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, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. 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, 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, and 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 listed by the Clay Mathematics Institute in 2000, six remain unsolved to date: The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman 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 structures—is unsolved.

Notebooks

Unsolved problems

Algebra

Group theory

Representation theory

Analysis

Combinatorics

Dynamical systems

Games and puzzles

Combinatorial games

Games with imperfect information

Geometry

Algebraic geometry

Covering and packing

Differential geometry

Discrete geometry

Euclidean geometry

Graph theory

Algebraic graph theory

Games on graphs

Graph coloring and labeling

Graph drawing and embedding

Restriction of graph parameters

Subgraphs

Word-representation of graphs

Miscellaneous graph theory

Model theory and formal languages

Probability theory

Number theory

General

Additive number theory

Algebraic number theory

Computational number theory

Diophantine approximation and transcendental number theory

Diophantine equations

Prime numbers

Set theory

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

Topology

Problems solved since 1995

Algebra

Analysis

Combinatorics

Dynamical systems

Game theory

Geometry

21st century

20th century

Graph theory

Group theory

Number theory

21st century

20th century

Ramsey theory

Theoretical computer science

Topology

Uncategorised

2010s

2000s

Books discussing problems solved since 1995

Books discussing unsolved problems