riemann hypothesis: do the nontrivial zeros of all
automorphic Lfunctions lie on the critical line with real ?

Generalized Riemann hypothesis: do the nontrivial zeros of all
Dirichlet Lfunctions lie on the critical line with real ?
 Riemann hypothesis: do the nontrivial zeros of the Riemann zeta function lie on the critical line with real ?

Generalized Riemann hypothesis: do the nontrivial zeros of all
Dirichlet Lfunctions lie on the critical line with real ?
 André–Oort conjecture: is every irreducible component of the Zariski closure of a set of special points in a Shimura variety a special subvariety?
 Beilinson's conjectures
 Brocard's problem: are there any integer solutions to other than ?
 Carmichael's totient function conjecture: do all values of Euler's totient function have multiplicity greater than ?
 CasasAlvero conjecture: if a polynomial of degree defined over a field of characteristic has a factor in common with its first through th derivative, then must be the th power of a linear polynomial?
 Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but nonrepeating.
 Congruent number problem (a corollary to Birch and SwinnertonDyer conjecture, per Tunnell's theorem): determine precisely what rational numbers are congruent numbers.
 Erdős–Moser problem: is the only solution to the Erdős–Moser equation?
 Erdős–Straus conjecture: for every , there are positive integers such that .
 Erdős–Ulam problem: is there a dense set of points in the plane all at rational distances from oneanother?
 Exponent pair conjecture: for all , is the pair an exponent pair?
 The Gauss circle problem: how far can the number of integer points in a circle centered at the origin be from the area of the circle?
 Goormaghtigh conjecture on solutions to where and .
 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 , there is some constant such that either or .
 Hilbert–Pólya conjecture: the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a selfadjoint operator.
 Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function^{ [139]}
 Lehmer's totient problem: if divides , must be prime?
 Leopoldt's conjecture: a padic analogue of the regulator of an algebraic number field does not vanish.

Lindelöf hypothesis that for all ,
 The density hypothesis for zeroes of the Riemann zeta function
 Littlewood conjecture: for any two real numbers , , where is the distance from to the nearest integer.
 Mahler's 3/2 problem that no real number has the property that the fractional parts of are less than for all positive integers .
 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.
 Newman's conjecture: the partition function satisfies any arbitrary congruence infinitely often.
 Pillai's conjecture: for any , the equation has finitely many solutions when are not both .

Piltz divisor problem on bounding
 Dirichlet's divisor problem: the specific case of the Piltz divisor problem for
 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 is at most plus the length of the shortest addition chain producing .
 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?^{ [140]}
 The uniqueness conjecture for Markov numbers^{ [141]} that every Markov number is the largest number in exactly one normalized solution to the Markov Diophantine equation.
 Vojta's conjecture on heights of points on algebraic varieties over algebraic number fields.
 Are there infinitely many perfect numbers?
 Do any odd perfect numbers exist?
 Do quasiperfect numbers exist?
 Do any nonpower 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 Taxicab(5, 2, n) exist for n > 1?
 Is there a covering system with odd distinct moduli?^{ [142]}
 Is a normal number (i.e., is each digit 0–9 equally frequent)?^{ [143]}
 Is 10 a solitary number?
 Can a 3×3 magic square be constructed from 9 distinct perfect square numbers?^{ [144]}
 Which integers can be written as the sum of three perfect cubes?^{ [145]}
 Can every integer be written as a sum of four perfect cubes?
 Find the value of the De Bruijn–Newman constant.
Additive number theory
 Beal's conjecture: for all integral solutions to where , all three numbers must share some prime factor.
 Erdős conjecture on arithmetic progressions that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long arithmetic progressions.
 Erdős–Turán conjecture on additive bases: if is an additive basis of order , then the number of ways that positive integers can be expressed as the sum of two numbers in must tend to infinity as tends to infinity.
 Fermat–Catalan conjecture: there are finitely many distinct solutions to the equation with being positive coprime integers and being positive integers satisfying .
 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 is the sum of two prime numbers.
 Lander, Parkin, and Selfridge conjecture: if the sum of th powers of positive integers is equal to a different sum of th powers of positive integers, then .
 Lemoine's conjecture: all odd integers greater than 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
 Pollock's conjectures
 Skolem problem: can an algorithm determine if a constantrecursive sequence contains a zero?
 The values of g(k) and G(k) in Waring's problem
 Do the Ulam numbers have a positive density?
 Determine growth rate of r_{k}(N) (see Szemerédi's theorem)
Algebraic number theory
 Class number problem: are there infinitely many real quadratic number fields with unique factorization?
 Fontaine–Mazur conjecture: actually numerous conjectures, all proposed by JeanMarc Fontaine and Barry Mazur.
 Gan–Gross–Prasad conjecture: a restriction problem in representation theory of real or padic Lie groups.
 Greenberg's conjectures
 Hermite's problem: is it possible, for any natural number , to assign a sequence of natural numbers to each real number such that the sequence for is eventually periodic if and only if is algebraic of degree ?
 Kummer–Vandiver conjecture: primes do not divide the class number of the maximal real subfield of the th cyclotomic field.
 Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant is within a constant multiple of
 Selberg's 1/4 conjecture: the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least .
 Stark conjectures (including Brumer–Stark conjecture)
 Characterize all algebraic number fields that have some power basis.
Computational number theory
 Can integer factorization be done in polynomial time?
Prime numbers
 Agoh–Giuga conjecture on the Bernoulli numbers that is prime if and only if
 Artin's conjecture on primitive roots that if an integer is neither a perfect square nor , then it is a primitive root modulo infinitely many prime numbers
 Brocard's conjecture: there are always at least prime numbers between consecutive squares of prime numbers, aside from and .
 Bunyakovsky conjecture: if an integercoefficient polynomial has a positive leading coefficient, is irreducible over the integers, and has no common factors over all where is a positive integer, then 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 with each , there are infinitely many for which all forms are prime, unless there is some congruence condition preventing it.
 Dubner's conjecture: every number greater than is the sum of two primes which both have twins.
 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 and , does not divide
 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.
 Goldbach conjecture: all even natural numbers greater than are the sum of two prime numbers.
 Landau's problems
 Problems associated to Linnik's theorem
 New Mersenne conjecture: for any odd natural number , if any two of the three conditions or , is prime, and is prime are true, then the third condition is true.
 Polignac's conjecture: for all positive even numbers , there are infinitely many prime gaps of size .
 Schinzel's hypothesis H that for every finite collection of nonconstant irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers for which are all primes, or there is some fixed divisor which, for all , divides some .
 Selfridge's conjecture: is 78,557 the lowest Sierpiński number?
 Twin prime conjecture: there are infinitely many twin primes.
 Does the converse of Wolstenholme's theorem hold for all natural numbers?
 Are all Euclid numbers squarefree?
 Are all Fermat numbers squarefree?
 Are all Mersenne numbers of prime index squarefree?
 Are there any composite c satisfying 2^{c − 1} ≡ 1 (mod c^{2})?
 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 ?
 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 and simultaneously?^{ [146]}
 Does every prime number appear in the Euclid–Mullin sequence?
 Find the smallest Skewes' 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 FibonacciWieferich primes, and when a = 2, this is the PellWieferich primes)
 For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p^{2})?^{ [147]}
 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 −4k^{4} for integer k, are there infinitely many repunit primes to base b?
 For any given integers , with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form with integer n ≥ 1?
 Is every Fermat number composite for ?
 Is 509,203 the lowest Riesel number?
Set theory
Note: These conjectures are about models of ZermeloFrankel 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 nonwellfounded 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 for every singular cardinal ?
 Does the generalized continuum hypothesis imply the existence of an ℵ_{2}Suslin tree?
 If ℵ_{ω} is a strong limit cardinal, is (see Singular cardinals hypothesis)? The best bound, ℵ_{ω4}, was obtained by Shelah using his PCF theory.
 The problem of finding the ultimate core model, one that contains all large cardinals.
 Woodin's Ωconjecture: if there is a proper class of Woodin cardinals, then Ωlogic 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 ℵ_{ω}?
 Is OCA (the open coloring axiom) consistent with ?
 Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
Topology
 Baum–Connes conjecture: the assembly map is an isomorphism.
 Bing–Borsuk conjecture: every 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^{ [148]}
 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.^{ [149]}
 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 twodimensional aspherical CW complex is aspherical.
 Zeeman conjecture: given a finite contractible twodimensional CW complex , is the space collapsible?
Problems solved since 1995
This section
duplicates the scope of other articles. (August 2022) 
Analysis
 Kadison–Singer problem ( Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)^{ [150]}^{ [151]} (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic and conjectures, BourgainTzafriri conjecture and conjecture)
 Ahlfors measure conjecture ( Ian Agol, 2004)^{ [152]}
 Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)^{ [153]}
Combinatorics
 Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)^{ [154]}
 McMullen's gconjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)^{ [155]}^{ [156]}
 Hirsch conjecture ( Francisco Santos Leal, 2010)^{ [157]}^{ [158]}
 Stanley–Wilf conjecture ( Gábor Tardos and Adam Marcus, 2004)^{ [159]} (and also the Alon–Friedgut conjecture)
 Kemnitz's conjecture ( Christian Reiher, 2003, Carlos di Fiore, 2003)^{ [160]}
 Cameron–Erdős conjecture ( Ben J. Green, 2003, Alexander Sapozhenko, 2003)^{ [161]}^{ [162]}
Dynamical systems
 Painlevé conjecture (Jinxin Xue, 2014)^{ [163]}^{ [164]}
Game theory
 The angel problem (Various independent proofs, 2006)^{ [165]}^{ [166]}^{ [167]}^{ [168]}
Geometry
21st century
 Yau's conjecture ( Antoine Song, 2018)^{ [169]}^{ [170]}
 Pentagonal tiling (Michaël Rao, 2017)^{ [171]}
 Willmore conjecture ( Fernando Codá Marques and André Neves, 2012)^{ [172]}
 Erdős distinct distances problem ( Larry Guth, Nets Hawk Katz, 2011)^{ [173]}
 Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)^{ [174]}
 Tameness conjecture ( Ian Agol, 2004)^{ [152]}
 Ending lamination theorem ( Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)^{ [175]}
 Carpenter's rule problem ( Robert Connelly, Erik Demaine, Günter Rote, 2003)^{ [176]}
 Nagata's conjecture (Ivan Shestakov, Ualbai Umirbaev, 2003)^{ [177]}
 Double bubble conjecture ( Michael Hutchings, Frank Morgan, Manuel Ritoré, Antonio Ros, 2002)^{ [178]}
20th century
 Honeycomb conjecture ( Thomas Callister Hales, 1999)^{ [179]}
 Bogomolov conjecture ( Emmanuel Ullmo, 1998, ShouWu Zhang, 1998)^{ [180]}^{ [181]}
 Kepler conjecture (Samuel Ferguson, Thomas Callister Hales, 1998)^{ [182]}
 Dodecahedral conjecture ( Thomas Callister Hales, Sean McLaughlin, 1998)^{ [183]}
Graph theory
 Blankenship–Oporowski conjecture on the book thickness of subdivisions ( Vida Dujmović, David Eppstein, Robert Hickingbotham, Pat Morin, and David Wood, 2021)^{ [184]}
 Ringel's conjecture on graceful labeling of trees (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)^{ [185]}^{ [186]}
 Disproof of Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)^{ [187]}
 Babai's problem (Alireza Abdollahi, Maysam Zallaghi, 2015)^{ [188]}
 Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
 Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)^{ [189]}
 Erdős–Menger conjecture ( Ron Aharoni, Eli Berger 2007)^{ [190]}
 Road coloring conjecture ( Avraham Trahtman, 2007)^{ [191]}
 Robertson–Seymour theorem ( Neil Robertson, Paul Seymour, 2004)^{ [192]}
 Strong perfect graph conjecture ( Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)^{ [193]}
Group theory
 Hanna Neumann conjecture (Joel Friedman, 2011, Igor Mineyev, 2011)^{ [194]}^{ [195]}
 Density theorem (Hossein Namazi, Juan Souto, 2010)^{ [196]}
 Full classification of finite simple groups ( Koichiro Harada, Ronald Solomon, 2008)
Number theory
21st century
 DuffinSchaeffer conjecture ( Dimitris Koukoulopoulos, James Maynard, 2019)
 Main conjecture in Vinogradov's meanvalue theorem ( Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)^{ [197]}
 Goldbach's weak conjecture ( Harald Helfgott, 2013)^{ [198]}^{ [199]}^{ [200]}
 Existence of bounded gaps between primes ( Yitang Zhang, Polymath8, James Maynard, 2013)^{ [201]}^{ [202]}^{ [203]}
 Sidon set problem (Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa, 2010)^{ [204]}
 Serre's modularity conjecture ( Chandrashekhar Khare and JeanPierre Wintenberger, 2008)^{ [205]}^{ [206]}^{ [207]}
 Green–Tao theorem ( Ben J. Green and Terence Tao, 2004)^{ [208]}
 Catalan's conjecture ( Preda Mihăilescu, 2002)^{ [209]}
 Erdős–Graham problem ( Ernest S. Croot III, 2000)^{ [210]}
20th century
 Lafforgue's theorem ( Laurent Lafforgue, 1998)^{ [211]}
 Fermat's Last Theorem ( Andrew Wiles and Richard Taylor, 1995)^{ [212]}^{ [213]}
Ramsey theory
 Burr–Erdős conjecture (Choongbum Lee, 2017)^{ [214]}
 Boolean Pythagorean triples problem ( Marijn Heule, Oliver Kullmann, Victor W. Marek, 2016)^{ [215]}^{ [216]}
Theoretical computer science
 Sensitivity conjecture for Boolean functions ( Hao Huang, 2019)^{ [217]}
Topology
 Deciding whether the Conway knot is a slice knot ( Lisa Piccirillo, 2020)^{ [218]}^{ [219]}
 Virtual Haken conjecture ( Ian Agol, Daniel Groves, Jason Manning, 2012)^{ [220]} (and by work of Daniel Wise also virtually fibered conjecture)
 Hsiang–Lawson's conjecture ( Simon Brendle, 2012)^{ [221]}
 Ehrenpreis conjecture ( Jeremy Kahn, Vladimir Markovic, 2011)^{ [222]}
 Atiyah conjecture (Austin, 2009)^{ [223]}
 Cobordism hypothesis ( Jacob Lurie, 2008)^{ [224]}
 Spherical space form conjecture ( Grigori Perelman, 2006)
 Poincaré conjecture ( Grigori Perelman, 2002)^{ [225]}
 Geometrization conjecture, proven by Grigori Perelman^{ [225]} in a series of preprints in 2002–2003.^{ [226]}
 Disproof of the Ganea conjecture (Iwase, 1997)^{ [227]}
Uncategorised
21st century
2010s
 Erdős discrepancy problem ( Terence Tao, 2015)^{ [228]}
 Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)^{ [229]}
 Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4manifolds satisfying certain properties ( Jeff Cheeger, Aaron Naber, 2014)^{ [230]}
 Gaussian correlation inequality ( Thomas Royen, 2014)^{ [231]}
 Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman, Aleksandar Nikolov, 2011)^{ [232]}
 Bloch–Kato conjecture ( Vladimir Voevodsky, 2011)^{ [233]} (and Quillen–Lichtenbaum conjecture and by work of Thomas Geisser and Marc Levine (2001) also Beilinson–Lichtenbaum conjecture^{ [234]}^{ [235]}^{: 359 }^{ [236]})
2000s
 Kauffman–Harary conjecture (Thomas Mattman, Pablo Solis, 2009)^{ [237]}
 Surface subgroup conjecture ( Jeremy Kahn, Vladimir Markovic, 2009)^{ [238]}
 Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Zhiqin Lu, 2007)^{ [239]}
 Nirenberg–Treves conjecture ( Nils Dencker, 2005)^{ [240]}^{ [241]}
 Lax conjecture ( Adrian Lewis, Pablo Parrilo, Motakuri Ramana, 2005)^{ [242]}
 The Langlands–Shelstad fundamental lemma ( Ngô Bảo Châu and Gérard Laumon, 2004)^{ [243]}
 Milnor conjecture ( Vladimir Voevodsky, 2003)^{ [244]}
 Kirillov's conjecture (Ehud Baruch, 2003)^{ [245]}
 Kouchnirenko’s conjecture (Bertrand Haas, 2002)^{ [246]}
 n! conjecture ( Mark Haiman, 2001)^{ [247]} (and also Macdonald positivity conjecture)
 Kato's conjecture ( Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philipp Tchamitchian, 2001)^{ [248]}
 Deligne's conjecture on 1motives (Luca BarbieriViale, Andreas Rosenschon, Morihiko Saito, 2001)^{ [249]}
 Modularity theorem ( Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, 2001)^{ [250]}
 Erdős–Stewart conjecture ( Florian Luca, 2001)^{ [251]}
 Berry–Robbins problem ( Michael Atiyah, 2000)^{ [252]}
20th century
 Torsion conjecture ( Loïc Merel, 1996)^{ [253]}
 Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen, 1996)^{ [254]}
See also
 List of conjectures
 List of unsolved problems in statistics
 List of unsolved problems in computer science
 List of unsolved problems in physics
 Lists of unsolved problems
 Open Problems in Mathematics
 The Great Mathematical Problems
 Scottish Book
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Further reading
Books discussing problems solved since 1995
 Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate. ISBN 9781841157917.
 O'Shea, Donal (2007). The Poincaré Conjecture. Penguin. ISBN 9781846140129.
 Szpiro, George G. (2003). Kepler's Conjecture. Wiley. ISBN 9780471086017.
 Ronan, Mark (2006). Symmetry and the Monster. Oxford. ISBN 9780192807229.
Books discussing unsolved problems
 Chung, Fan; Graham, Ron (1999). Erdös on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 9781568811116.
 Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 9780387975061.
 Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 9780387208602.
 Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 9780883853153.
 du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 9780060935580.
 Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 9780309085496.
 Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 9780760786598.
 Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 9780691117485.
 Ji, Lizhen; Poon, YatSun; Yau, ShingTung (2013). Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics). International Press of Boston. ISBN 9781571462787.
 Waldschmidt, Michel (2004). "Open Diophantine Problems" (PDF). Moscow Mathematical Journal. 4 (1): 245–305. arXiv: math/0312440. doi: 10.17323/16094514200441245305. ISSN 16093321. S2CID 11845578. Zbl 1066.11030.
 Mazurov, V. D.; Khukhro, E. I. (1 Jun 2015). "Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version)". arXiv: 1401.0300v6 [ math.GR].
External links
 24 Unsolved Problems and Rewards for them
 List of links to unsolved problems in mathematics, prizes and research
 Open Problem Garden
 AIM Problem Lists
 Unsolved Problem of the Week Archive. MathPro Press.
 Ball, John M. "Some Open Problems in Elasticity" (PDF).
 Constantin, Peter. "Some open problems and research directions in the mathematical study of fluid dynamics" (PDF).
 Serre, Denis. "Five Open Problems in Compressible Mathematical Fluid Dynamics" (PDF).
 Unsolved Problems in Number Theory, Logic and Cryptography
 200 open problems in graph theory
 The Open Problems Project (TOPP), discrete and computational geometry problems
 Kirby's list of unsolved problems in lowdimensional topology
 Erdös' Problems on Graphs
 Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory
 Open problems from the 12th International Conference on Fuzzy Set Theory and Its Applications
 List of open problems in inner model theory
 Aizenman, Michael. "Open Problems in Mathematical Physics".
 Barry Simon's 15 Problems in Mathematical Physics