Many math problems have been proposed but not yet solved. These problems come from many different areas of math, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete, and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems are studied using methods from more than one field. Prizes are sometimes given for solving problems that have been unsolved for a long time, and certain lists of unsolved problems, like the Millennium Prize Problems, are widely recognized.

This list includes notable unsolved problems from previously published lists, including those considered authoritative. The problems listed vary greatly in difficulty and importance.

Notable lists

For more than 100 years, many mathematicians and groups have created and shared lists of mathematical problems that have not yet been solved. Some of these lists offer prizes for people who find solutions, including the Millennium Prize Problems, which provide a reward of one million dollars for each solved problem.

In 2000, the Clay Mathematics Institute listed seven Millennium Prize Problems. As of now, six of these problems remain unsolved. The seventh problem, called the Poincaré conjecture, was solved by Grigori Perelman in 2003. However, a related question about whether a four-dimensional topological sphere can have different smooth structures is still unsolved.

Unsolved problems

The following statements are written using the language of axiomatic set theory. Unless otherwise noted, these statements are based on Zermelo-Frankel set theory, which may include the Axiom of Choice. It is important to note that the independence of these statements might not be an open question in other set theories that allow for different types of models, such as constructive or non-wellfounded set theories.