Category: Unsolved problems in mathematics

Köthe conjecture
In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2022. 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 i
Map folding
In the mathematics of paper folding, map folding and stamp folding are two problems of counting the number of ways that a piece of paper can be folded. In the stamp folding problem, the paper is a str
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.
1/3–2/3 conjecture
In order theory, a branch of mathematics, the 1/3–2/3 conjecture states that, if one is comparison sorting a set of items then, no matter what comparisons may have already been performed, it is always
Szilassi polyhedron
In geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces.
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 additi
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 varia
Sum of four cubes problem
The sum of four cubes problem asks whether every integer is the sum of four cubes of integers. It is conjectured the answer is affirmative, but this conjecture has been neither proved nor disproved. S
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.
Divisor summatory function
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The
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 conven
Thomson problem
The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of n electrons constrained to the surface of a unit sphere that repel each other with a fo
Burnside problem
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the
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 Andr
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 comp
Birch–Tate conjecture
The Birch–Tate conjecture is a conjecture in mathematics (more specifically in algebraic K-theory) proposed by both Bryan John Birch and John Tate.
Constant problem
In mathematics, the constant problem is the problem of deciding whether a given expression is equal to zero.
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 no
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,
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
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 . This problem, first posed in the
List of problems in loop theory and quasigroup theory
In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public
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 theo
Crouzeix's conjecture
Crouzeix's conjecture is an unsolved (as of 2018) problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it refines Crouzeix's theorem, which states: where the set is the field of
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 NSW primes were first described by , Daniel Shanks and Hugh C. Williams in 1981 during t
Littlewood conjecture
In mathematics, the Littlewood conjecture is an open problem (as of May 2021) in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α a
Hadamard's maximal determinant problem
Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equ
Dixmier problem
No description available.
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 Kap
Union-closed sets conjecture
In combinatorics, the union-closed sets conjecture is a problem, posed by Péter Frankl in 1979 and is still open. A family of sets is said to be union-closed if the union of any two sets from the fami
Pell number
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This se
Discrepancy theory
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the
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
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 c
Kurosh problem
In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases
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 Riema
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 desc
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 op
Discrepancy of hypergraphs
Discrepancy of hypergraphs is an area of discrepancy theory.
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 . This number is approximated by the
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 differe
Büchi's problem
In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such
Simon problems
In mathematics, the Simon problems (or Simon's problems) are a series of fifteen questions posed in the year 2000 by Barry Simon, an American mathematical physicist. Inspired by other collections of m
Lucas number
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci num
Whitehead conjecture
The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcompl
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 be a group, and let be a finite syste
Smale's problems
Smale's problems are a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, t
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–Beltra
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 2n + 1. Equivalently, n is the sum of its non-trivial divisors
Navier–Stokes existence and smoothness
The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion
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 th
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) conjectur
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-functions lie on
Berman–Hartmanis conjecture
In structural complexity theory, the Berman–Hartmanis conjecture is an unsolved conjecture named after Leonard C. Berman and Juris Hartmanis that states that all NP-complete languages look alike, in t
Dixmier conjecture
In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. in 2005, and independently and Kontsevich in 2007, sh
List of unsolved problems in statistics
There are many longstanding unsolved problems in mathematics for which a solution has still not yet been found. The notable unsolved problems in statistics are generally of a different flavor; accordi
Naimark's problem
Naimark's problem is a question in functional analysis asked by Naimark. It asks whether every C*-algebra that has only one irreducible -representation up to unitary equivalence is isomorphic to the -
Bloch's theorem (complex variables)
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 invers
P versus NP problem
The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.
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 me
Bing–Borsuk conjecture
In mathematics, the Bing–Borsuk conjecture states that every -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and
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 th
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 measurem
Jacobson's conjecture
In abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring. It has only been proven for special typ
Open Problems in Mathematics
Open Problems in Mathematics is a book, edited by John Forbes Nash Jr. and Michael Th. Rassias, published in 2016 by Springer (ISBN 978-3-319-32160-8). The book consists of seventeen expository articl
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 cohom
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
Mean value problem
In mathematics, the mean value problem was posed by Stephen Smale in 1981. This problem is still open in full generality. The problem asks: For a given complex polynomial of degree and a complex numbe
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 term
List of unsolved problems in mathematics
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, alg
Baum–Connes conjecture
In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of p
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. T
Dixmier Problem
No description available.
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 subm
M/G/k queue
In queueing theory, a discipline within the mathematical theory of probability, an M/G/k queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a Gene