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.
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 Kourovka Notebook (Russian: Коуровская тетрадь) is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.[11]
The Sverdlovsk Notebook (Russian: Свердловская тетрадь) is a collection of unsolved problems in semigroup theory, first published in 1969 and updated many times since.[12][13][14]
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.
Crouzeix's conjecture: the matrix norm of a complex function [math]\displaystyle{ f }[/math] applied to a complex matrix [math]\displaystyle{ A }[/math] is at most twice the supremum of [math]\displaystyle{ |f(z)| }[/math] over the field of values of [math]\displaystyle{ A }[/math].
Köthe conjecture: if a ring has no nil ideal other than [math]\displaystyle{ \{0\} }[/math], then it has no nil one-sided ideal other than [math]\displaystyle{ \{0\} }[/math].
Pierce–Birkhoff conjecture: every piecewise-polynomial [math]\displaystyle{ f:\mathbb{R}^{n}\rightarrow\mathbb{R} }[/math] is the maximum of a finite set of minimums of finite collections of polynomials.
Rota's basis conjecture: for matroids of rank [math]\displaystyle{ n }[/math] with [math]\displaystyle{ n }[/math] disjoint bases [math]\displaystyle{ B_{i} }[/math], it is possible to create an [math]\displaystyle{ n \times n }[/math] matrix whose rows are [math]\displaystyle{ B_{i} }[/math] and whose columns are also bases.
Serre's positivity conjecture that if [math]\displaystyle{ R }[/math] is a commutative regular local ring, and [math]\displaystyle{ P, Q }[/math] are prime ideals of [math]\displaystyle{ R }[/math], then [math]\displaystyle{ \dim (R/P) + \dim (R/Q) = \dim (R) }[/math] implies [math]\displaystyle{ \chi(R/P, R/Q) \gt 0 }[/math].
Wild problems: problems involving classification of pairs of [math]\displaystyle{ n\times n }[/math] matrices under simultaneous conjugation.
Zariski–Lipman conjecture: for a complex algebraic variety[math]\displaystyle{ V }[/math] with coordinate ring [math]\displaystyle{ R }[/math], if the derivations of [math]\displaystyle{ R }[/math] are a free module over [math]\displaystyle{ R }[/math], then [math]\displaystyle{ V }[/math] is smooth.
Zauner's conjecture: do SIC-POVMs exist in all dimensions?
Zilber–Pink conjecture that if [math]\displaystyle{ X }[/math] is a mixed Shimura variety or semiabelian variety defined over [math]\displaystyle{ \mathbb{C} }[/math], and [math]\displaystyle{ V \subseteq X }[/math] is a subvariety, then [math]\displaystyle{ V }[/math] contains only finitely many atypical subvarieties.
The free Burnside group [math]\displaystyle{ B(2,3) }[/math] is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups [math]\displaystyle{ B(m,n) }[/math] are finite remains open.
Andrews–Curtis conjecture: every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on relators and conjugations of relators
Burnside problem: for which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems[19]
Herzog–Schönheim conjecture: if a finite system of left cosets of subgroups of a group [math]\displaystyle{ G }[/math] form a partition of [math]\displaystyle{ G }[/math], 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
The area of the blue region converges to the Euler–Mascheroni constant, which may or may not be a rational number.
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 [math]\displaystyle{ \mathbb{C} }[/math]
Fuglede's conjecture on whether nonconvex sets in [math]\displaystyle{ \mathbb{R} }[/math] and [math]\displaystyle{ \mathbb{R}^{2} }[/math] are spectral if and only if they tile by translation.
The mean value problem: given a complexpolynomial[math]\displaystyle{ f }[/math] of degree[math]\displaystyle{ d \ge 2 }[/math] and a complex number [math]\displaystyle{ z }[/math], is there a critical point[math]\displaystyle{ c }[/math] of [math]\displaystyle{ f }[/math] such that [math]\displaystyle{ |f(z)-f(c)| \le |f'(z)||z-c| }[/math]?
The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy[23]
Schanuel's conjecture on the transcendence degree of exponentials of linearly independent irrationals[20]
Sendov's conjecture: if a complex polynomial with degree at least [math]\displaystyle{ 2 }[/math] has all roots in the closed unit disk, then each root is within distance [math]\displaystyle{ 1 }[/math] from some critical point.
Vitushkin's conjecture on compact subsets of [math]\displaystyle{ \mathbb{C} }[/math] with analytic capacity [math]\displaystyle{ 0 }[/math]
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?[27]
The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
The lonely runner conjecture – if [math]\displaystyle{ k }[/math] runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance [math]\displaystyle{ 1/k }[/math] from each other runner) at some time?[28]
Map folding – various problems in map folding and stamp folding.
No-three-in-line problem – how many points can be placed in the [math]\displaystyle{ n \times n }[/math] grid so that no three of them lie on a line?
The sunflower conjecture – can the number of [math]\displaystyle{ k }[/math] size sets required for the existence of a sunflower of [math]\displaystyle{ r }[/math] sets be bounded by an exponential function in [math]\displaystyle{ k }[/math] for every fixed [math]\displaystyle{ r\gt 2 }[/math]?
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[30]
Fatou conjecture that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters.
Furstenberg conjecture – is every invariant and ergodic measure for the [math]\displaystyle{ \times 2,\times 3 }[/math] action on the circle either Lebesgue or atomic?
Does every positive integer generate a juggler sequence terminating at 1?
Lyapunov function: Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion?
Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also Hales-Jewett theorem and nd game)[40]
Chess:
What is the outcome of a perfectly played game of chess? (See also first-move advantage in chess)
Nagata–Biran conjecture that if [math]\displaystyle{ X }[/math] is a smooth algebraic surface and [math]\displaystyle{ L }[/math] is an ample line bundle on [math]\displaystyle{ X }[/math] of degree [math]\displaystyle{ d }[/math], then for sufficiently large [math]\displaystyle{ r }[/math], the Seshadri constant satisfies [math]\displaystyle{ \varepsilon(p_1,\ldots,p_r;X,L) = d/\sqrt{r} }[/math].
Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a boundedn-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?[44]
The Erdős–Oler conjecture: when [math]\displaystyle{ n }[/math] is a triangular number, packing [math]\displaystyle{ n-1 }[/math] circles in an equilateral triangle requires a triangle of the same size as packing [math]\displaystyle{ n }[/math] circles[45]
Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets[47]
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.
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.[50]
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[51]
The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds[52]
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.
The big-line-big-clique conjecture on the existence of either many collinear points or many mutually visible points in large planar point sets[53]
The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies[54]
The Atiyah conjecture on configurations on the invertibility of a certain [math]\displaystyle{ n }[/math]-by-[math]\displaystyle{ n }[/math] matrix depending on [math]\displaystyle{ n }[/math] points in [math]\displaystyle{ \mathbb{R}^{3} }[/math][63]
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[64]
Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?[65]
Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?[66]
Ehrhart's volume conjecture: a convex body [math]\displaystyle{ K }[/math] in [math]\displaystyle{ n }[/math] dimensions containing a single lattice point in its interior as its center of mass cannot have volume greater than [math]\displaystyle{ (n+1)^{n}/n! }[/math]
Falconer's conjecture: sets of Hausdorff dimension greater than [math]\displaystyle{ d/2 }[/math] in [math]\displaystyle{ \mathbb{R}^d }[/math] must have a distance set of nonzero Lebesgue measure[68]
The values of the Hermite constants for dimensions other than 1–8 and 24
Inscribed square problem, also known as Toeplitz' conjecture and the square peg problem – does every Jordan curve have an inscribed square?[69]
The Kakeya conjecture – do [math]\displaystyle{ n }[/math]-dimensional sets that contain a unit line segment in every direction necessarily have Hausdorff dimension and Minkowski dimension equal to [math]\displaystyle{ n }[/math]?[70]
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[71]
Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs[82]
Meyniel's conjecture that cop number is [math]\displaystyle{ O(\sqrt n) }[/math][83]
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.
The 1-factorization conjecture that if [math]\displaystyle{ n }[/math] is odd or even and [math]\displaystyle{ k \geq n, n - 1 }[/math] respectively, then a [math]\displaystyle{ k }[/math]-regular graph with [math]\displaystyle{ 2n }[/math] vertices is 1-factorable.
The overfull conjecture that a graph with maximum degree [math]\displaystyle{ \Delta(G) \geq n/3 }[/math] is class 2 if and only if it has an overfull subgraph[math]\displaystyle{ S }[/math] satisfying [math]\displaystyle{ \Delta(S) = \Delta(G) }[/math].
The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree[92]
Degree diameter problem: given two positive integers [math]\displaystyle{ d, k }[/math], what is the largest graph of diameter [math]\displaystyle{ k }[/math] such that all vertices have degrees at most [math]\displaystyle{ d }[/math]?
Jørgensen's conjecture that every 6-vertex-connected K6-minor-free graph is an apex graph[102]
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.[112]
Sumner's conjecture: does every [math]\displaystyle{ (2n-2) }[/math]-vertex tournament contain as a subgraph every [math]\displaystyle{ n }[/math]-vertex oriented tree?[116]
Tuza's conjecture: if the maximum number of disjoint triangles is [math]\displaystyle{ \nu }[/math], can all triangles be hit by a set of at most [math]\displaystyle{ 2\nu }[/math] edges?[117]
Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs[123])
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[124]
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?[127]
The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in [math]\displaystyle{ \aleph_0 }[/math] is a simple algebraic group over an algebraically closed field.
The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for [math]\displaystyle{ \aleph_1 }[/math]-saturated models of a countable theory.[131]
Shelah's categoricity conjecture for [math]\displaystyle{ L_{\omega_1,\omega} }[/math]: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[131]
Shelah's eventual categoricity conjecture: For every cardinal [math]\displaystyle{ \lambda }[/math] there exists a cardinal [math]\displaystyle{ \mu(\lambda) }[/math] such that if an AEC K with LS(K)<= [math]\displaystyle{ \lambda }[/math] is categorical in a cardinal above [math]\displaystyle{ \mu(\lambda) }[/math] then it is categorical in all cardinals above [math]\displaystyle{ \mu(\lambda) }[/math].[131][132]
The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
The stable forking conjecture for simple theories[133]
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?[134]
The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[135]
Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality [math]\displaystyle{ \aleph_{\omega_1} }[/math] does it have a model of cardinality continuum?[136]
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 [math]\displaystyle{ \aleph_n }[/math], is it categorical in every cardinal?[137][138]
Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?[139]
Is the theory of the field of Laurent series over [math]\displaystyle{ \mathbb{Z}_p }[/math]decidable? of the field of polynomials over [math]\displaystyle{ \mathbb{C} }[/math]?
Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[140]
Determine the structure of Keisler's order.[141][142]
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 is odd.
Brocard's problem: are there any integer solutions to [math]\displaystyle{ n! + 1 = m^{2} }[/math] other than [math]\displaystyle{ n = 4, 5, 7 }[/math]?
Büchi's problem on sufficiently large sequences of square numbers with constant second difference.
Casas-Alvero conjecture: if a polynomial of degree [math]\displaystyle{ d }[/math] defined over a field[math]\displaystyle{ K }[/math] of characteristic[math]\displaystyle{ 0 }[/math] has a factor in common with its first through [math]\displaystyle{ d - 1 }[/math]-th derivative, then must [math]\displaystyle{ f }[/math] be the [math]\displaystyle{ d }[/math]-th power of a linear polynomial?
Erdős–Moser problem: is [math]\displaystyle{ 1^{1} + 2^{1} = 3^{1} }[/math] the only solution to the Erdős–Moser equation?
Erdős–Straus conjecture: for every [math]\displaystyle{ n \geq 2 }[/math], there are positive integers [math]\displaystyle{ x, y, z }[/math] such that [math]\displaystyle{ 4/n = 1/x + 1/y + 1/z }[/math].
Exponent pair conjecture: for all [math]\displaystyle{ \epsilon \gt 0 }[/math], is the pair [math]\displaystyle{ (\epsilon, 1/2 + \epsilon) }[/math] 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 [math]\displaystyle{ (x^{m} - 1)/(x - 1) = (y^{n} - 1)/(y - 1) }[/math] where [math]\displaystyle{ x \gt y \gt 1 }[/math] and [math]\displaystyle{ m, n \gt 2 }[/math].
Riemann hypothesis: do the nontrivial zeros of the Riemann zeta function lie on the critical line [math]\displaystyle{ 1/2 + it }[/math] with real [math]\displaystyle{ t }[/math]?
Hall's conjecture: for any [math]\displaystyle{ \epsilon \gt 0 }[/math], there is some constant [math]\displaystyle{ c(\epsilon) }[/math] such that either [math]\displaystyle{ y^{2} = x^{3} }[/math] or [math]\displaystyle{ |y^{2} - x^{3}| \gt c(\epsilon)x^{1/2 - \epsilon} }[/math].
Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function[143]
Lehmer's totient problem: if [math]\displaystyle{ \phi(n) }[/math] divides [math]\displaystyle{ n - 1 }[/math], must [math]\displaystyle{ n }[/math] be prime?
Littlewood conjecture: for any two real numbers [math]\displaystyle{ \alpha, \beta }[/math], [math]\displaystyle{ \liminf_{n \rightarrow \infty} n\,\Vert n\alpha\Vert\,\Vert n\beta\Vert = 0 }[/math], where [math]\displaystyle{ \Vert x\Vert }[/math] is the distance from [math]\displaystyle{ x }[/math] to the nearest integer.
Mahler's 3/2 problem that no real number [math]\displaystyle{ x }[/math] has the property that the fractional parts of [math]\displaystyle{ x(3/2)^{n} }[/math] are less than [math]\displaystyle{ 1/2 }[/math] for all positive integers [math]\displaystyle{ n }[/math].
n conjecture: a generalization of the abc conjecture to more than three integers.
abc conjecture: for any [math]\displaystyle{ \epsilon \gt 0 }[/math], [math]\displaystyle{ \text{rad}(abc)^{1+\epsilon} \lt c }[/math] is true for only finitely many positive [math]\displaystyle{ a, b, c }[/math] such that [math]\displaystyle{ a + b = c }[/math].
Szpiro's conjecture: for any [math]\displaystyle{ \epsilon \gt 0 }[/math], there is some constant [math]\displaystyle{ C(\epsilon) }[/math] such that, for any elliptic curve [math]\displaystyle{ E }[/math] defined over [math]\displaystyle{ \mathbb{Q} }[/math] with minimal discriminant [math]\displaystyle{ \Delta }[/math] and conductor [math]\displaystyle{ f }[/math], we have [math]\displaystyle{ |\Delta| \leq C(\epsilon) \cdot f^{6+\epsilon} }[/math].
Pillai's conjecture: for any [math]\displaystyle{ A, B, C }[/math], the equation [math]\displaystyle{ Ax^{m} - By^{n} = C }[/math] has finitely many solutions when [math]\displaystyle{ m, n }[/math] are not both [math]\displaystyle{ 2 }[/math].
Piltz divisor problem on bounding [math]\displaystyle{ \Delta_{k}(x) = D_{k}(x) - xP_{k}(log(x)) }[/math]
Dirichlet's divisor problem: the specific case of the Piltz divisor problem for [math]\displaystyle{ k = 1 }[/math]
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 [math]\displaystyle{ 2^{n} - 1 }[/math] is at most [math]\displaystyle{ n - 1 }[/math] plus the length of the shortest addition chain producing [math]\displaystyle{ n }[/math].
Beal's conjecture: for all integral solutions to [math]\displaystyle{ A^{x} + B^{y} = C^{z} }[/math] where [math]\displaystyle{ x, y, z \gt 2 }[/math], all three numbers [math]\displaystyle{ A, B, C }[/math] must share some prime factor.
Erdős–Heilbronn conjecture that [math]\displaystyle{ |2^\wedge A| \ge \min\{p,2|A|-3\} }[/math] if [math]\displaystyle{ p }[/math] is a prime and [math]\displaystyle{ A }[/math] is a nonempty subset of the field [math]\displaystyle{ \mathbb{Z}/p\mathbb{Z} }[/math].
Erdős–Turán conjecture on additive bases: if [math]\displaystyle{ B }[/math] is an additive basis of order [math]\displaystyle{ 2 }[/math], then the number of ways that positive integers [math]\displaystyle{ n }[/math] can be expressed as the sum of two numbers in [math]\displaystyle{ B }[/math] must tend to infinity as [math]\displaystyle{ n }[/math] tends to infinity.
Fermat–Catalan conjecture: there are finitely many distinct solutions [math]\displaystyle{ (a^{m}, b^{n}, c^{k}) }[/math] to the equation [math]\displaystyle{ a^{m} + b^{n} = c^{k} }[/math] with [math]\displaystyle{ a, b, c }[/math] being positive coprime integers and [math]\displaystyle{ m, n, k }[/math] being positive integers satisfying [math]\displaystyle{ 1/m + 1/n + 1/k \lt 1 }[/math].
Lander, Parkin, and Selfridge conjecture: if the sum of [math]\displaystyle{ m }[/math][math]\displaystyle{ k }[/math]-th powers of positive integers is equal to a different sum of [math]\displaystyle{ n }[/math][math]\displaystyle{ k }[/math]-th powers of positive integers, then [math]\displaystyle{ m + n \geq k }[/math].
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 [math]\displaystyle{ \{1, \ldots, 2n\} }[/math]
Hermite's problem: is it possible, for any natural number [math]\displaystyle{ n }[/math], to assign a sequence of natural numbers to each real number such that the sequence for [math]\displaystyle{ x }[/math] is eventually periodic if and only if [math]\displaystyle{ x }[/math] is algebraic of degree [math]\displaystyle{ n }[/math]?
Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant [math]\displaystyle{ X }[/math] is within a constant multiple of [math]\displaystyle{ \sqrt{X}/\ln{X} }[/math]
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.
Agoh–Giuga conjecture on the Bernoulli numbers that [math]\displaystyle{ p }[/math] is prime if and only if [math]\displaystyle{ pB_{p-1} \equiv -1 \pmod p }[/math]
Agrawal's conjecture that given coprime positive integers[math]\displaystyle{ n }[/math] and [math]\displaystyle{ r }[/math], if [math]\displaystyle{ (X - 1)^{n} \equiv X^{n} - 1 \pmod{n, X^{r} - 1} }[/math], then either [math]\displaystyle{ n }[/math] is prime or [math]\displaystyle{ n^{2} \equiv 1 \pmod{r} }[/math]
Brocard's conjecture: there are always at least [math]\displaystyle{ 4 }[/math]prime numbers between consecutive squares of prime numbers, aside from [math]\displaystyle{ 2^{2} }[/math] and [math]\displaystyle{ 3^{2} }[/math].
Bunyakovsky conjecture: if an integer-coefficient polynomial [math]\displaystyle{ f }[/math] has a positive leading coefficient, is irreducible over the integers, and has no common factors over all [math]\displaystyle{ f(x) }[/math] where [math]\displaystyle{ x }[/math] is a positive integer, then [math]\displaystyle{ f(x) }[/math] is prime infinitely often.
Dickson's conjecture: for a finite set of linear forms [math]\displaystyle{ a_{1} + b_{1}n, \ldots, a_{k} + b_{k}n }[/math] with each [math]\displaystyle{ b_{i} \geq 1 }[/math], there are infinitely many [math]\displaystyle{ n }[/math] for which all forms are prime, unless there is some congruence condition preventing it.
Dubner's conjecture: every even number greater than [math]\displaystyle{ 4208 }[/math] is the sum of two primes which both have a twin.
Feit–Thompson conjecture: for all distinct prime numbers[math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math], [math]\displaystyle{ (p^{q} - 1)/(p - 1) }[/math] does not divide [math]\displaystyle{ (q^{p} - 1)/(q - 1) }[/math]
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?
Goldbach conjecture: all even natural numbers greater than [math]\displaystyle{ 2 }[/math] are the sum of two prime numbers.
Legendre's conjecture: for every positive integer [math]\displaystyle{ n }[/math], there is a prime between [math]\displaystyle{ n^{2} }[/math] and [math]\displaystyle{ (n+1)^{2} }[/math].
New Mersenne conjecture: for any odd natural number[math]\displaystyle{ p }[/math], if any two of the three conditions [math]\displaystyle{ p = 2^{k} \pm 1 }[/math] or [math]\displaystyle{ p = 4^{k} \pm 3 }[/math], [math]\displaystyle{ 2^{p} - 1 }[/math] is prime, and [math]\displaystyle{ (2^{p} + 1)/3 }[/math] is prime are true, then the third condition is true.
Polignac's conjecture: for all positive even numbers [math]\displaystyle{ n }[/math], there are infinitely many prime gaps of size [math]\displaystyle{ n }[/math].
Schinzel's hypothesis H that for every finite collection [math]\displaystyle{ \{f_{1}, \ldots, f_{k}\} }[/math] of nonconstant irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers [math]\displaystyle{ n }[/math] for which [math]\displaystyle{ f_{1}(n), \ldots, f_{k}(n) }[/math] are all primes, or there is some fixed divisor [math]\displaystyle{ m \gt 1 }[/math] which, for all [math]\displaystyle{ n }[/math], divides some [math]\displaystyle{ f_{i}(n) }[/math].
Can a prime p satisfy [math]\displaystyle{ 2^{p-1}\equiv 1\pmod{p^2} }[/math] and [math]\displaystyle{ 3^{p-1}\equiv 1\pmod{p^2} }[/math] simultaneously?[150]
For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)
For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[151]
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 [math]\displaystyle{ k\geq 1, b\geq 2, c\neq 0 }[/math], with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form [math]\displaystyle{ (k\times b^n+c)/\text{gcd}(k+c,b-1) }[/math] with integer n ≥ 1?
Is every Fermat number[math]\displaystyle{ 2^{2^n} + 1 }[/math] composite for [math]\displaystyle{ n \gt 4 }[/math]?
Note: These conjectures are about models of Zermelo-Frankel set theory with choice, and may not be able to be expressed in models of other set theories such as the various constructive set theories or non-wellfounded set theory.
(Woodin) Does the generalized continuum hypothesis below a strongly compact cardinal imply the generalized continuum hypothesis everywhere?
Does the generalized continuum hypothesis entail [math]\displaystyle{ {\diamondsuit(E^{\lambda^+} {\operatorname{cf}(\lambda)}}) }[/math] for every singular cardinal [math]\displaystyle{ \lambda }[/math]?
Does the generalized continuum hypothesis imply the existence of an ℵ2-Suslin tree?
If ℵω is a strong limit cardinal, is [math]\displaystyle{ 2^{\aleph_\omega} \lt \aleph_{\omega_1} }[/math] (see Singular cardinals hypothesis)? The best bound, ℵω4, was obtained by Shelah using his PCF theory.
The problem of finding the ultimate core model, one that contains all large cardinals.
This section duplicates the scope of other sections. Please discuss this issue on the talk page and edit it to conform with Wikipedia's Manual of Style by replacing the section with a link and a summary of the repeated material, or by spinning off the repeated text into an article in its own right.(August 2022)
Ricci flow, here illustrated with a 2D manifold, was the key tool in Grigori Perelman's solution of the Poincaré conjecture.
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)[164][165]
Ringel's conjecture that the complete graph [math]\displaystyle{ K_{2n+1} }[/math] can be decomposed into [math]\displaystyle{ 2n+1 }[/math] copies of any tree with [math]\displaystyle{ n }[/math] edges (Richard Montgomery, Benny Sudakov, Alexey Pokrovskiy, 2020)[201][202]
Disproof of Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)[203]
Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (Jeff Cheeger, Aaron Naber, 2014)[256]
↑A disproof has been announced, with a preprint made available on arXiv.[154]
References
↑Friedl, Stefan (2014). "Thurston's vision and the virtual fibering theorem for 3-manifolds". Jahresbericht der Deutschen Mathematiker-Vereinigung116 (4): 223–241. doi:10.1365/s13291-014-0102-x.
↑Thurston, William P. (1982). "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. New Series 6 (3): 357–381. doi:10.1090/S0273-0979-1982-15003-0.
↑Dowling, T. A. (February 1973). "A class of geometric lattices based on finite groups". Journal of Combinatorial Theory. Series B 14 (1): 61–86. doi:10.1016/S0095-8956(73)80007-3.
↑Smyth, Chris (2008), "The Mahler measure of algebraic numbers: a survey", in McKee, James; Smyth, Chris, Number Theory and Polynomials, London Mathematical Society Lecture Note Series, 352, Cambridge University Press, pp. 322–349, ISBN978-0-521-71467-9
↑Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order12 (4): 327–349, doi:10.1007/BF01110378.
↑"Some remarks on the lonely runner conjecture". Contributions to Discrete Mathematics13 (2): 1–31. 2018. doi:10.11575/cdm.v13i2.62728.
↑González-Jiménez, Enrique; Xarles, Xavier (2014). "On a conjecture of Rudin on squares in arithmetic progressions". LMS Journal of Computation and Mathematics17 (1): 58–76. doi:10.1112/S1461157013000259.
↑Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", American Mathematical Monthly100 (10): 916–925, doi:10.2307/2324212
↑Conway, John H.; Neil J.A. Sloane (1999), Sphere Packings, Lattices and Groups (3rd ed.), New York: Springer-Verlag, pp. 21–22, ISBN978-0-387-98585-5
↑The Reinhardt conjecture as an optimal control problem, 2017
↑Ghosh, Subir Kumar; Goswami, Partha P. (2013), "Unsolved problems in visibility graphs of points, segments, and polygons", ACM Computing Surveys46 (2): 22:1–22:29, doi:10.1145/2543581.2543589
↑Boltjansky, V.; Gohberg, I. (1985), "11. Hadwiger's Conjecture", Results and Problems in Combinatorial Geometry, Cambridge University Press, pp. 44–46.
↑Morris, Walter D.; Soltan, Valeriu (2000), "The Erdős-Szekeres problem on points in convex position—a survey", Bull. Amer. Math. Soc.37 (4): 437–458, doi:10.1090/S0273-0979-00-00877-6; Suk, Andrew (2016), "On the Erdős–Szekeres convex polygon problem", J. Amer. Math. Soc.30 (4): 1047–1053, doi:10.1090/jams/869
↑"The number of faces of centrally-symmetric polytopes", Graphs and Combinatorics5 (1): 389–391, 1989, doi:10.1007/BF01788696.
↑Moreno, José Pedro; Prieto-Martínez, Luis Felipe (2021). "El problema de los triángulos de Kobon" (in es). La Gaceta de la Real Sociedad Matemática Española24 (1): 111–130.
↑Brass, Peter; Moser, William; Pach, János (2005), "5.1 The Maximum Number of Unit Distances in the Plane", Research problems in discrete geometry, Springer, New York, pp. 183–190, ISBN978-0-387-23815-9
↑"Improved bounds for planar k-sets and related problems", Discrete & Computational Geometry19 (3): 373–382, 1998, doi:10.1007/PL00009354; Tóth, Gábor (2001), "Point sets with many k-sets", Discrete & Computational Geometry26 (2): 187–194, doi:10.1007/s004540010022.
↑Atiyah, Michael (2001), "Configurations of points", Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences359 (1784): 1375–1387, doi:10.1098/rsta.2001.0840, ISSN1364-503X, Bibcode: 2001RSPTA.359.1375A
↑Howards, Hugh Nelson (2013), "Forming the Borromean rings out of arbitrary polygonal unknots", Journal of Knot Theory and Its Ramifications22 (14): 1350083, 15, doi:10.1142/S0218216513500831
↑Arutyunyants, G.; Iosevich, A. (2004), "Falconer conjecture, spherical averages and discrete analogs", in Pach, János, Towards a Theory of Geometric Graphs, Contemp. Math., 342, Amer. Math. Soc., Providence, RI, pp. 15–24, doi:10.1090/conm/342/06127, ISBN978-0-8218-3484-8
↑Mahler, Kurt (1939). "Ein Minimalproblem für konvexe Polygone". Mathematica (Zutphen) B: 118–127.
↑Norwood, Rick; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser", Discrete & Computational Geometry7 (2): 153–162, doi:10.1007/BF02187832
↑Steininger, Jakob; Yurkevich, Sergey (December 27, 2021), An algorithmic approach to Rupert's problem
↑Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338
↑Ghomi, Mohammad (2018-01-01). "D "urer's Unfolding Problem for Convex Polyhedra". Notices of the American Mathematical Society65 (1): 25–27. doi:10.1090/noti1609. ISSN0002-9920.
↑Whyte, L. L. (1952), "Unique arrangements of points on a sphere", The American Mathematical Monthly59 (9): 606–611, doi:10.2307/2306764
↑Pleanmani, Nopparat (2019), "Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph", Discrete Mathematics, Algorithms and Applications11 (6): 1950068, 7, doi:10.1142/s179383091950068x
↑Baird, William; Bonato, Anthony (2012), "Meyniel's conjecture on the cop number: a survey", Journal of Combinatorics3 (2): 225–238, doi:10.4310/JOC.2012.v3.n2.a6
↑Bousquet, Nicolas; Bartier, Valentin (2019), "Linear Transformations Between Colorings in Chordal Graphs", in Bender, Michael A.; Svensson, Ola; Herman, Grzegorz, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, LIPIcs, 144, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 24:1–24:15, doi:10.4230/LIPIcs.ESA.2019.24, ISBN978-3-95977-124-5
↑"To the Moon and beyond", Graph Theory: Favorite Conjectures and Open Problems, II, Problem Books in Mathematics, Springer International Publishing, 2018, pp. 115–133, doi:10.1007/978-3-319-97686-0_11
↑Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, 1998, pp. 97–99.
↑"Extending the Gyárfás-Sumner conjecture", Journal of Combinatorial Theory, Series B 105: 11–16, 2014, doi:10.1016/j.jctb.2013.11.002
↑Toft, Bjarne (1996), "A survey of Hadwiger's conjecture", Congressus Numerantium115: 249–283.
↑Croft, Hallard T.; Falconer, Kenneth J. (1991), Unsolved Problems in Geometry, Springer-Verlag, Problem G10.
↑Fulek, Radoslav; Pach, János (2011), "A computational approach to Conway's thrackle conjecture", Computational Geometry44 (6–7): 345–355, doi:10.1016/j.comgeo.2011.02.001.
↑Gupta, Anupam; Newman, Ilan; Rabinovich, Yuri (2004), "Cuts, trees and [math]\displaystyle{ \ell_1 }[/math]-embeddings of graphs", Combinatorica24 (2): 233–269, doi:10.1007/s00493-004-0015-x
↑Hartsfield, Nora; Ringel, Gerhard (2013), Pearls in Graph Theory: A Comprehensive Introduction, Dover Books on Mathematics, Courier Dover Publications, p. 247, ISBN978-0-486-31552-2.
↑Pach, János; Sharir, Micha (2009), "5.1 Crossings—the Brick Factory Problem", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, 152, American Mathematical Society, pp. 126–127.
↑Jaeger, F. (1985), "A survey of the cycle double cover conjecture", Annals of Discrete Mathematics 27 – Cycles in Graphs, North-Holland Mathematics Studies, 27, pp. 1–12, doi:10.1016/S0304-0208(08)72993-1, ISBN978-0-444-87803-8.
↑Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics20 (2): P7, doi:10.37236/3252.
↑Ramachandran, S. (1981), "On a new digraph reconstruction conjecture", Journal of Combinatorial Theory, Series B 31 (2): 143–149, doi:10.1016/S0095-8956(81)80019-6
↑"A proof of Sumner's universal tournament conjecture for large tournaments", Proceedings of the London Mathematical Society, Third Series 102 (4): 731–766, 2011, doi:10.1112/plms/pdq035.
↑Tuza, Zsolt (1990). "A conjecture on triangles of graphs". Graphs and Combinatorics6 (4): 373–380. doi:10.1007/BF01787705.
↑Brešar, Boštjan; Dorbec, Paul; Goddard, Wayne; Hartnell, Bert L.; Henning, Michael A.; Klavžar, Sandi; Rall, Douglas F. (2012), "Vizing's conjecture: a survey and recent results", Journal of Graph Theory69 (1): 46–76, doi:10.1002/jgt.20565.
↑Shelah, Saharon (2009). Classification theory for abstract elementary classes. College Publications. ISBN978-1-904987-71-0.
↑Peretz, Assaf (2006). "Geometry of forking in simple theories". Journal of Symbolic Logic71 (1): 347–359. doi:10.2178/jsl/1140641179.
↑Cherlin, Gregory; Shelah, Saharon (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory. Series B 97 (3): 293–333. doi:10.1016/j.jctb.2006.05.008.
↑Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
↑Shelah, Saharon (2009). "Introduction to classification theory for abstract elementary classes". arXiv:0903.3428 [math.LO].
↑Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
↑Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
↑Keisler, HJ (1967). "Ultraproducts which are not saturated". J. Symb. Log.32 (1): 23–46. doi:10.2307/2271240.
↑Malliaris, Maryanthe; Shelah, Saharon (10 August 2012). "A Dividing Line Within Simple Unstable Theories". arXiv:1208.2140 [math.LO]. Malliaris, M.; Shelah, S. (2012). "A Dividing Line within Simple Unstable Theories". arXiv:1208.2140 [math.LO].
↑Guan, Qi'an; Zhou, Xiangyu (2015). "A solution of an [math]\displaystyle{ L^2 }[/math] extension problem with optimal estimate and applications". Annals of Mathematics181 (3): 1139–1208. doi:10.4007/annals.2015.181.3.6.
↑Merel, Loïc (1996). ""Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]". Inventiones Mathematicae124 (1): 437–449. doi:10.1007/s002220050059. Bibcode: 1996InMat.124..437M.
↑Cohen, Stephen D.; Fried, Michael D. (1995), "Lenstra's proof of the Carlitz–Wan conjecture on exceptional polynomials: an elementary version", Finite Fields and Their Applications1 (3): 372–375, doi:10.1006/ffta.1995.1027
↑Kurdyka, Krzysztof; Mostowski, Tadeusz; Parusiński, Adam (2000). "Proof of the gradient conjecture of R. Thom". Annals of Mathematics152 (3): 763–792. doi:10.2307/2661354.
↑Stanley, Richard P. (1994), "A survey of Eulerian posets", in Bisztriczky, T.; McMullen, P.; Schneider, R. et al., Polytopes: abstract, convex and computational (Scarborough, ON, 1993), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 440, Dordrecht: Kluwer Academic Publishers, pp. 301–333. See in particular p. 316.
↑Chung, Fan; Greene, Curtis; Hutchinson, Joan (April 2015). "Herbert S. Wilf (1931–2012)". Notices of the AMS62 (4): 358. doi:10.1090/noti1247. ISSN1088-9477. OCLC34550461. "The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004.".
↑Green, Ben (2004). "The Cameron–Erdős conjecture". The Bulletin of the London Mathematical Society36 (6): 769–778. doi:10.1112/S0024609304003650.
↑"News from 2007". AMS. 31 December 2007. https://www.ams.org/news?news_id=155. "The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture...""
↑Brown, Aaron; Fisher, David; Hurtado, Sebastian (2017-10-07). "Zimmer's conjecture for actions of SL(𝑚,ℤ)". arXiv:1710.02735 [math.DS].
↑Xue, Jinxin (2014). "Noncollision Singularities in a Planar Four-body Problem". arXiv:1409.0048 [math.DS].
↑Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics179 (2): 683–782. doi:10.4007/annals.2014.179.2.6.
↑Guth, Larry; Katz, Nets Hawk (2015). "On the Erdos distinct distance problem in the plane". Annals of Mathematics181 (1): 155–190. doi:10.4007/annals.2015.181.1.2.
↑Brock, Jeffrey F.; Canary, Richard D. (2012). "The classification of Kleinian surface groups, II: The Ending Lamination Conjecture". Annals of Mathematics176 (1): 1–149. doi:10.4007/annals.2012.176.1.1.
↑Faber, C.; Pandharipande, R. (2003), "Hodge integrals, partition matrices, and the [math]\displaystyle{ \lambda_g }[/math] conjecture", Ann. of Math., 2 157 (1): 97–124, doi:10.4007/annals.2003.157.97
↑Shestakov, Ivan P.; Umirbaev, Ualbai U. (2004). "The tame and the wild automorphisms of polynomial rings in three variables". Journal of the American Mathematical Society17 (1): 197–227. doi:10.1090/S0894-0347-03-00440-5.
↑Hutchings, Michael; Morgan, Frank; Ritoré, Manuel; Ros, Antonio (2002). "Proof of the double bubble conjecture". Annals of Mathematics. Second Series 155 (2): 459–489. doi:10.2307/3062123.
↑Teixidor i Bigas, Montserrat; Russo, Barbara (1999). "On a conjecture of Lange". Journal of Algebraic Geometry8 (3): 483–496. ISSN1056-3911. Bibcode: 1997alg.geom.10019R.
↑Ullmo, E (1998). "Positivité et Discrétion des Points Algébriques des Courbes". Annals of Mathematics147 (1): 167–179. doi:10.2307/120987.
↑Zhang, S.-W. (1998). "Equidistribution of small points on abelian varieties". Annals of Mathematics147 (1): 159–165. doi:10.2307/120986.
↑Hales, Thomas; Adams, Mark; Bauer, Gertrud; Dang, Dat Tat; Harrison, John; Hoang, Le Truong; Kaliszyk, Cezary; Magron, Victor et al. (2017). "A formal proof of the Kepler conjecture". Forum of Mathematics, Pi5: e2. doi:10.1017/fmp.2017.1.
↑Zang, Wenan; Jing, Guangming; Chen, Guantao (2019-01-29). "Proof of the Goldberg–Seymour Conjecture on Edge-Colorings of Multigraphs". arXiv:1901.10316v1 [math.CO].
↑Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra43 (12): 5159–5167. doi:10.1080/00927872.2014.967398.
↑Huh, June (2012). "Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs". Journal of the American Mathematical Society25 (3): 907–927. doi:10.1090/S0894-0347-2012-00731-0.
↑Chalopin, Jérémie; Gonçalves, Daniel (2009). "Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009". in Mitzenmacher, Michael. ACM. pp. 631–638. doi:10.1145/1536414.1536500.
↑Klin, M. H., M. Muzychuk and R. Poschel: The isomorphism problem for circulant graphs via Schur ring theory, Codes and Association Schemes, American Math. Society, 2001.
↑Mineyev, Igor (2012). "Submultiplicativity and the Hanna Neumann conjecture". Annals of Mathematics. Second Series 175 (1): 393–414. doi:10.4007/annals.2012.175.1.11.
↑Pila, Jonathan; Shankar, Ananth; Tsimerman, Jacob; Esnault, Hélène; Groechenig, Michael (2021-09-17). "Canonical Heights on Shimura Varieties and the André-Oort Conjecture". arXiv:2109.08788 [math.NT].
↑Bourgain, Jean; Ciprian, Demeter; Larry, Guth (2015). "Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three". Annals of Mathematics184 (2): 633–682. doi:10.4007/annals.2016.184.2.7. Bibcode: 2015arXiv151201565B.
↑Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
↑Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
↑Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
↑"Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer". Theory and Applications of Satisfiability Testing – SAT 2016. Lecture Notes in Computer Science. 9710. Springer, [Cham]. 2016. pp. 228–245. doi:10.1007/978-3-319-40970-2_15. ISBN978-3-319-40969-6.
↑"Embedded minimal tori in [math]\displaystyle{ S^3 }[/math] and the Lawson conjecture". Acta Mathematica211 (2): 177–190. 2013. doi:10.1007/s11511-013-0101-2.
↑"The good pants homology and the Ehrenpreis conjecture". Annals of Mathematics182 (1): 1–72. 2015. doi:10.4007/annals.2015.182.1.1.
↑Austin, Tim (December 2013). "Rational group ring elements with kernels having irrational dimension". Proceedings of the London Mathematical Society107 (6): 1424–1448. doi:10.1112/plms/pdt029. Bibcode: 2009arXiv0909.2360A.
↑Duncan, John F. R.; Griffin, Michael J.; Ono, Ken (1 December 2015). "Proof of the umbral moonshine conjecture". Research in the Mathematical Sciences2 (1): 26. doi:10.1186/s40687-015-0044-7. Bibcode: 2015arXiv150301472D.
↑Cheeger, Jeff; Naber, Aaron (2015). "Regularity of Einstein Manifolds and the Codimension 4 Conjecture". Annals of Mathematics182 (3): 1093–1165. doi:10.4007/annals.2015.182.3.5.
↑Newman, Alantha; Nikolov, Aleksandar (2011). "A counterexample to Beck's conjecture on the discrepancy of three permutations". arXiv:1104.2922 [cs.DM].
↑Geisser, Thomas; Levine, Marc (2001). "The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky". Journal für die Reine und Angewandte Mathematik2001 (530): 55–103. doi:10.1515/crll.2001.006.
↑Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics175 (3): 1127–1190. doi:10.4007/annals.2012.175.3.4.
↑Lu, Zhiqin (September 2011). "Normal Scalar Curvature Conjecture and its applications". Journal of Functional Analysis261 (5): 1284–1308. doi:10.1016/j.jfa.2011.05.002.
↑Lewis, A. S.; Parrilo, P. A.; Ramana, M. V. (2005). "The Lax conjecture is true". Proceedings of the American Mathematical Society133 (9): 2495–2499. doi:10.1090/S0002-9939-05-07752-X.
↑Baruch, Ehud Moshe (2003). "A proof of Kirillov's conjecture". Annals of Mathematics. Second Series 158 (1): 207–252. doi:10.4007/annals.2003.158.207.
↑Haiman, Mark (2001). "Hilbert schemes, polygraphs and the Macdonald positivity conjecture". Journal of the American Mathematical Society14 (4): 941–1006. doi:10.1090/S0894-0347-01-00373-3.
↑Auscher, Pascal; Hofmann, Steve; Lacey, Michael; McIntosh, Alan; Tchamitchian, Ph. (2002). "The solution of the Kato square root problem for second order elliptic operators on [math]\displaystyle{ \mathbb{R}^n }[/math]". Annals of Mathematics. Second Series 156 (2): 633–654. doi:10.2307/3597201.
↑Barbieri-Viale, Luca; Rosenschon, Andreas; Saito, Morihiko (2003). "Deligne's Conjecture on 1-Motives". Annals of Mathematics158 (2): 593–633. doi:10.4007/annals.2003.158.593.
↑"The geometry of classical particles". Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer. Surveys in Differential Geometry. 7. Somerville, Massachusetts: International Press. 2000. pp. 1–15. doi:10.4310/SDG.2002.v7.n1.a1.
Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN978-0-7607-8659-8.
Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN978-0-691-11748-5.
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. ISBN978-1-57146-278-7.
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].