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Many
mathematical problems A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more ...
have been stated but not yet solved. These problems come from many areas of mathematics, such as
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
,
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
,
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, algebraic, differential,
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
and Euclidean geometries,
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
,
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
,
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
,
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
,
Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a ...
,
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s, and
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the
Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. According ...
, receive considerable attention. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative. Although this list may never be comprehensive, the problems listed here vary widely in both difficulty and importance.


Lists of unsolved problems in mathematics

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


Millennium Prize Problems

Of the original seven
Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. According ...
listed by the Clay Mathematics Institute in 2000, six remain unsolved to date: *
Birch and Swinnerton-Dyer conjecture In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory an ...
*
Hodge conjecture In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjectu ...
* Navier–Stokes existence and smoothness * P versus NP *
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
*
Yang–Mills existence and mass gap The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 f ...
The seventh problem, the
Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
, was solved by
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
in 2003. However, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a ''four''-dimensional topological sphere can have two or more inequivalent
smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. Definition A smooth structure on a manifold M is ...
s—is unsolved.


Unsolved problems


Algebra

* Birch–Tate conjecture on the relation between the order of the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of the Steinberg group of the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
of a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
to the field's
Dedekind zeta function In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
. *
Bombieri–Lang conjecture In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type. Statement The weak Bombie ...
s on densities of rational points of algebraic surfaces and
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
defined on
number fields In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
and their field extensions. * Connes embedding problem in
Von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algeb ...
theory * Crouzeix's conjecture: the
matrix norm In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Preliminaries Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m rows ...
of a complex function f applied to a complex matrix A is at most twice the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of , f(z), over the field of values of A. *
Eilenberg–Ganea conjecture The Eilenberg–Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group ''G'' has cohomological dimension 2, then it h ...
: a group with
cohomological dimension In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. Cohomologica ...
2 also has a 2-dimensional
Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
K(G, 1). *
Farrell–Jones conjecture In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms. The motivation is the interest in the target of ...
on whether certain
assembly map In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geo ...
s are
isomorphisms In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. ** Bost conjecture: a specific case of the Farrell–Jones conjecture *
Finite lattice representation problem In mathematics, the finite lattice representation problem, or finite congruence lattice problem, asks whether every finite lattice is isomorphic to the congruence lattice of some finite algebra. Background A lattice is called algebraic if ...
: is every finite
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
isomorphic to the
congruence lattice In mathematics, a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation. Quotient algebras are also called factor algebras. Here, the congruence relation must be an equivalence relation ...
of some finite
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
? *
Goncharov conjecture In mathematics, the Goncharov conjecture is a conjecture introduced by suggesting that the cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually ...
on the
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of certain motivic complexes. * Green's conjecture: the
Clifford index In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve ''C''. Statement A divisor on a Riemann surface ''C'' is a formal sum \textstyle D = \sum_P ...
of a non-
hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dist ...
is determined by the extent to which it, as a
canonical curve In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, ...
, has linear syzygies. *
Grothendieck–Katz p-curvature conjecture In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density the ...
: a conjectured local–global principle for linear ordinary differential equations. *
Hadamard conjecture In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows ...
: for every positive integer k, a
Hadamard matrix In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows ...
of order 4k exists. ** Williamson conjecture: the problem of finding Williamson matrices, which can be used to construct Hadamard matrices. * Hadamard's maximal determinant problem: what is the largest
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of a matrix with entries all equal to 1 or –1? *
Hilbert's fifteenth problem Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. The problem is to put Schubert's enumerative calculus on a rigorous foundation. Introduction Schubert calculus is the ...
: put
Schubert calculus In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of ...
on a rigorous foundation. *
Hilbert's sixteenth problem Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the ''Problem of the topolo ...
: what are the possible configurations of the connected components of M-curves? *
Homological conjectures in commutative algebra In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectures relating various homological properties of a ...
* Jacobson's conjecture: the intersection of all powers of the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...
of a left-and-right
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
is precisely 0. *
Kaplansky's conjectures The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures. Group rings Let be a fie ...
*
Köthe conjecture In mathematics, the Köthe conjecture is a problem in ring theory, open . It is formulated in various ways. Suppose that ''R'' is a ring. One way to state the conjecture is that if ''R'' has no nil ideal, other than , then it has no nil one-sided ...
: if a ring has no
nil ideal In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent., p. 194 The nilradical of a commutative ring is an example of a nil ideal; in fact, it is ...
other than \, then it has no nil one-sided ideal other than \. *
Monomial conjecture In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following: Let ''A'' be a Noetherian local ring of Krull dimension ''d'' and let ''x''1, ..., ''x'd'' be a system of parameters for '' ...
on
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
s * Existence of
perfect cuboid In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. A perfect Euler brick ...
s and associated cuboid conjectures *
Pierce–Birkhoff conjecture In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a Supremum, maximum of finite Infimum, minima of finite collections of polynomials. It was first stated, albeit in non-Mathemati ...
: every piecewise-polynomial f:\mathbb^\rightarrow\mathbb is the maximum of a finite set of minimums of finite collections of polynomials. *
Rota's basis conjecture In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota. It states that, if ''X'' is either a vector space of dimension ''n'' or more generally a matr ...
: for matroids of rank n with n disjoint bases B_, it is possible to create an n \times n matrix whose rows are B_ and whose columns are also bases. *
Sendov's conjecture In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov. The ...
: if a complex polynomial with degree at least 2 has all roots in the closed unit disk, then each root is within distance 1 from some critical point. * Serre's conjecture II: if G is a simply connected
semisimple algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
over a perfect
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of
cohomological dimension In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. Cohomologica ...
at most 2, then the
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...
set H^(F, G) is zero. *
Serre's multiplicity conjectures In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection n ...
*
Uniform boundedness conjecture for rational points In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field K and a positive integer g \geq 2 that there exists a number N(K,g) depending only on K and g such that for any algebraic curve C ...
: do
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
g \geq 2 over
number fields In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
K have at most some bounded number N(K, g) of K-
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
s? * Wild problems: problems involving classification of pairs of n\times n matrices under simultaneous conjugation. * Zariski–Lipman conjecture: for a
complex algebraic variety In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex number In mathematics, a complex number is an element of a number system that extends the real numbers ...
V with
coordinate ring In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...
R, if the
derivations Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
of R are a
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
over R, then V is
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
. * Zauner's conjecture: do
SIC-POVM A symmetric, informationally complete, positive operator-valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain d ...
s exist in all dimensions? * Zilber–Pink conjecture that if X is a mixed
Shimura variety In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties a ...
or semiabelian variety defined over \mathbb, and V \subseteq X is a subvariety, then V contains only finitely many atypical subvarieties.


Representation theory

*
Arthur's conjectures In mathematics, the Arthur conjectures are some conjectures about automorphic representations of reductive groups over the adeles and unitary representations of reductive groups over local field In mathematics, a field ''K'' is called a (non-Arch ...
*
Dade's conjecture In finite group theory, Dade's conjecture is a conjecture relating the numbers of characters of blocks of a finite group to the numbers of characters of blocks of local subgroups, introduced by Everett C. Dade Everett Clarence Dade is a mathema ...
relating the numbers of
characters Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of blocks of a finite group to the numbers of characters of blocks of local
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation âˆ—, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation âˆ—. More precisely, ''H'' is a subgroup ...
s. * Demazure conjecture on representations of
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
s over the integers. *
McKay conjecture In mathematics, specifically in the field of group theory, the McKay conjecture is a conjecture of equality between the number of irreducible complex character theory, characters of degree not divisible by a prime number p to that of the centralize ...
: in a group G, the number of irreducible complex characters of degree not divisible by a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
p is equal to the number of irreducible complex characters of the
normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
of any Sylow p-subgroup within G.


Notebook problems

* The Dniester Notebook () lists several hundred unsolved problems in algebra, particularly
ring theory In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...
and modulus theory. * The Erlagol Notebook () lists unsolved problems in algebra and model theory.


Analysis

* The
Brennan conjecture The Brennan conjecture is a mathematical hypothesis (in complex analysis) for estimating (under specified conditions) the integral powers of the moduli of the derivatives of conformal maps into the open unit disk. The conjecture was formulated by Ja ...
: estimating the integral of powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of \mathbb * The
four exponentials conjecture In mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials. ...
: the transcendence of at least one of four exponentials of combinations of irrationals * Goodman's conjecture on the coefficients of multivalent functions *
Invariant subspace problem In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many vari ...
– does every
bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
on a complex
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
send some non-trivial closed subspace to itself? * Kung–Traub conjecture on the optimal order of a multipoint iteration without memory *
Lehmer's conjecture Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant \mu>1 such that every polynomial with integer coeffic ...
on the Mahler measure of non-cyclotomic polynomials * The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy *
Schanuel's conjecture In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers. Statement The con ...
on the transcendence degree of exponentials of linearly independent irrationals * Vitushkin's conjecture on compact subsets of \mathbb with analytic capacity 0 * Are \gamma (the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...
),\pi + e, \pi - e, \pi e, \pi/e, \pi^e, \pi^, \pi^, e^, \ln\pi, 2^e, e^e,
Catalan's constant In mathematics, Catalan's constant , is defined by : G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots, where is the Dirichlet beta function. Its numerical value is approximately : It is not known whether is irra ...
, or
Khinchin's constant In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers ''x'', coefficients ''a'i'' of the continued fraction expansion of ''x'' have a finite geometric mean that is independent of the value of ''x'' and is kno ...
rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers? * What is the exact value of
Landau's constants In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit disk. It gives a lower bound on the size of a disk in which an inverse to a holomorphic function exists. It is named ...
, including Bloch's constant? * How are suspended infinite-infinitesimals paradoxes justified? * Regularity of solutions of
Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
* Convergence of Flint Hills series * Regularity of solutions of Vlasov–Maxwell equations


Combinatorics

* The 1/3–2/3 conjecture – does every finite
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
that is not
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
contain two elements ''x'' and ''y'' such that the probability that ''x'' appears before ''y'' in a random
linear extension In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extens ...
is between 1/3 and 2/3? * The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition *
Problems in Latin squares In mathematics, the theory of Latin squares is an active research area with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here appeared in, for ins ...
– open questions concerning
Latin squares In combinatorics and in experimental design, a Latin square is an ''n'' Ã— ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 Latin s ...
* The lonely runner conjecture – if k runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance 1/k from each other runner) at some time? *
No-three-in-line problem The no-three-in-line problem in discrete geometry asks how many points can be placed in the n\times n grid so that no three points lie on the same line. The problem concerns lines of all slopes, not only those aligned with the grid. It was introd ...
– how many points can be placed in the n \times n grid so that no three of them lie on a line? * Rudin's conjecture on the number of squares in finite
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
s * The
sunflower conjecture The common sunflower (''Helianthus annuus'') is a large annual forb of the genus ''Helianthus'' grown as a crop for its edible oily seeds. Apart from cooking oil production, it is also used as livestock forage (as a meal or a silage plant), as ...
: can the number of k size sets required for the existence of a sunflower of r sets be bounded by an exponential function in k for every fixed r>2? * Frankl's
union-closed sets conjecture The union-closed sets conjecture is an open problem in combinatorics posed by Péter Frankl in 1979. A family of sets is said to be ''union-closed'' if the union of any two sets from the family belongs to the family. The conjecture states: For e ...
– for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets * Give a combinatorial interpretation of the
Kronecker coefficient In mathematics, Kronecker coefficients ''g''λ''μν'' describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representations. They play an important role ...
s * The values of the
Dedekind number File:Monotone Boolean functions 0,1,2,3.svg, 400px, The free distributive lattices of monotonic Boolean functions on 0, 1, 2, and 3 arguments, with 2, 3, 6, and 20 elements respectively (move mouse over right diagram to see description) circle 6 ...
s M(n) for n \ge 9 * The values of the
Ramsey numbers In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To demonstrate the theorem for two colours (say ...
, particularly R(5, 5) * The values of the
Van der Waerden number Van der Waerden's theorem states that for any positive integers ''r'' and ''k'' there exists a positive integer ''N'' such that if the integers are colored, each with one of ''r'' different colors, then there are at least ''k'' integers in arithme ...
s * Finding a function to model n-step
self-avoiding walk In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) ...
s


Dynamical systems

*
Arnold–Givental conjecture The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. It gives a lower bound in terms of the Betti numbers of a Lagrangian submanifold on the number of intersection points ...
and
Arnold conjecture The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry. Statement Let (M, \omega) be a compact symplectic manifold. For any smooth f ...
– relating symplectic geometry to Morse theory. * Berry–Tabor conjecture in
quantum chaos Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mech ...
* Banach's problem – is there an
ergodic system Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expre ...
with simple Lebesgue spectrum? * Birkhoff conjecture – if a
billiard table A billiard table or billiards table is a bounded table on which cue sports are played. In the modern era, all billiards tables (whether for carom billiards, pool, pyramid or snooker) provide a flat surface usually made of quarried slate, that ...
is strictly convex and integrable, is its boundary necessarily an ellipse? *
Collatz conjecture The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...
(''aka'' the 3n + 1 conjecture) * Eden's conjecture that the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of the local Lyapunov dimensions on the global
attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
is achieved on a stationary point or an unstable periodic orbit embedded into the attractor. * Eremenko's conjecture: every component of the
escaping set In mathematics, and particularly complex dynamics, the escaping set of an entire function Æ’ consists of all points that tend to infinity under the repeated application of Æ’. That is, a complex number z_0\in\mathbb belongs to the escaping set if ...
of an
entire Entire may refer to: * Entire function, a function that is holomorphic on the whole complex plane * Entire (animal), an indication that an animal is not neutered * Entire (botany) This glossary of botanical terms is a list of definitions of ...
transcendental function is unbounded. * Fatou conjecture that a quadratic family of maps from the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
to itself is hyperbolic for an open dense set of parameters. * Furstenberg conjecture – is every invariant and ergodic measure for the \times 2,\times 3 action on the circle either Lebesgue or atomic? *
Kaplan–Yorke conjecture In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents. By arranging the Lyapunov exponents in order from largest to smallest \lambda_1\geq\lambda_2\geq\dots\geq\lambda_n, let ''j'' be ...
on the dimension of an
attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
in terms of its Lyapunov exponents * Margulis conjecture – measure classification for diagonalizable actions in higher-rank groups. *
MLC conjecture The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This ...
– is the Mandelbrot set locally connected? * Many problems concerning an
outer billiard Outer billiards is a dynamical system based on a convex shape in the plane. Classically, this system is defined for the Euclidean plane but one can also consider the system in the hyperbolic plane or in other spaces that suitably generalize the pla ...
, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits. * Quantum unique ergodicity conjecture on the distribution of large-frequency
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
on a negatively-curved
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
* Rokhlin's multiple mixing problem – are all strongly mixing systems also strongly 3-mixing? * Weinstein conjecture – does a regular compact contact type
level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
of a
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow? * Does every positive integer generate a
juggler sequence In number theory, a juggler sequence is an integer sequence that starts with a positive integer ''a''0, with each subsequent term in the sequence defined by the recurrence relation: a_= \begin \left \lfloor a_k^ \right \rfloor, & \text a_k \text ...
terminating at 1? * Lyapunov function: Lyapunov's second method for stability – For what classes of
ODEs Odes may refer to: *The plural of ode, a type of poem * ''Odes'' (Horace), a collection of poems by the Roman author Horace, circa 23 BCE *Odes of Solomon, a pseudepigraphic book of the Bible *Book of Odes (Bible), a Deuterocanonical book of the ...
, 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 A reversible cellular automaton is a cellular automaton in which every configuration has a unique predecessor. That is, it is a regular grid of cells, each containing a state drawn from a finite set of states, with a rule for updating all cells s ...
in three or more dimensions locally reversible?


Games and puzzles


Combinatorial games

* Is there a non-terminating game of
beggar-my-neighbour Beggar-my-neighbour, also known as Strip Jack naked, Beat your neighbour out of doors, or Beat Jack out of doors, or Beat Your Neighbour is a simple card game. It is somewhat similar in nature to the children's card game War, and has spawned a ...
? *
Sudoku Sudoku (; ja, 数独, sÅ«doku, digit-single; originally called Number Place) is a logic-based, combinatorial number-placement puzzle. In classic Sudoku, the objective is to fill a 9 Ã— 9 grid with digits so that each column, each row ...
: ** How many puzzles have exactly one solution? *** How many puzzles with exactly one solution are minimal? ** What is the maximum number of givens for a minimal puzzle? *
Tic-tac-toe variants Tic-tac-toe is an instance of an m,n,k-game, where two players alternate taking turns on an ''m''×''n'' board until one of them gets ''k'' in a row. Harary's generalized tic-tac-toe is an even broader generalization. The game can also be general ...
: **Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy? * What is the
Turing completeness In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Tu ...
status of all unique elementary cellular automata?


Games with imperfect information

*
Rendezvous problem The rendezvous dilemma is a logical dilemma, typically formulated in this way: :Two people have a date in a park they have never been to before. Arriving separately in the park, they are both surprised to discover that it is a huge area and conseq ...


Geometry


Algebraic geometry

*
Abundance conjecture In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for every projective variety X with Kawamata log terminal singularities over a field k if the canonic ...
: if the
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...
of a
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
with Kawamata log terminal singularities is nef, then it is semiample. * Bass conjecture on the finite generation of certain algebraic K-groups. *
Deligne conjecture Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord P ...
: any one of numerous named for
Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
. * Dixmier conjecture: any
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
of a
Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form : f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X). More prec ...
is an automorphism. * Fröberg conjecture on the Hilbert functions of a set of forms. *
Fujita conjecture In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved . It is named after Takao Fujita, who formulated it in 1985. Statement In complex geometry, the conjecture states that for a p ...
regarding the line bundle K_ \otimes L^ constructed from a
positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
L on a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
M and the
canonical line bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...
K_ of M * Hartshorne's conjectures *
Jacobian conjecture In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an ''n''-dimensional space to itself has Jacobian determinant which is a non-zero co ...
: if a
polynomial mapping In algebra, a polynomial map or polynomial mapping P: V \to W between vector spaces over an infinite field ''k'' is a polynomial in linear functionals with coefficients in ''k''; i.e., it can be written as :P(v) = \sum_ \lambda_(v) \cdots \lambda_( ...
over a characteristic-0 field has a constant nonzero
Jacobian determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
, then it has a regular (i.e. with polynomial components) inverse function. *
Manin conjecture In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators in 1989 when they initiated a pr ...
on the distribution of
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
s of bounded
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...
in certain subsets of
Fano varieties In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has ...
* Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and
Donaldson–Thomas theory In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual num ...
* Nagata's conjecture on curves, specifically the minimal degree required for a
plane algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
to pass through a collection of very general points with prescribed
multiplicities In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multip ...
. * Nagata–Biran conjecture that if X is a smooth algebraic surface and L is an
ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
on X of degree d, then for sufficiently large r, the
Seshadri constant In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle ''L'' at a point ''P'' on an algebraic variety. It was introduced by Jean-Pierre Demailly, Demailly to measure a certain ''rate of growth'', of the tensor powers of ' ...
satisfies \varepsilon(p_1,\ldots,p_r;X,L) = d/\sqrt. * Nakai conjecture: if a
complex algebraic variety In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex number In mathematics, a complex number is an element of a number system that extends the real numbers ...
has a ring of differential operators generated by its contained
derivations Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
, then it must be
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
. * Parshin's conjecture: the higher algebraic K-groups of any
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
defined over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
must vanish up to torsion. *
Section conjecture In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism \pi_1(X)\to \operatorname(k), where X is a complete smooth curve of genus at least 2 over a fi ...
on splittings of
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
s from
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
s of complete smooth curves over finitely-generated
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
k to the
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of k. * Standard conjectures on algebraic cycles *
Tate conjecture In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The c ...
on the connection between
algebraic cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the a ...
s on
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
and Galois representations on étale cohomology groups. *
Virasoro conjecture In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra. The Virasoro conjecture is named af ...
: a certain
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
encoding the
Gromov–Witten invariant In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic man ...
s of a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
is fixed by an action of half of the
Virasoro algebra In mathematics, the Virasoro algebra (named after the physicist Miguel Ãngel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string the ...
. * Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of
varieties Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
at singular points * Are infinite sequences of
flips Flip, FLIP, or flips may refer to: People * Flip (nickname), a list of people * Lil' Flip (born 1981), American rapper * Flip Simmons, Australian actor and musician * Flip Wilson, American comedian Arts and entertainment Fictional characters * ...
possible in dimensions greater than 3? * Resolution of singularities in characteristic p


Covering and packing

* Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded ''n''-dimensional set. * The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be? * The Erdős–Oler conjecture: when n is a
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
, packing n-1 circles in an equilateral triangle requires a triangle of the same size as packing n circles * The
kissing number problem In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of ...
for dimensions other than 1, 2, 3, 4, 8 and 24 * Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets *
Sphere packing In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing p ...
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 Square packing in a square is a packing problem where the objective is to determine how many squares of side one (unit squares) can be packed into a square of side . If is an integer, the answer is , but the precise, or even asymptotic, amount ...
: what is the asymptotic growth rate of wasted space? *
Ulam's packing conjecture Ulam's packing conjecture, named for Stanislaw Ulam, is a conjecture about the highest possible packing density of identical convex solids in three-dimensional Euclidean space. The conjecture says that the optimal density for packing congruent ...
about the identity of the worst-packing convex solid


Differential geometry

* The spherical Bernstein's problem, a generalization of
Bernstein's problem In differential geometry, Bernstein's problem is as follows: if the graph of a function on R''n''−1 is a minimal surface in R''n'', does this imply that the function is linear? This is true in dimensions ''n'' at most 8, but false in dimens ...
*
Carathéodory conjecture In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924.''Sitzungsberichte der Berliner Mathematis ...
: any convex, closed, and twice-differentiable surface in three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
admits at least two
umbilical point In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are eq ...
s. *
Cartan–Hadamard conjecture In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, k ...
: can the classical
isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
for subsets of Euclidean space be extended to spaces of nonpositive curvature, known as Cartan–Hadamard manifolds? * Chern's conjecture (affine geometry) that the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
of a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
affine manifold In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset of ^n, with monodromy acting by affine t ...
vanishes. *
Chern's conjecture for hypersurfaces in spheres Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates from the Chern's unanswered question: Consider closed minimal submanifolds M^n immersed ...
, a number of closely-related conjectures. * Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed. * The
filling area conjecture In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points. Definitions ...
, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length * The
Hopf conjecture In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf. Positively or negatively curved Riemannian manifolds The Hopf conjecture is an open problem in g ...
s relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds *
Yau's conjecture In differential geometry, Yau's conjecture from 1982, is a mathematical conjecture which states that a closed Riemannian 3-manifold has an infinite number of smooth closed immersed minimal surfaces. It is named after Shing-Tung Yau. It was the ...
: a closed Riemannian 3-manifold has an infinite number of
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
closed immersed
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
s. *
Yau's conjecture on the first eigenvalue In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks: Is it true that the first eigenvalue for the Laplace–Beltrami operator on an embedded minimal hypersurfac ...
that the first
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
for the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
on an embedded minimal hypersurface of S^ is n.


Discrete geometry

* The
Hadwiger conjecture There are several conjectures known as the Hadwiger conjecture or Hadwiger's conjecture. They include: * Hadwiger conjecture (graph theory), a relationship between the number of colors needed by a given graph and the size of its largest clique mino ...
on covering ''n''-dimensional convex bodies with at most 2''n'' smaller copies * Solving the
happy ending problem In mathematics, the "happy ending problem" (so named by Paul Erdős because it led to the marriage of George Szekeres and Esther Klein) is the following statement: This was one of the original results that led to the development of Ramsey t ...
for arbitrary n *Improving lower and upper bounds for the
Heilbronn triangle problem In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conjectured that, no matter how points are placed ...
. * Kalai's 3''d'' conjecture on the least possible number of faces of
centrally symmetric In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
polytopes In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an - ...
.. * The Kobon triangle problem on triangles in line arrangements * The
Kusner conjecture In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to each other. Equilateral dimension has also been called " metric dimension", but the term "metric ...
: at most 2d points can be equidistant in L^1 spaces * The McMullen problem on projectively transforming sets of points into
convex position In discrete and computational geometry, a set of points in the Euclidean plane or a higher-dimensional Euclidean space is said to be in convex position or convex independent if none of the points can be represented as a convex combination of the o ...
* Opaque forest problem on finding opaque sets for various planar shapes * How many unit distances can be determined by a set of points in the Euclidean plane? * Finding matching upper and lower bounds for ''k''-sets and halving lines *
Tripod packing In combinatorics, tripod packing is a problem of finding many disjoint tripods in a three-dimensional grid, where a tripod is an infinite polycube, the union of the grid cubes along three positive axis-aligned rays with a shared apex. Several pr ...
: how many tripods can have their apexes packed into a given cube?


Euclidean geometry

* The
Atiyah conjecture on configurations In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by stating that a certain ''n'' by ''n'' matrix depending on ''n'' points in R3 is always non-singular In the mathematical field of algebraic geometry, a sin ...
on the invertibility of a certain n-by-n matrix depending on n points in \mathbb^ *
Bellman's lost in a forest problem Bellman's lost-in-a-forest problem is an unsolved minimization problem in geometry, originating in 1955 by the American applied mathematician Richard E. Bellman Richard Ernest Bellman (August 26, 1920 â€“ March 19, 1984) was an American a ...
– find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation *
Borromean rings In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the t ...
— are there three unknotted space curves, not all three circles, which cannot be arranged to form this link? * Danzer's problem and Conway's dead fly problem – do
Danzer set In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded density. Several variations of this problem remain unsolved. Density One way t ...
s of bounded density or bounded separation exist? *
Dissection into orthoschemes In geometry, it is an unsolved conjecture of Hugo Hadwiger that every simplex can be dissected into orthoschemes, using a number of orthoschemes bounded by a function of the dimension of the simplex. If true, then more generally every convex poly ...
– is it possible for
simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
of every dimension? *
Ehrhart's volume conjecture In the geometry of numbers, Ehrhart's volume conjecture gives an upper bound on the volume of a convex body containing only one lattice point in its interior. It is a kind of converse to Minkowski's theorem, which guarantees that a centrally symme ...
: a convex body K in n dimensions containing a single lattice point in its interior as its
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
cannot have volume greater than (n+1)^/n! * The – does there exist a two-dimensional shape that forms the
prototile In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation. Definition A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint in ...
for an
aperiodic tiling An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- peri ...
, but not for any periodic tiling? * Falconer's conjecture: sets of Hausdorff dimension greater than d/2 in \mathbb^d must have a distance set of nonzero Lebesgue measure * The values of the
Hermite constant In mathematics, the Hermite constant, named after Charles Hermite, determines how short an element of a lattice in Euclidean space can be. The constant ''γn'' for integers ''n'' > 0 is defined as follows. For a lattice ''L'' in Euclidean space ...
s for dimensions other than 1–8 and 24 *
Inscribed square problem The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: ''Does every plane simple closed curve contain all four vertices of some square?'' This is true if the curve is ...
, also known as
Toeplitz' conjecture The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: ''Does every Jordan curve, plane simple closed curve contain all four vertices of some Square (geometry), square? ...
and the square peg problem – does every
Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
have an inscribed square? * The Kakeya conjecture – do n-dimensional sets that contain a unit line segment in every direction necessarily have
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
and
Minkowski dimension Minkowski, MiÅ„kowski or Minkovski (Slavic feminine: Minkowska, MiÅ„kowska or Minkovskaya; plural: Minkowscy, MiÅ„kowscy; he, מינקובסקי, russian: МинковÑкий) is a surname of Polish origin. It may refer to: * Minkowski or MiÅ„ko ...
equal to n? * The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the
Weaire–Phelan structure In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...
as a solution to the Kelvin problem *
Lebesgue's universal covering problem Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover every planar set of diameter one. The diameter of a set by definition is the least upper bound of the distance ...
on the minimum-area convex shape in the plane that can cover any shape of diameter one * Mahler's conjecture on the product of the volumes of a
centrally symmetric In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
convex body In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non-empty interior. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in ...
and its
polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates * Polar climate, the c ...
. *
Moser's worm problem Moser's worm problem (also known as mother worm's blanket problem) is an unsolved problem in geometry formulated by the Austrian-Canadian mathematician Leo Moser in 1966. The problem asks for the region of smallest area that can accommodate every p ...
– what is the smallest area of a shape that can cover every unit-length curve in the plane? * The
moving sofa problem In mathematics, the moving sofa problem or sofa problem is a two-dimensional idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area that can be maneuvered through an L-shaped planar region ...
– what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor? * Does every convex polyhedron have Rupert's property? * Shephard's problem (a.k.a. Dürer's conjecture) – does every
convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
have a
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
, or simple edge-unfolding? * Is there a non-convex polyhedron without self-intersections with more than seven faces, all of which share an edge with each other? * The
Thomson problem The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. ...
– what is the minimum energy configuration of n mutually-repelling particles on a unit sphere? * Convex
uniform 5-polytope In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets. The complete set of convex uniform 5-polytopes ...
s – find and classify the complete set of these shapes


Graph theory


Graph coloring and labeling

*
Cereceda's conjecture In the mathematics of graph coloring, Cereceda’s conjecture is an unsolved problem on the distance between pairs of colorings of sparse graphs. It states that, for two different colorings of a graph of degeneracy , both using at most colors, ...
on the diameter of the space of colorings of degenerate graphs * The Erdős–Faber–Lovász conjecture on coloring unions of cliques * The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree * The
Hadwiger conjecture There are several conjectures known as the Hadwiger conjecture or Hadwiger's conjecture. They include: * Hadwiger conjecture (graph theory), a relationship between the number of colors needed by a given graph and the size of its largest clique mino ...
relating coloring to clique minors * The
Hadwiger–Nelson problem In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required to color the plane such that no two points at distance 1 from each other have the same color. ...
on the chromatic number of unit distance graphs * Jaeger's Petersen-coloring conjecture: every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph * The list coloring conjecture: for every graph, the list chromatic index equals the chromatic index * The
total coloring conjecture Total may refer to: Mathematics * Total, the summation of a set of numbers * Total order, a partial order without incomparable pairs * Total relation, which may also mean ** connected relation (a binary relation in which any two elements are compa ...
of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree


Graph drawing

* The Albertson conjecture: the crossing number can be lower-bounded by the crossing number of a
complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...
with the same
chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices o ...
*
Conway's thrackle conjecture A thrackle is an embedding of a graph in the plane, such that each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. ...
that
thrackle A thrackle is an embedding of a graph in the plane, such that each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. ...
s cannot have more edges than vertices *
Harborth's conjecture In mathematics, Harborth's conjecture states that every planar graph has a planar drawing in which every edge is a straight segment of integer length.. This conjecture is named after Heiko Harborth, and (if true) would strengthen Fáry's theorem ...
: every planar graph can be drawn with integer edge lengths * Negami's conjecture on projective-plane embeddings of graphs with planar covers * The strong Papadimitriou–Ratajczak conjecture: every polyhedral graph has a convex greedy embedding * Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz? *
Universal point set In graph drawing, a universal point set of order ''n'' is a set ''S'' of points in the Euclidean plane with the property that every ''n''-vertex planar graph has a Fáry's theorem, straight-line drawing in which the vertices are all placed at poin ...
s of subquadratic size for planar graphs


Paths and cycles in graphs

*
Barnette's conjecture Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that eve ...
: every cubic bipartite three-connected planar graph has a Hamiltonian cycle * Gilbert–Pollack conjecture on the Steiner ratio of the Euclidean plane that the Steiner ratio is \sqrt/2 * Chvátal's toughness conjecture, that there is a number such that every -tough graph is Hamiltonian * The cycle double cover conjecture: every bridgeless graph has a family of cycles that includes each edge twice * The
Erdős–Gyárfás conjecture In graph theory, the unproven Erdős–Gyárfás conjecture, made in 1995 by the prolific mathematician Paul Erdős and his collaborator András Gyárfás, states that every graph with minimum degree 3 contains a simple cycle whose length is a ...
on cycles with power-of-two lengths in cubic graphs * The
linear arboricity In graph theory, a branch of mathematics, the linear arboricity of an undirected graph is the smallest number of linear forests its edges can be partitioned into. Here, a linear forest is an acyclic graph with maximum degree two; that is, it is a ...
conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree * The Lovász conjecture on Hamiltonian paths in symmetric graphs * The
Oberwolfach problem The Oberwolfach problem is an unsolved problem in mathematics that may be formulated either as a problem of scheduling seating assignments for diners, or more abstractly as a problem in graph theory, on the edge cycle covers of complete graphs. I ...
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. * Szymanski's conjecture: every
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
on the n-dimensional doubly-
directed Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''D ...
hypercube graph In graph theory, the hypercube graph is the graph formed from the vertices and edges of an -dimensional hypercube. For instance, the cube graph is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. has vertices, e ...
can be routed with edge-disjoint paths.


Word-representation of graphs

*Are there any graphs on ''n'' vertices whose representation requires more than floor(''n''/2) copies of each letter? *Characterise (non-) word-representable planar graphs *Characterise word-representable graphs in terms of (induced) forbidden subgraphs. *Characterise word-representable near-triangulations containing the complete graph ''K''4 (such a characterisation is known for ''K''4-free planar graphs) *Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter *Is it true that out of all bipartite graphs,
crown graph In graph theory, a branch of mathematics, a crown graph on vertices is an undirected graph with two sets of vertices and and with an edge from to whenever . The crown graph can be viewed as a complete bipartite graph from which the edges ...
s require longest word-representants? *Is the
line graph In the mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edges of . is constructed in the following way: for each edge in , make a vertex in ; for every ...
of a non- word-representable graph always non- word-representable? *Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?


Miscellaneous graph theory

* Babai's problem: which groups are Babai invariant groups? * Brouwer's conjecture on upper bounds for sums of
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of Laplacians of graphs in terms of their number of edges *
Conway's 99-graph problem In graph theory, Conway's 99-graph problem is an unsolved problem asking whether there exists an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices ...
: does there exist a
strongly regular graph In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have comm ...
with parameters (99,14,1,2)? *
Degree diameter problem In graph theory, the degree diameter problem is the problem of finding the largest possible graph (in terms of the size of its vertex set ) of diameter such that the largest degree of any of the vertices in is at most . The size of is bounde ...
: given two positive integers d, k, what is the largest graph of diameter k such that all vertices have degrees at most d? * The
Erdős–Hajnal conjecture In graph theory, a branch of mathematics, the Erdős–Hajnal conjecture states that families of graphs defined by forbidden induced subgraphs have either large cliques or large independent sets. It is named for Paul Erdős and András Hajnal. ...
on large cliques or independent sets in graphs with a forbidden induced subgraph * The
GNRS conjecture In theoretical computer science and metric geometry, the GNRS conjecture connects the theory of graph minors, the stretch factor of embeddings, and the approximation ratio of multi-commodity flow problems. It is named after Anupam Gupta, Ilan Newma ...
on whether minor-closed graph families have \ell_1 embeddings with bounded distortion * Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs * The
implicit graph conjecture In the study of graph algorithms, an implicit graph representation (or more simply implicit graph) is a graph whose vertices or edges are not represented as explicit objects in a computer's memory, but rather are determined algorithmically from so ...
on the existence of implicit representations for slowly-growing hereditary families of graphs * Jørgensen's conjecture that every 6-vertex-connected ''K''6-minor-free graph is an
apex graph In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is ''an'' apex, not ''the'' apex because an apex graph may have mo ...
* Meyniel's conjecture that
cop number In graph theory, a branch of mathematics, the cop number or copnumber of an undirected graph is the minimum number of cops that suffices to ensure a win (i.e., a capture of the robber) in a certain pursuit–evasion game on the graph. Rules In th ...
is O(\sqrt n) * Does a
Moore graph In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). If the degree of such a graph is and its diameter is , its girth must ...
with girth 5 and degree 57 exist? * What is the largest possible
pathwidth In graph theory, a path decomposition of a graph is, informally, a representation of as a "thickened" path graph, and the pathwidth of is a number that measures how much the path was thickened to form . More formally, a path-decomposition ...
of an -vertex
cubic graph In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bi ...
? * The
reconstruction conjecture Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to KellyKelly, P. J.A congruence theorem for trees ''Pacific J. Math.'' 7 (1957), 961–968. and Ulam.Ulam, S. M., ...
and
new digraph reconstruction conjecture The reconstruction conjecture of Stanisław Ulam is one of the best-known open problems in graph theory. Using the terminology of Frank Harary it can be stated as follows: If ''G'' and ''H'' are two graphs on at least three vertices and ƒ is a bij ...
on whether a graph is uniquely determined by its vertex-deleted subgraphs. *
Ryser's conjecture In graph theory, Ryser's conjecture is a conjecture relating the maximum matching size and the minimum transversal size in hypergraphs. This conjecture first appeared in 1971 in the Ph.D. thesis of J. R. Henderson, whose advisor was Herbert John R ...
relating the maximum matching size and minimum transversal size in
hypergraph In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) wh ...
s * The second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one? * Sidorenko's conjecture on homomorphism densities of graphs in
graphon GraphOn GO-Global is a multi-user remote access application for Windows. Overview GO-Global allows multiple users to concurrently run Microsoft Windows applications installed on a Windows server or server farm  from network-connected lo ...
s * Do there exist infinitely many strongly regular
geodetic graph In graph theory, a geodetic graph is an undirected graph such that there exists a unique (unweighted) shortest path between each two vertices. Geodetic graphs were introduced in 1962 by Øystein Ore, who observed that they generalize a property of ...
s, or any strongly regular geodetic graphs that are not Moore graphs? * Sumner's conjecture: does every (2n-2)-vertex tournament contain as a subgraph every n-vertex oriented tree? * Tutte's conjectures: ** every bridgeless graph has a nowhere-zero 5-flow ** every
Petersen Petersen is a common Danish patronymic surname, meaning ''"son of Peter"''. There are other spellings. Petersen may refer to: People In arts and entertainment * Adolf Dahm-Petersen, Norwegian voice specialist * Anja Petersen, German operatic ...
- minor-free bridgeless graph has a nowhere-zero 4-flow *
Vizing's conjecture In graph theory, Vizing's conjecture concerns a relation between the domination number and the cartesian product of graphs. This conjecture was first stated by , and states that, if denotes the minimum number of vertices in a dominating set for ...
on the
domination number Domination or dominant may refer to: Society * World domination, which is mainly a conspiracy theory * Colonialism in which one group (usually a nation) invades another region for material gain or to eliminate competition * Chauvinism in which ...
of cartesian products of graphs * Woodall's conjecture that the minimum number of edges in a dicut of a
directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...
is equal to the maximum number of disjoint dijoins * Zarankiewicz problem: how many edges can there be in a bipartite graph on a given number of vertices with no complete bipartite subgraphs of a given size?


Group theory

*
Andrews–Curtis conjecture In mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relato ...
: every balanced
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
of the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
can be transformed into a trivial presentation by a sequence of
Nielsen transformation In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non-commutative analogue of row reduction and ...
s on relators and conjugations of relators * Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems *
Herzog–Schönheim conjecture In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974. Let G be a group, and let :A=\ be a finite system of left cosets of subgroups G_1,\ ...
: if a finite system of left
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of subgroups of a group G form a partition of G, then the finite indices of said subgroups cannot be distinct. * The
inverse Galois problem In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...
: is every finite group the Galois group of a Galois extension of the rationals? * Problems in loop theory and quasigroup theory consider generalizations of groups * Are there an infinite number of Leinster groups? * Does generalized moonshine exist? * For which positive integers ''m'', ''n'' is the
free Burnside group The Burnside problem asks whether a finitely generated group in which every element has finite Order (group theory), order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group ...
finite? In particular, is finite? * Is every finitely presented
periodic group In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For exa ...
finite? * Is every group surjunctive?


Notebook problems

* The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.


Model theory and formal languages

* The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in \aleph_0 is a simple algebraic group over an algebraically closed field. *
Generalized star height problem The generalized star-height problem in formal language theory is the open question whether all regular languages can be expressed using generalized regular expressions with a limited nesting depth of Kleene stars. Here, generalized regular expres ...
: can all
regular language In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to ...
s be expressed using generalized regular expressions with limited nesting depths of
Kleene star In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics, it is more commonly known as the free monoid ...
s? * For which number fields does
Hilbert's tenth problem Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equa ...
hold? * Kueker's conjecture * The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for \aleph_1-saturated models of a countable theory.Shelah S, ''Classification Theory'', North-Holland, 1990 * Shelah's categoricity conjecture for L_: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number. * Shelah's eventual categoricity conjecture: For every cardinal \lambda there exists a cardinal \mu(\lambda) such that if an AEC K with LS(K)<= \lambda is categorical in a cardinal above \mu(\lambda) then it is categorical in all cardinals above \mu(\lambda). * The stable field conjecture: every infinite field with a stable first-order theory is separably closed. * The stable forking conjecture for simple theories *
Tarski's exponential function problem In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred Tarski had previously shown that the theory of the real numbers (without the exponentia ...
: is the
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s with the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
decidable? * The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings? * The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum? *
Vaught conjecture The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite o ...
: the number of
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
models of a
first-order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
complete theory In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence \varphi, the theory T contains the sentence or its ...
in a countable
language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of met ...
is either finite, \aleph_, or 2^. * Assume K is the class of models of a countable first order theory omitting countably many
types Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type, collection of values used for computations. * File type * TYPE (DOS command), a command to display contents of a file. * Typ ...
. If K has a model of cardinality \aleph_ does it have a model of cardinality continuum? * Do the Henson graphs have the
finite model property In mathematical logic, a logic L has the finite model property (fmp for short) if any non-theorem of L is falsified by some ''finite'' model of L. Another way of putting this is to say that L has the fmp if for every formula ''A'' of L, ''A'' is an ...
? * Does a finitely presented homogeneous structure for a finite relational language have finitely many
reduct In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of the operation (mathematics), operations and relation (mathematics), relations of that structure. The opposite of "reduct" is "expansion. ...
s? * Does there exist an
o-minimal In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) which is totally ordered by < is called an o-minimal structure if and only if every
first order theory with a trans-exponential (rapid growth) function? * If the class of atomic models of a complete first order theory is categorical in the \aleph_n, is it categorical in every cardinal? * Is every infinite, minimal field of characteristic zero
algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.) * Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable? * Is the theory of the field of Laurent series over \mathbb_p decidable? of the field of polynomials over \mathbb? * Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property? * Determine the structure of Keisler's order.


Probability theory

* Ibragimov–Iosifescu conjecture for φ-mixing sequences


Number theory


General

* ''n'' conjecture: a generalization of the ''abc'' conjecture to more than three integers. ** ''abc'' conjecture: for any \epsilon > 0, \text(abc)^ < c is true for only finitely many positive a, b, c such that a + b = c. ** Szpiro's conjecture: for any \epsilon > 0, there is some constant C(\epsilon) such that, for any elliptic curve E defined over \mathbb with minimal discriminant \Delta and conductor f, we have , \Delta, \leq C(\epsilon) \cdot f^. *
Hardy–Littlewood zeta-function conjectures In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density of zeros of the Riemann zeta function. Con ...
* Hilbert's eleventh problem: classify
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
s over algebraic number fields. *
Hilbert's ninth problem Hilbert's ninth problem, from the list of 23 Hilbert's problems (1900), asked to find the most general reciprocity law for the norm residues of ''k''-th order in a general algebraic number field, where ''k'' is a power of a prime. Progress ma ...
: find the most general
reciprocity law In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an ir ...
for the norm residues of k-th order in a general algebraic number field, where k is a power of a prime. *
Hilbert's twelfth problem Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogue ...
: extend the
Kronecker–Weber theorem In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provides a partial conve ...
on
Abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable ...
s of \mathbb to any base number field. *
Grand Riemann hypothesis In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all automorphic L-function, automorphic L-function, ''L''-functions lie on the crit ...
: do the nontrivial zeros of all
automorphic L-function In mathematics, an automorphic ''L''-function is a function ''L''(''s'',π,''r'') of a complex variable ''s'', associated to an automorphic representation π of a reductive group ''G'' over a global field and a finite-dimensional complex representa ...
s lie on the critical line 1/2 + it with real t? **
Generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...
: do the nontrivial zeros of all
Dirichlet L-function In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By ...
s lie on the critical line 1/2 + it with real t? ***
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
: do the nontrivial zeros of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
lie on the critical line 1/2 + it with real t? * André–Oort conjecture: is every irreducible component of the
Zariski closure In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...
of a set of special points in a
Shimura variety In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties a ...
a special subvariety? * Beilinson's conjectures * Brocard's problem: are there any integer solutions to n! + 1 = m^ other than n = 4, 5, 7? *
Carmichael's totient function conjecture In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function ''φ''(''n''), which counts the number of integers less than and coprime to ''n''. It states that, for every ''n'' there ...
: do all values of
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
have
multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using mult ...
greater than 1? * Casas-Alvero conjecture: if a polynomial of degree d defined over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
K of characteristic 0 has a factor in common with its first through d - 1-th derivative, then must f be the d-th power of a linear polynomial? * Catalan–Dickson conjecture on aliquot sequences: no
aliquot sequence In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. Defi ...
s are infinite but non-repeating. *
Congruent number problem In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property. The sequence of (integer) cong ...
(a corollary to
Birch and Swinnerton-Dyer conjecture In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory an ...
, per
Tunnell's theorem In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution. Congruent number problem The congruent number problem asks which positive integ ...
): determine precisely what rational numbers are
congruent number In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property. The sequence of (integer) c ...
s. * Erdős–Moser problem: is 1^ + 2^ = 3^ the only solution to the
Erdős–Moser equation In number theory, the Erdős–Moser equation is :1^k+2^k+\cdots+m^k=(m+1)^k, where m and k are positive integers. The only known solution is 11 + 21 = 31, and Paul Erdős conjectured that no further solutions exist. Constraints on solutions ...
? *
Erdős–Straus conjecture The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer n that is 2 or more, there exist positive integers x, y, and z for which \frac=\frac+\frac+\frac. In other words, the number 4/n ...
: for every n \geq 2, there are positive integers x, y, z such that 4/n = 1/x + 1/y + 1/z. *
Erdős–Ulam problem In mathematics, the Erdős–Ulam problem asks whether the plane contains a dense set of points whose Euclidean distances are all rational numbers. It is named after Paul Erdős and Stanislaw Ulam. Large point sets with rational distances The Er ...
: is there a
dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the r ...
of points in the plane all at rational distances from one-another? * Exponent pair conjecture: for all \epsilon > 0, is the pair (\epsilon, 1/2 + \epsilon) an exponent pair? * The
Gauss circle problem In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is ...
: how far can the number of integer points in a circle centered at the origin be from the area of the circle? *
Goormaghtigh conjecture In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh. The conjecture is that the only non-trivial integer solutions of the exponential Diophantine equation :\frac = ...
on solutions to (x^ - 1)/(x - 1) = (y^ - 1)/(y - 1) where x > y > 1 and m, n > 2. * Grimm's conjecture: each element of a set of consecutive
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s can be assigned a distinct
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
that divides it. * Hall's conjecture: for any \epsilon > 0, there is some constant c(\epsilon) such that either y^ = x^ or , y^ - x^, > c(\epsilon)x^. *
Hilbert–Pólya conjecture In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theor ...
: the nontrivial zeros of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
correspond to
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of a
self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
. * Keating–Snaith conjecture concerning the asymptotics of an integral involving the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
*
Lehmer's totient problem In mathematics, Lehmer's totient problem asks whether there is any composite number ''n'' such that Euler's totient function φ(''n'') divides ''n'' âˆ’ 1. This is an unsolved problem. It is known that φ(''n'') = ''n'' âˆ’  ...
: if \phi(n) divides n - 1, must n be prime? *
Leopoldt's conjecture In algebraic number theory, Leopoldt's conjecture, introduced by , states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator defined using p-adic logarithms instead of the usual ...
: a
p-adic In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
analogue of the regulator of an algebraic number field does not vanish. *
Lindelöf hypothesis In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see ) about the rate of growth of the Riemann zeta function on the critical line. This hypothesis is implied by the Riemann hypothesis. It s ...
that for all \epsilon > 0, \zeta(1/2 + it) = o(t^) ** The density hypothesis for zeroes of the Riemann zeta function * Littlewood conjecture: for any two real numbers \alpha, \beta, \liminf_ n\,\Vert n\alpha\Vert\,\Vert n\beta\Vert = 0, where \Vert x\Vert is the distance from x to the nearest integer. * Mahler's 3/2 problem that no real number x has the property that the fractional parts of x(3/2)^ are less than 1/2 for all positive integers n. *
Montgomery's pair correlation conjecture In mathematics, Montgomery's pair correlation conjecture is a conjecture made by that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) is :1-\left(\frac\right)^ + \delta(u), which ...
: the normalized pair correlation function between pairs of zeros of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
is the same as the pair correlation function of random Hermitian matrices. * Newman's conjecture: the partition function satisfies any arbitrary congruence infinitely often. *
Pillai's conjecture Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are ...
: for any A, B, C, the equation Ax^ - By^ = C has finitely many solutions when m, n are not both 2. * Piltz divisor problem on bounding \Delta_(x) = D_(x) - xP_(log(x)) ** Dirichlet's divisor problem: the specific case of the Piltz divisor problem for k = 1 *
Ramanujan–Petersson conjecture In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight :\Delta(z)= \sum_\tau(n)q^n=q\prod_\left (1-q^n \right)^ = q-24q^2+252q^3- 1472q^4 + 4830q^5-\ ...
: a number of related conjectures that are generalizations of the original conjecture. *
Sato–Tate conjecture In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves ''Ep'' obtained from an elliptic curve ''E'' over the rational numbers by reduction modulo almost all prime numbers ''p''. Mikio Sato and J ...
: also a number of related conjectures that are generalizations of the original conjecture. *
Scholz conjecture In mathematics, the Scholz conjecture is a conjecture on the length of certain addition chains. It is sometimes also called the Scholz–Brauer conjecture or the Brauer–Scholz conjecture, after Arnold Scholz who formulated it in 1937 and Alfre ...
: the length of the shortest
addition chain In mathematics, an addition chain for computing a positive integer can be given by a sequence of natural numbers starting with 1 and ending with , such that each number in the sequence is the sum of two previous numbers. The ''length'' of an additi ...
producing 2^ - 1 is at most n - 1 plus the length of the shortest addition chain producing n. * Do
Siegel zero Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). Al ...
s exist? *
Singmaster's conjecture Singmaster's conjecture is a conjecture in combinatorial number theory, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on the multiplicities of entries in Pascal's triang ...
: is there a finite upper bound on the multiplicities of the entries greater than 1 in
Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
? * The uniqueness conjecture for Markov numbers that every
Markov number A Markov number or Markoff number is a positive integer ''x'', ''y'' or ''z'' that is part of a solution to the Markov Diophantine equation :x^2 + y^2 + z^2 = 3xyz,\, studied by . The first few Markov numbers are : 1, 2, 5, 13, 29, 34, 89 ...
is the largest number in exactly one normalized solution to the Markov Diophantine equation. *
Vojta's conjecture In mathematics, Vojta's conjecture is a conjecture introduced by about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distributi ...
on heights of points on
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
over algebraic number fields. * Are there infinitely many
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...
s? *Do any odd perfect numbers exist? *Do
quasiperfect number In mathematics, a quasiperfect number is a natural number ''n'' for which the sum of all its divisors (the divisor function ''σ''(''n'')) is equal to 2''n'' + 1. Equivalently, ''n'' is the sum of its non-trivial divisors (that is, its divisors excl ...
s exist? *Do any non-power of 2
almost perfect number In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number ''n'' such that the sum of all divisors of ''n'' (the sum-of-divisors function ''σ''(''n'')) is equal to 2''n'' â ...
s exist? *Are there 65, 66, or 67
idoneal number In mathematics, Leonhard Euler, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers ''D'' such that any integer expressible in only one way as ''x''2 Â± ''Dy''2 (where ''x''2 is relativel ...
s? * Are there any pairs of
amicable numbers Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(''a'')=''b'' and σ(''b'')=''a'', where σ(''n'') is equal to the sum of positive di ...
which have opposite parity? * Are there any pairs of
betrothed numbers Betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than the value of the other number. In other words, (''m'', ''n'') are a pair of betrothed numbers if ...
which have same parity? * Are there any pairs of
relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
amicable numbers Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(''a'')=''b'' and σ(''b'')=''a'', where σ(''n'') is equal to the sum of positive di ...
? * Are there infinitely many
amicable numbers Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(''a'')=''b'' and σ(''b'')=''a'', where σ(''n'') is equal to the sum of positive di ...
? * Are there infinitely many
betrothed numbers Betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than the value of the other number. In other words, (''m'', ''n'') are a pair of betrothed numbers if ...
? * Are there infinitely many Giuga numbers? * Does every
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
with an odd denominator have an
odd greedy expansion In number theory, the odd greedy expansion problem asks whether a greedy algorithm for finding Egyptian fractions with odd denominators always succeeds. , it remains unsolved. Description An Egyptian fraction represents a given rational number as ...
? * Do any
Lychrel number A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the ''196-algorithm'', after the most famous numb ...
s exist? * Do any odd
noncototient In mathematics, a noncototient is a positive integer ''n'' that cannot be expressed as the difference between a positive integer ''m'' and the number of coprime integers below it. That is, ''m'' − Ï†(''m'') = ''n'', where Ï ...
s exist? * Do any odd
weird number In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divis ...
s exist? * Do any Taxicab(5, 2, n) exist for ''n'' > 1? * Is there a
covering system In mathematics, a covering system (also called a complete residue system) is a collection :\ of finitely many residue classes a_i(\mathrm\ ) = \ whose union contains every integer. Examples and definitions The notion of covering system was ...
with odd distinct moduli? * Is \pi a
normal number In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to b ...
(i.e., is each digit 0–9 equally frequent)? * Is 10 a solitary number? * Can a 3×3
magic square In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
be constructed from 9 distinct perfect square numbers? * Which integers can be written as the sum of three perfect cubes? * Can every integer be written as a sum of four perfect cubes? * Find the value of the De Bruijn–Newman constant.


Additive number theory

* Beal's conjecture: for all integral solutions to A^ + B^ = C^ where x, y, z > 2, all three numbers A, B, C must share some prime factor. *
Erdős conjecture on arithmetic progressions Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additive bases). It states that if the sum o ...
that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
s. *
Erdős–Turán conjecture on additive bases The Erdős–Turán conjecture is an old unsolved problem in additive number theory (not to be confused with Erdős conjecture on arithmetic progressions) posed by Paul Erdős and Pál Turán in 1941. The question concerns subsets of the natur ...
: if B is an additive basis of order 2, then the number of ways that positive integers n can be expressed as the sum of two numbers in B must tend to infinity as n tends to infinity. *
Fermat–Catalan conjecture In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation has only finitely many solutions (''a'',''b'',''c'',''m'',''n'','' ...
: there are finitely many distinct solutions (a^, b^, c^) to the equation a^ + b^ = c^ with a, b, c being positive
coprime integers In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
and m, n, k being positive integers satisfying 1/m + 1/n + 1/k < 1. *
Gilbreath's conjecture Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive ter ...
on consecutive applications of the unsigned
forward difference A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
operator to the sequence of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. *
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold ...
: every even natural number greater than 2 is the sum of two
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. *
Lander, Parkin, and Selfridge conjecture The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. The conjecture is that if the sum of some ' ...
: if the sum of m k-th powers of positive integers is equal to a different sum of n k-th powers of positive integers, then m + n \geq k. * Lemoine's conjecture: all odd integers greater than 5 can be represented as the sum of an odd
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
and an even
semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime nu ...
. * Minimum overlap problem of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set \ *
Pollock's conjectures Pollock's conjectures are two closely related unproven conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician, but also a contributor of papers on mathematics to the R ...
*
Skolem problem In mathematics, the Skolem problem is the problem of determining whether the values of a constant-recursive sequence include the number zero. The problem can be formulated for recurrences over different types of numbers, including integers, ratio ...
: can an algorithm determine if a
constant-recursive sequence In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or more of its immediate predecessors. A constant ...
contains a zero? * The values of ''g''(''k'') and ''G''(''k'') in
Waring's problem In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural numb ...
* Do the
Ulam number In mathematics, the Ulam numbers comprise an integer sequence devised by and named after Stanislaw Ulam, who introduced it in 1964. The standard Ulam sequence (the (1, 2)-Ulam sequence) starts with ''U''1 = 1 and ''U''2 =&nbs ...
s have a positive density? * Determine growth rate of ''r''''k''(''N'') (see
Szemerédi's theorem In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers ''A'' with positive natural density contains a ''k''-ter ...
)


Algebraic number theory

*
Class number problem In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each ''n'' â‰¥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers ''d'') having c ...
: are there infinitely many real quadratic number fields with
unique factorization In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is a ...
? *
Fontaine–Mazur conjecture In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by about when ''p''-adic representations of Galois groups of number fields can be constructed from representations on étale cohomology In mathematics, the étale co ...
: actually numerous conjectures, all proposed by
Jean-Marc Fontaine Jean-Marc Fontaine (13 March 1944 – 29 January 2019) was a French mathematician. He was one of the founders of p-adic Hodge theory. He was a professor at Paris-Sud 11 University from 1988 to his death. Life In 1962 Fontaine entered the Écol ...
and Barry Mazur. *
Gan–Gross–Prasad conjecture In mathematics, the Gan–Gross–Prasad conjecture is a restriction problem in the representation theory of real or p-adic Lie groups posed by Gan Wee Teck, Benedict Gross, and Dipendra Prasad. The problem originated from a conjecture of Gro ...
: a
restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and logi ...
problem in representation theory of real or p-adic Lie groups. * Greenberg's conjectures * Hermite's problem: is it possible, for any natural number n, to assign a sequence of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s to each
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
such that the sequence for x is eventually periodic if and only if x is algebraic of degree n? *
Kummer–Vandiver conjecture In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime ''p'' does not divide the class number ''hK'' of the maximal real subfield K=\mathbb(\zeta_p)^+ of the ''p''-th cyclotomic field. The conjecture wa ...
: primes p do not divide the class number of the maximal real subfield of the p-th
cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
. * Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant X is within a constant multiple of \sqrt/\ln * Selberg's 1/4 conjecture: the
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of the Laplace operator on Maass wave forms of
congruence subgroup In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 Ã— 2 integer matrices of determinant 1, in which the ...
s are at least 1/4. *
Stark conjectures In number theory, the Stark conjectures, introduced by and later expanded by , give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension ''K''/''k'' ...
(including Brumer–Stark conjecture) * Characterize all algebraic number fields that have some power basis.


Computational number theory

* Can
integer factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...
be done in
polynomial time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
?


Prime numbers

* Agoh–Giuga conjecture on the
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s that p is prime if and only if pB_ \equiv -1 \pmod p *
Agrawal's conjecture In number theory, Agrawal's conjecture, due to Manindra Agrawal in 2002, forms the basis for the cyclotomic AKS test. Agrawal's conjecture states formally: Let n and r be two coprime integers, coprime positive integers. If :(X - 1)^n \equiv X^n - ...
that given coprime positive integers n and r, if (X - 1)^ \equiv X^ - 1 \pmod, then either n is prime or n^ \equiv 1 \pmod *
Artin's conjecture on primitive roots In number theory, Artin's conjecture on primitive roots states that a given integer ''a'' that is neither a square number nor −1 is a primitive root modulo infinitely many primes ''p''. The conjecture also ascribes an asymptotic density to the ...
that if an integer is neither a perfect square nor -1, then it is a primitive root modulo infinitely many
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s p *
Brocard's conjecture In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (''p'n'')2 and (''p'n''+1)2, where ''p'n'' is the ''n''th prime number, for every ''n'' ≥ 2. The conjecture is named after Hen ...
: there are always at least 4
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s between consecutive squares of prime numbers, aside from 2^ and 3^. *
Bunyakovsky conjecture The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial f(x) in one variable with integer coefficients to give infinitely many prime values in the sequencef(1), f(2), f(3),\ldots. It was stated in 1857 by the Ru ...
: if an integer-coefficient polynomial f has a positive leading coefficient, is irreducible over the integers, and has no common factors over all f(x) where x is a positive integer, then f(x) is prime infinitely often. *
Catalan's Mersenne conjecture In mathematics, a double Mersenne number is a Mersenne number of the form :M_ = 2^-1 where ''p'' is prime. Examples The first four terms of the sequence of double Mersenne numbers areChris Caldwell''Mersenne Primes: History, Theorems and Li ...
: some Catalan–Mersenne number is composite and thus all Catalan–Mersenne numbers are composite after some point. *
Dickson's conjecture In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by that for a finite set of linear forms , , ..., with , there are infinitely many positive integers for which they are all prime, unless there is a congrue ...
: for a finite set of linear forms a_ + b_n, \ldots, a_ + b_n with each b_ \geq 1, there are infinitely many n for which all forms are
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, unless there is some congruence condition preventing it. * Dubner's conjecture: every even number greater than 4208 is the sum of two
primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
which both have
twins Twins are two offspring produced by the same pregnancy.MedicineNet > Definition of TwinLast Editorial Review: 19 June 2000 Twins can be either ''monozygotic'' ('identical'), meaning that they develop from one zygote, which splits and forms two em ...
. *
Elliott–Halberstam conjecture In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who st ...
on the distribution of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s in
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
s. * Erdős–Mollin–Walsh conjecture: no three consecutive numbers are all powerful. *
Feit–Thompson conjecture In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by . The conjecture states that there are no distinct prime numbers ''p'' and ''q'' such that :\frac divides \frac. If the conjecture were true, it would ...
: for all distinct
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s p and q, (p^ - 1)/(p - 1) does not divide (q^ - 1)/(q - 1) * Fortune's conjecture that no
Fortunate number A Fortunate number, named after Reo Fortune, is the smallest integer ''m'' > 1 such that, for a given positive integer ''n'', ''p'n''# + ''m'' is a prime number, where the primorial ''p'n''# is the product of the first ''n'' prime numbers. ...
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 In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes. The conj ...
on the distribution of
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
divisors of
Mersenne numbers In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...
. * Goldbach conjecture: all even
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s greater than 2 are the sum of two
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. *
Landau's problems At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau ...
* Problems associated to
Linnik's theorem Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive ''c'' and ''L'' such that, if we denote p(''a'',''d'') the least prime in the arithme ...
* New Mersenne conjecture: for any odd
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
p, if any two of the three conditions p = 2^ \pm 1 or p = 4^ \pm 3, 2^ - 1 is prime, and (2^ + 1)/3 is prime are true, then the third condition is true. * Polignac's conjecture: for all positive even numbers n, there are infinitely many
prime gap A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ W ...
s of size n. *
Schinzel's hypothesis H In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory. It is a very broad generalization of widely open conjectures such as the twin prime conjecture. The hypothesis is named after Andrzej S ...
that for every finite collection \ of nonconstant
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted ...
s over the integers with positive leading coefficients, either there are infinitely many positive integers n for which f_(n), \ldots, f_(n) are all
primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, or there is some fixed divisor m > 1 which, for all n, divides some f_(n). * Selfridge's conjecture: is 78,557 the lowest Sierpiński number? *
Twin prime conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
: there are infinitely many
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
s. * Does the converse of Wolstenholme's theorem hold for all natural numbers? * Are all
Euclid number In mathematics, Euclid numbers are integers of the form , where ''p'n''# is the ''n''th primorial, i.e. the product of the first ''n'' prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theor ...
s
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...
? * Are all
Fermat number In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 42949672 ...
s
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...
? * Are all
Mersenne number In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
s of prime index
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...
? * Are there any composite ''c'' satisfying 2''c'' − 1 ≡ 1 (mod ''c''2)? * Are there any
Wall–Sun–Sun prime In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. Definition Let p be a prime number. When each term in the sequence of Fibonac ...
s? * Are there any
Wieferich prime In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Ar ...
s in base 47? * Are there infinitely many balanced primes? * Are there infinitely many Carol primes? * Are there infinitely many cluster primes? * Are there infinitely many
cousin prime In number theory, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six. The cousin primes (sequences and in O ...
s? * Are there infinitely many
Cullen prime Cullen may refer to: Places Canada *Cullen, Saskatchewan, a former hamlet in Benson No. 35 Rural Municipality Ireland *Cullen, County Cork, a village near Boherbue, County Cork *Cullen, County Tipperary, a small village in County Tipperary Scotl ...
s? * Are there infinitely many Euclid primes? * Are there infinitely many
Fibonacci prime A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime. The first Fibonacci primes are : : 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, .... Known Fibonacci primes It is not known whet ...
s? * Are there infinitely many Kummer primes? * Are there infinitely many Kynea primes? * Are there infinitely many
Lucas prime The Lucas numbers or Lucas series are an integer sequence named after the mathematician Édouard Lucas, François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers a ...
s? * Are there infinitely many
Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
s (
Lenstra–Pomerance–Wagstaff conjecture In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one. Original Mersenne conjecture The original, called Mersenne's conjectu ...
); equivalently, infinitely many even
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...
s? * Are there infinitely many
Newman–Shanks–Williams prime In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number ''p'' which can be written in the form :S_=\frac. NSW primes were first described by Morris Newman, Daniel Shanks and Hugh C. Williams in 1981 during the study of ...
s? * Are there infinitely many
palindromic prime In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such ...
s to every base? * Are there infinitely many Pell primes? * Are there infinitely many
Pierpont prime In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who use ...
s? * Are there infinitely many
prime quadruplet In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. Prim ...
s? * Are there infinitely many
prime triplet In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or . With the exceptions of and , this is the closest possible grouping of t ...
s? * Are there infinitely many
regular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli num ...
s, and if so is their relative density e^? * Are there infinitely many
sexy prime In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and . The term "sexy prime" is a pun stemming from the Latin word for six: . If o ...
s? * Are there infinitely many safe and Sophie Germain primes? * Are there infinitely many
Wagstaff prime In number theory, a Wagstaff prime is a prime number of the form : where ''p'' is an odd prime. Wagstaff primes are named after the mathematician Samuel S. Wagstaff Jr.; the prime pages credit François Morain for naming them in a lecture at the ...
s? * Are there infinitely many
Wieferich prime In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Ar ...
s? * Are there infinitely many
Wilson prime In number theory, a Wilson prime is a prime number p such that p^2 divides (p-1)!+1, where "!" denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p-1)!+1. Both are named for 18th-century E ...
s? * Are there infinitely many Wolstenholme primes? * Are there infinitely many
Woodall prime In number theory, a Woodall number (''W'n'') is any natural number of the form :W_n = n \cdot 2^n - 1 for some natural number ''n''. The first few Woodall numbers are: :1, 7, 23, 63, 159, 383, 895, … . History Woodall numbers were first st ...
s? * Can a prime ''p'' satisfy 2^\equiv 1\pmod and 3^\equiv 1\pmod simultaneously? * Does every prime number appear in the Euclid–Mullin sequence? * Find the smallest
Skewes' number In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which :\pi(x) > \operatorname(x), where is the prime-counting function ...
* 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 ''p''2)? * 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 −4''k''4 for integer ''k'', are there infinitely many
repunit In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recreat ...
primes to base ''b''? * For any given integers k\geq 1, b\geq 2, c\neq 0, with and are there infinitely many primes of the form (k\times b^n+c)/\text(k+c,b-1) with integer ''n'' ≥ 1? * Is every
Fermat number In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 42949672 ...
2^ + 1 composite for n > 4? * Is 509,203 the lowest
Riesel number In mathematics, a Riesel number is an odd natural number ''k'' for which k\times2^n-1 is composite for all natural numbers ''n'' . In other words, when ''k'' is a Riesel number, all members of the following set are composite: :\left\. If the for ...
?


Set theory

Note: These conjectures are about
models A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
of Zermelo-Frankel set theory with
choice A choice is the range of different things from which a being can choose. The arrival at a choice may incorporate motivators and models. For example, a traveler might choose a route for a journey based on the preference of arriving at a give ...
, 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 Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axio ...
. * ( Woodin) Does the
generalized continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
below a strongly compact cardinal imply the
generalized continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
everywhere? * Does the
generalized continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
entail ) for every
singular cardinal Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, s ...
\lambda? * Does the
generalized continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
imply the existence of an ℵ2-Suslin tree? * If ℵω is a strong limit cardinal, is 2^ < \aleph_ (see
Singular cardinals hypothesis In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal. According to Mitchell (1992), the si ...
)? The best bound, ℵω4, was obtained by
Shelah Shelah may refer to: * Shelah (son of Judah), a son of Judah according to the Bible * Shelah (name), a Hebrew personal name * Shlach, the 37th weekly Torah portion (parshah) in the annual Jewish cycle of Torah reading * Salih, a prophet described ...
using his PCF theory. * The problem of finding the ultimate
core model In set theory, the core model is a definable inner model of the von Neumann universe, universe of all Set (mathematics), sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a ...
, one that contains all
large cardinals In the mathematical field of set theory, a large cardinal property is a certain kind of property of Transfinite number, transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, big ...
. * Woodin's Ω-conjecture: if there is a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
of
Woodin cardinal In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number \lambda such that for all functions :f : \lambda \to \lambda there exists a cardinal \kappa < \lambda with : \ \subseteq \kappa and an
s, then
Ω-logic In set theory, Ω-logic is an infinitary logic and deductive system proposed by as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure H_. Just as the axiom of projective determinacy yields a canoni ...
satisfies an analogue of
Gödel's completeness theorem Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order theory: I ...
. * Does the
consistency In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
of the existence of a strongly compact cardinal imply the consistent existence of a
supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. Formal definition If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...
? * Does there exist a
Jónsson algebra Jónsson is a surname of Icelandic origin, meaning ''son of Jón''. In Icelandic names, the name is not strictly a surname, but a patronymic. The name refers to: * Arnar Jónsson (actor) (born 1943), Icelandic actor * Arnar Jónsson (basketball) (bo ...
on ℵω? * Is OCA (the
open coloring axiom The open coloring axiom (abbreviated OCA) is an axiom about coloring edges of a graph whose vertices are a subset of the real numbers: two different versions were introduced by and by . Statement Suppose that ''X'' is a subset of the reals, and ...
) consistent with 2^>\aleph_? * Without assuming the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, can a nontrivial elementary embedding ''V''→''V'' exist?


Topology

*
Baum–Connes conjecture In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the operator K-theory, K-theory of the reduced C*-algebra of a group theory, group and the K-homology of the classifying space of proper act ...
: the
assembly map In mathematics, assembly maps are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geo ...
is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. *
Bing–Borsuk conjecture In mathematics, the Bing–Borsuk conjecture states that every n-dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional v ...
: every n-dimensional homogeneous
absolute neighborhood retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deform ...
is a
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
. * Borel conjecture: Aspherical space, aspherical closed manifolds are determined up to homeomorphism by their
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
s. * Halperin conjecture on rational Serre spectral sequences of certain fibrations. * Hilbert–Smith conjecture: if a Locally compact space, locally compact topological group has a Continuous function, continuous, Group action#Types of actions, faithful group action on a
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
, then the group must be a Lie group. * Mazur's conjectures * Novikov conjecture on the Homotopy#Invariance, homotopy invariance of certain polynomials in the Pontryagin classes of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, arising from the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
. * Quadrisecants of wild knots: it has been conjectured that wild knots always have infinitely many quadrisecants. * Ravenel conjectures, Telescope conjecture: the last of Ravenel's conjectures in stable homotopy theory to be resolved. * Unknotting problem: can unknots be recognized in Time complexity#Polynomial time, polynomial time? * Volume conjecture relating quantum invariants of Knot (mathematics), knots to the hyperbolic geometry of their knot complements. * Whitehead conjecture: every Connectedness, connected CW complex#Inductive construction of CW complexes, subcomplex of a two-dimensional Aspherical space, aspherical CW complex is aspherical. * Zeeman conjecture: given a finite Contractible space, contractible two-dimensional CW complex K, is the space K \times [0, 1] Collapse (topology), collapsible?


Problems solved since 1995


Algebra

* Suita conjecture (Qi'an Guan and Xiangyu Zhou, 2015) * Torsion conjecture (Loïc Merel, 1996) * Carlitz–Wan conjecture (Hendrik Lenstra, 1995)


Analysis

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


Combinatorics

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


Dynamical systems

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


Game theory

* The angel problem (Various independent proofs, 2006)


Geometry


21st century

* Maximal rank conjecture (Eric Larson, 2018) * Weibel's conjecture (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018) *
Yau's conjecture In differential geometry, Yau's conjecture from 1982, is a mathematical conjecture which states that a closed Riemannian 3-manifold has an infinite number of smooth closed immersed minimal surfaces. It is named after Shing-Tung Yau. It was the ...
(Antoine Song, 2018) * Pentagonal tiling (Michaël Rao, 2017) * Willmore conjecture (Fernando Codá Marques and André Neves, 2012) * Erdős distinct distances problem (Larry Guth, Nets Katz, Nets Hawk Katz, 2011) * Squaring the plane, Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008) * Tameness conjecture (Ian Agol, 2004) * Ending lamination theorem (Jeffrey Brock, Jeffrey F. Brock, Richard Canary, Richard D. Canary, Yair Minsky, Yair N. Minsky, 2004) * Carpenter's rule problem (Robert Connelly, Erik Demaine, Günter Rote, 2003) * Lambda g conjecture (Carel Faber and Rahul Pandharipande, 2003) * Nagata's conjecture (Ivan Shestakov, Ualbai Umirbaev, 2003) * Double bubble conjecture (Michael Hutchings (mathematician), Michael Hutchings, Frank Morgan (mathematician), Frank Morgan, Manuel Ritoré, Antonio Ros, 2002)


20th century

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


Graph theory

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


Group theory

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


Number theory


21st century

*Duffin–Schaeffer conjecture, Duffin-Schaeffer conjecture (Dimitris Koukoulopoulos, James Maynard (mathematician), James Maynard, 2019) * Vinogradov's mean-value theorem#The conjectured form, Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015) * Goldbach's weak conjecture (Harald Helfgott, 2013) *Prime gap#Further results, Existence of bounded gaps between primes (Yitang Zhang, Polymath Project, Polymath8, James Maynard (mathematician), James Maynard, 2013) * Sidon sequence, Sidon set problem (Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa, 2010) * Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008) * Green–Tao theorem (Ben J. Green and Terence Tao, 2004) * Mihăilescu's theorem, Catalan's conjecture (Preda Mihăilescu, 2002) * Erdős–Graham problem (Ernest S. Croot III, 2000)


20th century

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


Ramsey theory

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


Theoretical computer science

*Decision tree model#Sensitivity conjecture, Sensitivity conjecture for Boolean functions (Hao Huang (mathematician), Hao Huang, 2019)


Topology

*Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020) * Virtual Haken conjecture (Ian Agol, Daniel Groves, Jason Manning, 2012) (and by work of Daniel Wise (mathematician), Daniel Wise also virtually fibered conjecture) * Hsiang–Lawson's conjecture (Simon Brendle, 2012) * Ehrenpreis conjecture (Jeremy Kahn, Vladimir Markovic, 2011) * Atiyah conjecture (Austin, 2009) * Cobordism hypothesis (Jacob Lurie, 2008) * Spherical space form conjecture (
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
, 2006) *
Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...
(
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
, 2002) * Geometrization conjecture, (
Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
, series of preprints in 2002–2003) * Nikiel's conjecture (Mary Ellen Rudin, 1999) * Disproof of the Ganea conjecture (Iwase, 1997)


Uncategorised


2010s

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


2000s

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


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

* * * *


Books discussing unsolved problems

* * * * * * * * * * *


External links


24 Unsolved Problems and Rewards for them

List of links to unsolved problems in mathematics, prizes and research

Open Problem Garden

AIM Problem Lists


MathPro Press. * * *
Unsolved Problems in Number Theory, Logic and Cryptography



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


* * Barry Simon'
15 Problems in Mathematical Physics
{{DEFAULTSORT:Unsolved problems in mathematics Unsolved problems in mathematics, Conjectures, Lists of unsolved problems, Mathematics Mathematics-related lists