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 long-standing 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. The list is not comprehensive, for at least the reason that entries may not be updated at the time of viewing. This list 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.

The Riemann zeta function, subject of the celebrated and influential unsolved problem known as the Riemann hypothesis

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 August, 2021:^[6]

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

The seventh problem, the Poincaré conjecture, has been solved;^[12] however, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.^[13]

Unsolved problems[]

Algebra[]

In the Bloch sphere representation of a qubit, a SIC-POVM forms a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.

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 p-curvature conjecture
• Hadamard conjecture
• Hadamard's maximal determinant problem
• 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 SIC-POVMs in all dimensions

Analysis[]

The area of the blue region converges to the Euler–Mascheroni constant, which may or may not be a rational number.

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
• ^[18]
• Lehmer's conjecture on the Mahler measure of non-cyclotomic 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 ${\displaystyle \gamma }$ (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]
• What is the exact value of Landau's constants, including Bloch's constant?

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 ${\displaystyle k+1}$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance ${\displaystyle 1/(k+1)}$ from each other runner) at some time?^[25]
• No-three-in-line problem
• Frankl's union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets^[26]

Other[]

• The values of the Dedekind numbers ${\displaystyle M(n)}$ for ${\displaystyle n\geq 9}$.^[27]
• Give a combinatorial interpretation of the Kronecker coefficients.^[28]
• The values of the Ramsey numbers, particularly ${\displaystyle R(5,5)}$
• Finding a function to model n-step self-avoiding walks.^[29]
• The values of the Van der Waerden numbers

Dynamical systems[]

A detail of the Mandelbrot set. It is not known whether the Mandelbrot set is locally connected or not.

Conjectures and problems[]

• Arnold–Givental conjecture and Arnold conjecture – relating symplectic geometry to Morse theory
• Quantum chaos: Berry–Tabor 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 ${\displaystyle \times 2,\times 3}$ action on the circle either Lebesgue or atomic?
• Kaplan–Yorke conjecture
• Margulis conjecture – Measure classification for diagonalizable actions in higher-rank groups
• MLC conjecture – Is the Mandelbrot set locally connected?
• Many problems concerning an outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.
• ^[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[]

• 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?^[32]

Games and puzzles[]

Combinatorial games[]

• Is there a non-terminating game of beggar-my-neighbour?
• 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]
• Tic-tac-toe variants:
  • Given a width of tic-tac-toe 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[]

• Rendezvous problem

Geometry[]

Algebraic geometry[]

Conjectures[]

• Abundance conjecture
• Bass conjecture
• Deligne conjecture
• Dixmier conjecture
• Fröberg conjecture
• Fujita conjecture
• ^[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
• Weight-monodromy conjecture
• ^[37]

Other[]

• Flip - Termination of flips
• Resolution of singularities in characteristic ${\displaystyle p}$

Covering and packing[]

Conjectures and problems[]

• Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set.
• The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?^[38]
• The Erdős–Oler conjecture that when ${\displaystyle n}$ is a triangular number, packing ${\displaystyle n-1}$ circles in an equilateral triangle requires a triangle of the same size as packing ${\displaystyle n}$ circles^[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 centrally-symmetric 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 worst-packing 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
• : 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 shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length^[45]
• The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds^[46]
• Yau's conjecture
• Yau's conjecture on the first eigenvalue

Discrete geometry[]

In three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.

Conjectures and problems[]

• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2ⁿ smaller copies^[47]
• Solving the happy ending problem for arbitrary ${\displaystyle n}$^[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 ${\displaystyle 2d}$ points can be equidistant in ${\displaystyle L^{1}}$ 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 k-sets 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 two-dimensional 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 ${\displaystyle d/2}$ in ${\displaystyle \mathbb {R} ^{d}}$ 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 ${\displaystyle n}$-dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to ${\displaystyle n}$?^[62]
• The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem^[63]
• Lebesgue's universal covering problem on the minimum-area 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 unit-length curve in the plane?^[66]
• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?^[67]
• Shephard's problem (a.k.a. Dürer's conjecture) – does every convex polyhedron have a net, or simple edge-unfolding?^[68][69]
• The Thomson problem – what is the minimum energy configuration of ${\displaystyle n}$ mutually-repelling 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 5-polytopes – find and classify the complete set of these shapes^[73]

Graph theory[]

Graph coloring and labeling[]

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

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 χ-boundedness 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 Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous 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 lower-bounded 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 projective-plane 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 three-connected planar graph has a Hamiltonian cycle^[90]
• Chvátal's toughness conjecture, that there is a number t such that every t-tough 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 power-of-two 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 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.^[96]
• Szymanski's conjecture

Word-representation 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-)word-representable planar graphs^{[97][98][99][100]}
• Characterise word-representable graphs in terms of (induced) forbidden subgraphs.^{[97][98][99][100]}
• Characterise word-representable near-triangulations containing the complete graph K₄ (such a characterisation is known for K₄-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 word-representants?^[103]
• Is the line graph of a non-word-representable graph always non-word-representable?^{[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 99-graph 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 minor-closed graph families have ${\displaystyle \ell _{1}}$ 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 slowly-growing hereditary families of graphs^[108]
• Jørgensen's conjecture that every 6-vertex-connected K₆-minor-free graph is an apex graph^[109]
• Meyniel's conjecture that cop number is ${\displaystyle O({\sqrt {n}})}$^[110]
• The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted 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 ${\displaystyle (2n-2)}$-vertex tournament contain as a subgraph every ${\displaystyle n}$-vertex oriented tree?^[115]
• Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow^[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 n-vertex cubic graph?^[119]

Group theory[]

The free Burnside group ${\displaystyle B(2,3)}$ is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups ${\displaystyle B(m,n)}$ are finite remains open.

Notebook problems[]

• The 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
• ^[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 first-order theory is stable in ${\displaystyle \aleph _{0}}$ 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 ${\displaystyle \aleph _{1}}$-saturated models of a countable theory.^[123]
• Shelah's categoricity conjecture for ${\displaystyle L_{\omega _{1},\omega }}$: 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 ${\displaystyle \lambda }$ there exists a cardinal ${\displaystyle \mu (\lambda )}$ such that if an AEC K with LS(K)<= ${\displaystyle \lambda }$ is categorical in a cardinal above ${\displaystyle \mu (\lambda )}$ then it is categorical in all cardinals above ${\displaystyle \mu (\lambda )}$.^[123][124]
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• The Stable Forking Conjecture for simple theories^[125]
• Tarski's exponential function problem
• The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?^[126]
• The universality spectrum problem: Is there a first-order 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 ${\displaystyle \aleph _{\omega _{1}}}$ 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 o-minimal first order theory with a trans-exponential (rapid growth) function?
• If the class of atomic models of a complete first order theory is categorical in the ${\displaystyle \aleph _{n}}$, 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 co-finite.)
• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?^[131]
• Is the theory of the field of Laurent series over ${\displaystyle \mathbb {Z} _{p}}$ decidable? of the field of polynomials over ${\displaystyle \mathbb {C} }$?
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?^[132]

Other[]

• Determine the structure of Keisler's order^[133][134]

Number theory[]

General[]

6 is a perfect number because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them are odd.

Conjectures, problems and hypotheses[]

• André–Oort conjecture
• Beilinson conjecture
• Brocard's problem: existence of integers, (n,m), such that n! + 1 = m² other than n = 4, 5, 7
• Carmichael's totient function conjecture
• Casas-Alvero conjecture
• Catalan–Dickson conjecture on aliquot sequences
• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem)
• Erdős–Moser problem: is 1¹ + 2¹ = 3¹ 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
  • Generalized Riemann hypothesis
    • Riemann hypothesis
• Grimm's conjecture
• Hall's conjecture
• Hardy–Littlewood zeta-function conjectures
• Hilbert's eleventh problem
• Hilbert's ninth problem
• Hilbert's twelfth problem
• Hilbert–Pólya 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 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
  • abc conjecture
  • Szpiro's conjecture
• Newman's conjecture
• Pillai's conjecture
• Piltz divisor problem, especially Dirichlet's divisor problem
• Ramanujan–Petersson conjecture
• Sato–Tate conjecture
• Scholz conjecture
• 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?^[136]
• The uniqueness conjecture for Markov numbers^[137]
• Vojta's conjecture

Open questions[]

• 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?
• Are there infinitely many perfect numbers?
• Does every rational number with an odd denominator have an odd greedy expansion?
• Do any Lychrel numbers exist?
• Do any odd perfect numbers exist?
• Do any odd noncototients exist?
• Do any odd superperfect 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
• Lang and Trotter's conjecture on supersingular primes
• 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[]

Goldbach's conjecture states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.

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 (so-called Selfridge's conjecture)?
• Twin prime conjecture
• Does the conjectural converse of Wolstenholme's theorem hold for all natural numbers?

Open questions[]

• Are all Euclid numbers square-free?
• Are all Fermat numbers square-free?
• Are all Mersenne numbers of prime index square-free?
• Are there any composite c satisfying 2^{c − 1} ≡ 1 (mod c²)?
• Are there any Wall–Sun–Sun primes?
• Are there any Wieferich primes in base 47?
• Are there infinitely many balanced primes?
• Are there infinitely many Carol primes?
• Are there infinitely many cousin primes?
• Are there infinitely many Cullen primes?
• Are there infinitely many Euclid primes?
• Are there infinitely many Fibonacci primes?
• Are there infinitely many Kummer primes?
• Are there infinitely many Kynea primes?
• Are there infinitely many Lucas primes?
• Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
• Are there infinitely many Newman–Shanks–Williams primes?
• Are there infinitely many palindromic primes to every base?
• Are there infinitely many Pell primes?
• Are there infinitely many Pierpont primes?
• Are there infinitely many prime quadruplets?
• Are there infinitely many prime triplets?
• Are there infinitely many regular primes, and if so is their relative density ${\displaystyle e^{-1/2}}$?
• Are there infinitely many sexy primes?
• Are there infinitely many safe and Sophie Germain primes?
• Are there infinitely many Wagstaff primes?
• Are there infinitely many Wieferich primes?
• Are there infinitely many Wilson primes?
• Are there infinitely many Wolstenholme primes?
• Are there infinitely many Woodall primes?
• Can a prime p satisfy 2^p − 1 ≡ 1 (mod p²) and 3^p − 1 ≡ 1 (mod p²) 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 Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)
• For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p²)?^[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 −4k⁴ 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×bⁿ+c)/gcd(k+c,b−1) with integer n ≥ 1?
• Is every Fermat number 2²ⁿ + 1 composite for ${\displaystyle n>4}$?
• Is 509,203 the lowest Riesel number?

Set theory[]

Note: These conjectures are about models of Zermelo-Frankel set theory with choice, and may not be able to be expressed in models of other set theories such as the various constructive set theories or non-wellfounded set theory.

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 ${\displaystyle {\diamondsuit (E_{\operatorname {cf} (\lambda )}^{\lambda ^{+}}})}$ for every singular cardinal ${\displaystyle \lambda }$?
• Does the Generalized Continuum Hypothesis imply the existence of an ℵ₂-Suslin tree?
• If ℵ_ω is a strong limit cardinal, then 2^ℵ_ω < 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 Ω-conjecture

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 ${\displaystyle 2^{\aleph _{0}}>\aleph _{2}}$?
• Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?

Topology[]

The unknotting problem asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot.

Conjectures and problems[]

• Baum–Connes conjecture
• Bing–Borsuk conjecture
• Borel conjecture
• Halperin conjecture
• Hilbert–Smith conjecture
• ^[143]
• Novikov conjecture
• Telescope conjectures
• Unknotting problem
• Volume conjecture
• Whitehead conjecture
• Zeeman conjecture

Problems solved since 1995[]

Ricci flow, here illustrated with a 2D manifold, was the key tool in Grigori Perelman's solution of the Poincaré conjecture.

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 ${\displaystyle KS_{r}}$ and ${\displaystyle KS'_{r}}$ conjectures, Bourgain-Tzafriri conjecture and ${\displaystyle R_{\epsilon }}$-conjecture)

Combinatorics[]

• Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)^[146]
• McMullen's g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)^[147][148]
• Hirsch conjecture (Francisco Santos Leal, 2010)^[149][150]

Dynamical systems[]

• Painlevé conjecture (Jinxin Xue, 2014)^[151][152]

Game theory[]

• The angel problem (Various independent proofs, 2006)^{[153][154][155][156]}

Geometry[]

21st century[]

• Yau's conjecture (Antoine Song, 2018)^[157][158]
• Pentagonal tiling (Michaël Rao, 2017)^[159]
• Erdős distinct distances problem (Larry Guth, Netz Hawk Katz, 2011)^[160]
• Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)^[161]

20th century[]

• Kepler conjecture (Ferguson, Hales, 1998)^[162]
• Dodecahedral conjecture (Hales, McLaughlin, 1998)^[163]

Graph theory[]

• Ringel's conjecture on graceful labeling of trees (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)^[164][165]
• Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)^[166]
• Babai's problem (Problem 3.3 in "Spectra of Cayley graphs") (Alireza Abdollahi, Maysam Zallaghi, 2015)^[167]
• Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
• Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)^[168]
• Erdős–Menger conjecture (Aharoni, Berger 2007)^[169]
• Road coloring conjecture (Avraham Trahtman, 2007)^[170]

Group theory[]

• Hanna Neumann conjecture (Mineyev, 2011)^[171]
• Density theorem (Namazi, Souto, 2010)^[172]
• Full classification of finite simple groups (Harada, Solomon, 2008)

Number theory[]

21st century[]

• Duffin-Schaeffer conjecture (Dimitris Koukoulopoulos, James Maynard, 2019)
• Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)^[173]
• Goldbach's weak conjecture (Harald Helfgott, 2013)^{[174][175][176]}
• Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)^{[177][178][179]}
• Existence of bounded gaps between primes (Yitang Zhang, Polymath8, James Maynard, 2013)^{[180][181][182]}

20th century[]

• Fermat's Last Theorem (Andrew Wiles and Richard Taylor, 1995)^[183][184]

Ramsey theory[]

• Burr–Erdős conjecture (Choongbum Lee, 2017)^[185]
• Boolean Pythagorean triples problem (Marijn Heule, Oliver Kullmann, Victor Marek, 2016)^[186][187]

Theoretical computer science[]

• Sensitivity conjecture for Boolean functions (Hao Huang, 2019) ^[188]

Topology[]

• Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020)^[189][190]
• Virtual Haken conjecture (Agol, Groves, Manning, 2012)^[191] (and by work of Wise also virtually fibered conjecture)
• Hsiang–Lawson's conjecture (Brendle, 2012)^[192]
• Ehrenpreis conjecture (Kahn, Markovic, 2011)^[193]
• Atiyah conjecture (Austin, 2009)^[194]
• Cobordism hypothesis (Jacob Lurie, 2008)^[195]
• Geometrization conjecture, proven by Grigori Perelman^[196] in a series of preprints in 2002–2003.^[197]
• Spherical space form conjecture (Grigori Perelman, 2006)

Uncategorised[]

21st century[]

2010s[]

• Erdős discrepancy problem (Terence Tao, 2015)^[198]
• Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)^[199]
• (Cheeger, Naber, 2014)^[200]
• Gaussian correlation inequality (Thomas Royen, 2014)^[201]
• Willmore conjecture (Fernando Codá Marques and André Neves, 2012)^[202]
• (Newman, Nikolov, 2011)^[203]
• Bloch–Kato conjecture (Voevodsky, 2011)^[204] (and Quillen–Lichtenbaum conjecture and by work of Geisser and Levine (2001) also Beilinson–Lichtenbaum conjecture^{[205][206][207]})
• Sidon set problem (J. Cilleruelo, I. Ruzsa and C. Vinuesa, 2010)^[208]

2000s[]

• (Matmann, Solis, 2009)^[209]
• Surface subgroup conjecture (Kahn, Markovic, 2009)^[210]
• and the (Lu, 2007)^[211]
• (Nils Dencker, 2005)^[212][213]
• Lax conjecture (Lewis, Parrilo, Ramana, 2005)^[214]
• The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)^[215]
• Tameness conjecture and Ahlfors measure conjecture (Ian Agol, 2004)^[216]
• Robertson–Seymour theorem (Robertson, Seymour, 2004)^[217]
• Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)^[218] (and also )
• Green–Tao theorem (Ben J. Green and Terence Tao, 2004)^[219]
• Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)^[220]
• Carpenter's rule problem (Connelly, Demaine, Rote, 2003)^[221]
• Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)^[222][223]
• Milnor conjecture (Vladimir Voevodsky, 2003)^[224]
• Kemnitz's conjecture (Reiher, 2003, di Fiore, 2003)^[225]
• Nagata's conjecture (Shestakov, Umirbaev, 2003)^[226]
• (Baruch, 2003)^[227]
• Poincaré conjecture (Grigori Perelman, 2002)^[196]
• Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)^[228]
• (Haas, 2002)^[229]
• Vaught conjecture (Knight, 2002)^[230]
• Double bubble conjecture (Hutchings, Morgan, Ritoré, Ros, 2002)^[231]
• Catalan's conjecture (Preda Mihăilescu, 2002)^[232]
• n! conjecture (Haiman, 2001)^[233] (and also Macdonald positivity conjecture)
• Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh and Tchamitchian, 2001)^[234]
• (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)^[235]
• Modularity theorem (Breuil, Conrad, Diamond and Taylor, 2001)^[236]
• (Florian Luca, 2001)^[237]
• Berry–Robbins problem (Atiyah, 2000)^[238]
• Erdős–Graham problem (Croot, 2000)^[239]

20th century[]

• Honeycomb conjecture (Thomas Hales, 1999)^[240]
• Gradient conjecture (Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusinski, 1999)^[241]
• Bogomolov conjecture (Emmanuel Ullmo, 1998, Shou-Wu Zhang, 1998)^[242][243]
• Lafforgue's theorem (Laurent Lafforgue, 1998)^[244]
• Ganea conjecture (Iwase, 1997)^[245]
• Torsion conjecture (Merel, 1996)^[246]
• (Chen, 1996)^[247]

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

References[]

Further reading[]

Books discussing problems solved since 1995[]

• Singh, Simon (2002). Fermat's Last Theorem. Fourth Estate. ISBN 978-1-84115-791-7.
• O'Shea, Donal (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-84614-012-9.
• Szpiro, George G. (2003). Kepler's Conjecture. Wiley. ISBN 978-0-471-08601-7.
• Ronan, Mark (2006). Symmetry and the Monster. Oxford. ISBN 978-0-19-280722-9.

Books discussing unsolved problems[]

• Chung, Fan; Graham, Ron (1999). Erdös on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 978-1-56881-111-6.
• Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 978-0-387-97506-1.
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 978-0-387-20860-2.
• Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 978-0-88385-315-3.
• du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 978-0-06-093558-0.
• Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 978-0-309-08549-6.
• Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8.
• Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 978-0-691-11748-5.
• Ji, Lizhen; Poon, Yat-Sun; Yau, Shing-Tung (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 978-1-57146-278-7.
• Waldschmidt, Michel (2004). "Open Diophantine Problems" (PDF). Moscow Mathematical Journal. 4 (1): 245–305. arXiv:math/0312440. doi:10.17323/1609-4514-2004-4-1-245-305. ISSN 1609-3321. 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 is a collection of unsolved problems in semigroup theory.^[1][2]
• Formulation of ${\displaystyle 50}$ unresloved problems for infinite Abelian groups are depicted in the book^[3]
• The list of ${\displaystyle 17}$ 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 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).
• . "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 low-dimensional 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