List of unsolved problems in mathematics
Many mathematical problems have not been solved yet. These unsolved problems occur in multiple domains, including theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, and partial differential equations. Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas. Prizes are often awarded for the solution to a longstanding problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention.
This article is a composite of notable unsolved problems derived from many sources, including but not limited to lists considered authoritative. It does not claim to be comprehensive, it may not always be quite up to date, and it includes problems which are considered by the mathematical community to be widely varying in both difficulty and centrality to the science as a whole.
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 unresolved or incompletely resolved 
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 set by the Clay Mathematics Institute in 2000, six have yet to be solved as of May, 2021:^{ [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, has been solved;^{ [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 still unsolved.^{ [13]}
Unsolved problems
Algebra
Notebook problems
 The Dneister Notebook (Dnestrovskaya Tetrad) collects several hundred unresolved problems in algebra, particularly ring theory and modulus theory.^{ [14]}
 The Erlagol Notebook (Erlagolskaya Tetrad) collects unresolved problems in algebra and model theory.^{ [15]}
Conjectures and problems
 Birch–Tate conjecture
 Bombieri–Lang conjecture
 Crouzeix's conjecture
 Demazure conjecture
 Eilenberg–Ganea conjecture
 Farrell–Jones conjecture
 Bost conjecture
 Finite lattice representation problem^{ [16]}
 Green's conjecture
 Grothendieck–Katz pcurvature conjecture
 Hadamard conjecture
 Hilbert's fifteenth problem
 Hilbert's sixteenth problem
 Homological conjectures in commutative algebra
 Jacobson's conjecture
 Kaplansky's conjectures
 Köthe conjecture
 Kummer–Vandiver conjecture
 Existence of perfect cuboids and associated cuboid conjectures
 Pierce–Birkhoff conjecture
 Rota's basis conjecture
 Sendov's conjecture
 Serre's conjecture II
 Serre's multiplicity conjectures
 Uniform boundedness conjecture for rational points
 Wild problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems
 Zariski–Lipman conjecture
 Zauner's conjecture: existence of SICPOVMs in all dimensions
Analysis
Conjectures and problems
 Brennan conjecture
 The four exponentials conjecture on the transcendence of at least one of four exponentials of combinations of irrationals^{ [17]}
 Invariant subspace problem
 Kung–Traub conjecture^{ [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
Open questions
 Are (the Euler–Mascheroni constant), π + e, π − e, πe, π/e, π^{e}, π^{ √2}, π^{π}, e^{π2}, ln π, 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?^{ [21]}^{ [22]}^{ [23]}
Other
 Regularity of solutions of Euler equations
 Convergence of Flint Hills series
 Regularity of solutions of Vlasov–Maxwell equations
Combinatorics
Conjectures and problems
 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]}
 Nothreeinline problem
 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]}
Other
 The values of the Dedekind numbers for .^{ [27]}
 Give a combinatorial interpretation of the Kronecker coefficients.^{ [28]}
 The values of the Ramsey numbers, particularly
 Finding a function to model nstep selfavoiding walks.^{ [29]}
 The values of the Van der Waerden numbers
Dynamical systems
Conjectures and problems
 Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theory
 Quantum chaos: Berry–Tabor conjecture
 Birkhoff conjecture: if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse?^{ [30]}
 Collatz conjecture (3n + 1 conjecture)
 Eremenko's conjecture that 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
 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.
 Painlevé conjecture
 Quantum unique ergodicity conjecture^{ [31]}
 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?
Open questions
 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?^{ [32]}
Games and puzzles
Combinatorial games

Sudoku:
 How many puzzles have exactly one solution?^{ [33]}
 How many puzzles with exactly one solution are minimal?^{ [33]}
 What is the maximum number of givens for a minimal puzzle?^{ [33]}

Tictactoe variants:
 Given a width of tictactoe board, what is the smallest dimension such that X is guaranteed a winning strategy?^{ [34]}
 What is the Turing completeness status of all unique elementary cellular automata?
Games with imperfect information
Geometry
Algebraic geometry
Conjectures
 Abundance conjecture
 Bass conjecture
 Deligne conjecture
 Dixmier conjecture
 Fröberg conjecture
 Fujita conjecture
 Hartshorne's conjectures^{ [35]}
 The Jacobian conjecture
 Manin conjecture
 Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory^{ [36]}
 Nakai conjecture
 Parshin's conjecture
 Section conjecture
 Standard conjectures on algebraic cycles
 Tate conjecture
 Virasoro conjecture
 Weightmonodromy conjecture
 Zariski multiplicity conjecture^{ [37]}
Other
 Flip  Termination of flips
 Resolution of singularities in characteristic
Covering and packing
Conjectures and problems
 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?^{ [38]}
 The Erdős–Oler conjecture that when is a triangular number, packing circles in an equilateral triangle requires a triangle of the same size as packing circles^{ [39]}
 The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24^{ [40]}
 Reinhardt's conjecture that the smoothed octagon has the lowest maximum packing density of all centrallysymmetric convex plane sets^{ [41]}
 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?^{ [42]}
 Ulam's packing conjecture about the identity of the worstpacking convex solid^{ [43]}
Differential geometry
Conjectures and problems
 The spherical Bernstein's problem, a possible generalization of the original Bernstein's problem
 Carathéodory conjecture
 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)
 Chern's conjecture for hypersurfaces in spheres
 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.^{ [44]}
 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^{ [45]}
 The Hopf conjectures relating the curvature and Euler characteristic of higherdimensional Riemannian manifolds^{ [46]}
 Yau's conjecture
 Yau's conjecture on the first eigenvalue
Discrete geometry
Conjectures and problems
 The Hadwiger conjecture on covering ndimensional convex bodies with at most 2^{n} smaller copies^{ [47]}
 Solving the happy ending problem for arbitrary ^{ [48]}
 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.^{ [49]}
 The Kobon triangle problem on triangles in line arrangements^{ [50]}
 The Kusner conjecture that at most points can be equidistant in spaces^{ [51]}
 The McMullen problem on projectively transforming sets of points into convex position^{ [52]}
 Opaque forest problem
Open questions
 How many unit distances can be determined by a set of n points in the Euclidean plane?^{ [53]}
Other
 Finding matching upper and lower bounds for ksets and halving lines^{ [54]}
 Tripod packing^{ [55]}
Euclidean geometry
Conjectures and problems
 The Atiyah conjecture on configurations^{ [56]}
 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^{ [57]}
 Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?^{ [58]}
 Ehrhart's volume conjecture
 The einstein problem – does there exist a twodimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?^{ [59]}
 Falconer's conjecture that sets of Hausdorff dimension greater than in must have a distance set of nonzero Lebesgue measure^{ [60]}
 Inscribed square problem, also known as Toeplitz' conjecture – does every Jordan curve have an inscribed square?^{ [61]}
 The Kakeya conjecture – do dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to ?^{ [62]}
 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^{ [63]}
 Lebesgue's universal covering problem on the minimumarea convex shape in the plane that can cover any shape of diameter one^{ [64]}
 Mahler's conjecture on the product of the volumes of a centrally symmetric convex body and its polar.^{ [65]}
 Moser's worm problem – what is the smallest area of a shape that can cover every unitlength curve in the plane?^{ [66]}
 The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unitwidth Lshaped corridor?^{ [67]}
 Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net, or simple edgeunfolding?^{ [68]}^{ [69]}
 The Thomson problem – what is the minimum energy configuration of mutuallyrepelling particles on a unit sphere?^{ [70]}
Open questions
 Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?^{ [71]}
 Dissection into orthoschemes – is it possible for simplices of every dimension?^{ [72]}
Other
 Uniform 5polytopes – find and classify the complete set of these shapes^{ [73]}
Graph theory
Graph coloring and labeling
Conjectures and problems
 Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs^{ [74]}
 The Erdős–Faber–Lovász conjecture on coloring unions of cliques^{ [75]}
 The Gyárfás–Sumner conjecture on χboundedness of graphs with a forbidden induced tree^{ [76]}
 The Hadwiger conjecture relating coloring to clique minors^{ [77]}
 The Hadwiger–Nelson problem on the chromatic number of unit distance graphs^{ [78]}
 Jaeger's Petersencoloring conjecture that every bridgeless cubic graph has a cyclecontinuous mapping to the Petersen graph^{ [79]}
 The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index^{ [80]}
 The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree^{ [81]}
Graph drawing
Conjectures and problems
 The Albertson conjecture that the crossing number can be lowerbounded by the crossing number of a complete graph with the same chromatic number^{ [82]}
 The Blankenship–Oporowski conjecture on the book thickness of subdivisions^{ [83]}
 Conway's thrackle conjecture^{ [84]}
 Harborth's conjecture that every planar graph can be drawn with integer edge lengths^{ [85]}
 Negami's conjecture on projectiveplane embeddings of graphs with planar covers^{ [86]}
 The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding^{ [87]}
 Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?^{ [88]}
Other
 Universal point sets of subquadratic size for planar graphs^{ [89]}
Paths and cycles in graphs
Conjectures and problems
 Barnette's conjecture that every cubic bipartite threeconnected planar graph has a Hamiltonian cycle^{ [90]}
 Chvátal's toughness conjecture, that there is a number t such that every ttough graph is Hamiltonian^{ [91]}
 The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice^{ [92]}
 The Erdős–Gyárfás conjecture on cycles with poweroftwo lengths in cubic graphs^{ [93]}
 The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree^{ [94]}
 The Lovász conjecture on Hamiltonian paths in symmetric graphs^{ [95]}
 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.^{ [96]}
 Szymanski's conjecture
Wordrepresentation of graphs
 Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter?^{ [97]}^{ [98]}^{ [99]}^{ [100]}
 Characterise (non) wordrepresentable planar graphs^{ [97]}^{ [98]}^{ [99]}^{ [100]}
 Characterise wordrepresentable graphs in terms of (induced) forbidden subgraphs.^{ [97]}^{ [98]}^{ [99]}^{ [100]}
 Characterise wordrepresentable neartriangulations containing the complete graph K_{4} (such a characterisation is known for K_{4}free planar graphs^{ [101]})
 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^{ [102]}
 Is it true that out of all bipartite graphs, crown graphs require longest wordrepresentants?^{ [103]}
 Is the line graph of a non wordrepresentable graph always non wordrepresentable?^{ [97]}^{ [98]}^{ [99]}^{ [100]}
 Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?^{ [97]}^{ [98]}^{ [99]}^{ [100]}
Miscellaneous graph theory
Conjectures and problems
 Brouwer's conjecture
 Conway's 99graph problem: does there exist a strongly regular graph with parameters (99,14,1,2)?^{ [104]}
 The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph^{ [105]}
 The GNRS conjecture on whether minorclosed graph families have embeddings with bounded distortion^{ [106]}
 Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs^{ [107]}
 The implicit graph conjecture on the existence of implicit representations for slowlygrowing hereditary families of graphs^{ [108]}
 Jørgensen's conjecture that every 6vertexconnected K_{6}minorfree graph is an apex graph^{ [109]}
 Meyniel's conjecture that cop number is ^{ [110]}
 The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertexdeleted subgraphs.^{ [111]}^{ [112]}
 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?^{ [113]}
 Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?^{ [114]}
 Sumner's conjecture: does every vertex tournament contain as a subgraph every vertex oriented tree?^{ [115]}
 Tutte's conjectures that every bridgeless graph has a nowherezero 5flow and every Petersen minorfree bridgeless graph has a nowherezero 4flow^{ [116]}
 Vizing's conjecture on the domination number of cartesian products of graphs^{ [117]}
 Zarankiewicz problem
Open questions
 Does a Moore graph with girth 5 and degree 57 exist?^{ [118]}
 What is the largest possible pathwidth of an nvertex cubic graph?^{ [119]}
Group theory
Notebook problems
 The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.^{ [120]}
Conjectures and problems
 Andrews–Curtis conjecture
 Guralnick–Thompson conjecture^{ [121]}
 Herzog–Schönheim conjecture
 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
Open questions
 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?
Model theory and formal languages
Conjectures and problems
 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
 For which number fields does Hilbert's tenth problem hold?
 Kueker's conjecture^{ [122]}
 The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for saturated models of a countable theory.^{ [123]}
 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.^{ [123]}
 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 .^{ [123]}^{ [124]}
 The stable field conjecture: every infinite field with a stable firstorder theory is separably closed.
 The Stable Forking Conjecture for simple theories^{ [125]}
 Tarski's exponential function problem
 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?^{ [126]}
 The universality spectrum problem: Is there a firstorder theory whose universality spectrum is minimum?^{ [127]}
 Vaught's conjecture
Open questions
 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?^{ [128]}
 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?^{ [129]}^{ [130]}
 Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or cofinite.)
 (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of wellordering consistently decidable?^{ [131]}
 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?^{ [132]}
Other
 Determine the structure of Keisler's order^{ [133]}^{ [134]}
Number theory
General
Conjectures, problems and hypotheses
 André–Oort conjecture
 Beilinson conjecture
 Brocard's problem: existence of integers, (n,m), such that n! + 1 = m^{2} other than n = 4, 5, 7
 Carmichael's totient function conjecture
 CasasAlvero conjecture
 Catalan–Dickson conjecture on aliquot sequences
 Congruent number problem (a corollary to Birch and SwinnertonDyer conjecture, per Tunnell's theorem)
 Erdős–Moser problem: is 1^{1} + 2^{1} = 3^{1} the only solution to the Erdős–Moser equation?
 Erdős–Straus conjecture
 Erdős–Ulam problem
 Exponent pair conjecture (Van der Corput's method)
 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
 Grand Riemann hypothesis
 Grimm's conjecture
 Hall's conjecture
 Hardy–Littlewood zetafunction conjectures
 Hilbert's eleventh problem
 Hilbert's ninth problem
 Hilbert's twelfth problem
 Hilbert–Pólya conjecture
 Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function^{ [135]}
 Lehmer's totient problem: if φ(n) divides n − 1, must n be prime?
 Leopoldt's conjecture
 Lindelöf hypothesis and its consequence, the density hypothesis for zeroes of the Riemann zeta function (see Bombieri–Vinogradov theorem)
 Littlewood conjecture
 Mahler's 3/2 problem
 Montgomery's pair correlation conjecture
 n conjecture
 Newman's conjecture
 Pillai's conjecture
 Piltz divisor problem, especially Dirichlet's divisor problem
 Ramanujan–Petersson conjecture
 Sato–Tate conjecture
 Scholz conjecture
 Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?^{ [136]}
 The uniqueness conjecture for Markov numbers^{ [137]}
 Vojta's conjecture
Open questions
 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 perfect numbers?
 Do any Lychrel numbers exist?
 Do any odd perfect numbers exist?
 Do any odd weird numbers exist?
 Do any Taxicab(5, 2, n) exist for n > 1?
 Do quasiperfect numbers exist?
 Is there a covering system with odd distinct moduli?^{ [138]}
 Is π a normal number (its digits are "random")?^{ [139]}
 Is 10 a solitary number?
 Which integers can be written as the sum of three perfect cubes?^{ [140]}
Other
 Find the value of the De Bruijn–Newman constant
Additive number theory
Conjectures and problems
 Beal's conjecture
 Erdős conjecture on arithmetic progressions
 Erdős–Turán conjecture on additive bases
 Fermat–Catalan conjecture
 Gilbreath's conjecture
 Goldbach's conjecture
 Lander, Parkin, and Selfridge conjecture
 Lemoine's conjecture
 Minimum overlap problem
 Pollock's conjectures
 Skolem problem
 The values of g(k) and G(k) in Waring's problem
Open questions
 Do the Ulam numbers have a positive density?
Other
 Determine growth rate of r_{k}(N) (see Szemerédi's theorem)
Algebraic number theory
Conjectures and problems
 Class number problem: are there infinitely many real quadratic number fields with unique factorization?
 Fontaine–Mazur conjecture
 Gan–Gross–Prasad conjecture
 Greenberg's conjectures
 Hermite's problem
 Kummer–Vandiver conjecture
 Selberg's 1/4 conjecture
 Stark conjectures (including Brumer–Stark conjecture)
Other
 Characterize all algebraic number fields that have some power basis.
Computational number theory
 Integer factorization: Can integer factorization be done in polynomial time?
Prime numbers
Conjectures, problems and hypotheses
 Agoh–Giuga conjecture
 Artin's conjecture on primitive roots
 Brocard's conjecture
 Bunyakovsky conjecture
 Catalan's Mersenne conjecture
 Dickson's conjecture
 Dubner's conjecture
 Elliott–Halberstam conjecture
 Erdős–Mollin–Walsh conjecture
 Feit–Thompson conjecture
 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
 Goldbach conjecture
 Landau's problems
 Problems associated to Linnik's theorem
 New Mersenne conjecture
 Polignac's conjecture
 Schinzel's hypothesis H
 Is 78,557 the lowest Sierpiński number (socalled Selfridge's conjecture)?
 Twin prime conjecture
 Does the conjectural converse of Wolstenholme's theorem hold for all natural numbers?
Open questions
 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 Carol primes?
 Are there infinitely many cousin primes?
 Are there infinitely many Cullen primes?
 Are there infinitely many Fibonacci 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 regular primes, and if so is their relative density ?
 Are there infinitely many sexy primes?
 Are there infinitely many 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} ≡ 1 (mod p^{2}) and 3^{p − 1} ≡ 1 (mod p^{2}) simultaneously?^{ [141]}
 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})?^{ [142]}
 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 k ≥ 1, b ≥ 2, c ≠ 0, with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form (k×b^{n}+c)/gcd(k+c,b−1) with integer n ≥ 1?
 Is every Fermat number 2^{2n} + 1 composite for ?
 Is 509,203 the lowest Riesel number?
Set theory
Conjectures, problems, and hypotheses
 ( 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, then 2^{ℵω} < ℵ_{ω1} (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
Open questions
 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 ( Open coloring axiom) consistent with ?
 Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
Topology
Conjectures and problems
 Baum–Connes conjecture
 Bing–Borsuk conjecture
 Borel conjecture
 Halperin conjecture
 Hilbert–Smith conjecture
 Mazur's conjectures^{ [143]}
 Novikov conjecture
 Telescope conjectures
 Unknotting problem
 Volume conjecture
 Whitehead conjecture
 Zeeman conjecture
Problems solved since 1995
Algebra
 Connes embedding problem (Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, Henry Yuen, 2020)
Analysis
 Kadison–Singer problem ( Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)^{ [144]}^{ [145]} (and the Feichtinger's conjecture, Anderson’s paving conjectures, Weaver’s discrepancy theoretic and conjectures, BourgainTzafriri conjecture and conjecture)
Combinatorics
 Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)^{ [146]}
 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)^{ [147]}^{ [148]}
 Hirsch conjecture ( Francisco Santos Leal, 2010)^{ [149]}^{ [150]}
Game theory
 The angel problem (Various independent proofs, 2006)^{ [151]}^{ [152]}^{ [153]}^{ [154]}
Geometry
21st century
 Yau's conjecture ( Antoine Song, 2018)^{ [155]}^{ [156]}
 Pentagonal tiling (Michaël Rao, 2017)^{ [157]}
 Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)^{ [158]}
 Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)^{ [159]}
20th century
 Kepler conjecture (Ferguson, Hales, 1998)^{ [160]}
 Dodecahedral conjecture (Hales, McLaughlin, 1998)^{ [161]}
Graph theory
 Ringel's conjecture on graceful labeling of trees (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)^{ [162]}^{ [163]}
 Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)^{ [164]}
 Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (Alireza Abdollahi, Maysam Zallaghi, 2015)^{ [165]}
 Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
 Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)^{ [166]}
 Erdős–Menger conjecture (Aharoni, Berger 2007)^{ [167]}
 Road coloring conjecture ( Avraham Trahtman, 2007)^{ [168]}
Group theory
 Hanna Neumann conjecture (Mineyev, 2011)^{ [169]}
 Density theorem (Namazi, Souto, 2010)^{ [170]}
 Full classification of finite simple groups (Harada, 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)^{ [171]}
 Goldbach's weak conjecture ( Harald Helfgott, 2013)^{ [172]}^{ [173]}^{ [174]}
 Serre's modularity conjecture ( Chandrashekhar Khare and JeanPierre Wintenberger, 2008)^{ [175]}^{ [176]}^{ [177]}
20th century
 Fermat's Last Theorem ( Andrew Wiles and Richard Taylor, 1995)^{ [178]}^{ [179]}
Ramsey theory
 Burr–Erdős conjecture (Choongbum Lee, 2017)^{ [180]}
 Boolean Pythagorean triples problem ( Marijn Heule, Oliver Kullmann, Victor Marek, 2016)^{ [181]}^{ [182]}
Theoretical computer science
 Sensitivity conjecture for Boolean functions ( Hao Huang, 2019) ^{ [183]}
Topology
 Deciding whether the Conway knot is a slice knot ( Lisa Piccirillo, 2020)^{ [184]}^{ [185]}
 Virtual Haken conjecture (Agol, Groves, Manning, 2012)^{ [186]} (and by work of Wise also virtually fibered conjecture)
 Hsiang–Lawson's conjecture (Brendle, 2012)^{ [187]}
 Ehrenpreis conjecture (Kahn, Markovic, 2011)^{ [188]}
 Atiyah conjecture (Austin, 2009)^{ [189]}
 Cobordism hypothesis ( Jacob Lurie, 2008)^{ [190]}
 Geometrization conjecture, proven by Grigori Perelman^{ [191]} in a series of preprints in 2002–2003.^{ [192]}
 Spherical space form conjecture ( Grigori Perelman, 2006)
Uncategorised
21st century
2010s
 Erdős discrepancy problem ( Terence Tao, 2015)^{ [193]}
 Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)^{ [194]}
 Anderson conjecture (Cheeger, Naber, 2014)^{ [195]}
 Gaussian correlation inequality ( Thomas Royen, 2014)^{ [196]}
 Willmore conjecture ( Fernando Codá Marques and André Neves, 2012)^{ [197]}
 Beck's 3permutations conjecture (Newman, Nikolov, 2011)^{ [198]}
 Bloch–Kato conjecture (Voevodsky, 2011)^{ [199]} (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture^{ [200]}^{ [201]}^{ [202]})
 Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)^{ [203]}
2000s
 Kauffman–Harary conjecture (Matmann, Solis, 2009)^{ [204]}
 Surface subgroup conjecture (Kahn, Markovic, 2009)^{ [205]}
 Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Lu, 2007)^{ [206]}
 Nirenberg–Treves conjecture ( Nils Dencker, 2005)^{ [207]}^{ [208]}
 Lax conjecture (Lewis, Parrilo, Ramana, 2005)^{ [209]}
 The Langlands–Shelstad fundamental lemma ( Ngô Bảo Châu and Gérard Laumon, 2004)^{ [210]}
 Tameness conjecture and Ahlfors measure conjecture ( Ian Agol, 2004)^{ [211]}
 Robertson–Seymour theorem (Robertson, Seymour, 2004)^{ [212]}
 Stanley–Wilf conjecture ( Gábor Tardos and Adam Marcus, 2004)^{ [213]} (and also Alon–Friedgut conjecture)
 Green–Tao theorem ( Ben J. Green and Terence Tao, 2004)^{ [214]}
 Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)^{ [215]}
 Carpenter's rule problem (Connelly, Demaine, Rote, 2003)^{ [216]}
 Cameron–Erdős conjecture ( Ben J. Green, 2003, Alexander Sapozhenko, 2003)^{ [217]}^{ [218]}
 Milnor conjecture ( Vladimir Voevodsky, 2003)^{ [219]}
 Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)^{ [220]}
 Nagata's conjecture (Shestakov, Umirbaev, 2003)^{ [221]}
 Kirillov's conjecture (Baruch, 2003)^{ [222]}
 Poincaré conjecture ( Grigori Perelman, 2002)^{ [191]}
 Strong perfect graph conjecture ( Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)^{ [223]}
 Kouchnirenko’s conjecture (Haas, 2002)^{ [224]}
 Vaught conjecture (Knight, 2002)^{ [225]}
 Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)^{ [226]}
 Catalan's conjecture ( Preda Mihăilescu, 2002)^{ [227]}
 n! conjecture (Haiman, 2001)^{ [228]} (and also Macdonald positivity conjecture)
 Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)^{ [229]}
 Deligne's conjecture on 1motives (Luca BarbieriViale, Andreas Rosenschon, Morihiko Saito, 2001)^{ [230]}
 Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)^{ [231]}
 Erdős–Stewart conjecture (Florian Luca, 2001)^{ [232]}
 Berry–Robbins problem (Atiyah, 2000)^{ [233]}
 Erdős–Graham problem (Croot, 2000)^{ [234]}
20th century
 Honeycomb conjecture (Thomas Hales, 1999)^{ [235]}
 Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)^{ [236]}
 Bogomolov conjecture ( Emmanuel Ullmo, 1998, ShouWu Zhang, 1998)^{ [237]}^{ [238]}
 Lafforgue's theorem (Laurent Lafforgue, 1998)^{ [239]}
 Ganea conjecture (Iwase, 1997)^{ [240]}
 Torsion conjecture (Merel, 1996)^{ [241]}
 Harary's conjecture (Chen, 1996)^{ [242]}
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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 ^ Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3adic exercises", Journal of the American Mathematical Society, 14 (4): 843–939, doi: 10.1090/S0894034701003708, ISSN 08940347, MR 1839918
 ^ Luca, Florian (2000). "On a conjecture of Erdős and Stewart" (PDF). Mathematics of Computation. 70 (234): 893–897. Bibcode: 2001MaCom..70..893L. doi: 10.1090/s0025571800011789. Archived (PDF) from the original on 20160402. Retrieved 20160318.
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 ^ Croot, Ernest S., III (2000), Unit Fractions, Ph.D. thesis, University of Georgia, Athens. Croot, Ernest S., III (2003), "On a coloring conjecture about unit fractions", Annals of Mathematics, 157 (2): 545–556, arXiv: math.NT/0311421, Bibcode: 2003math.....11421C, doi: 10.4007/annals.2003.157.545, S2CID 13514070
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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].
 The Sverdlovsk Notebook is a collection of unsolved problems in semigroup theory.^{ [1]}^{ [2]}
 Formulation of unresloved problems for infinite Abelian groups are depicted in the book^{ [3]}
 The list of unresolved problems for Combinatorial Geometry are depicted in the book.^{ [4]}
 Several dozens of unresolved problems for Combinatorial Geometry are depicted in the book.^{ [5]}
 Many unresolved problems for Graph theory are depicted in the article.^{ [6]}
 The list of several unresolved problems converning Maler Conjecture are depicted in the book.^{ [7]}
External links
 24 Unsolved Problems and Rewards for them
 List of links to unsolved problems in mathematics, prizes and research
 Open Problem Garden The collection of open problems in mathematics build on the principle of user editable ("wiki") site
 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
 ^ The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1979
 ^ The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1989
 ^ Fuks 1974, p. 47, 88, 116, 134, 158, 159, 186, 210, 242, 243, 292, 318.
 ^ Boltiansky 1965, p. 83.
 ^ Grunbaum 1971, p. 6.
 ^ V. G. Vizing Some unresolved problems for Graph theory // Russian Mathematical Surveys, 23:6(144) (1968), 117–134; Russian Math. Surveys, 23:6 (1968), 125–141
 ^ Sprinjuk 1967, p. 150—154.