Millennium Prize Problems

This article is about the math prizes. For the technology prize, see Millennium Technology Prize.

The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged to pay one million US dollars for the first correct solution to each problem.

The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on the official website of the Clay Mathematics Institute, these seven problems are officially called the Millennium Problems.

As of 2026, the only Millennium Prize problem to have been officially declared solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010. However, he declined the award because it was not also offered to Richard S. Hamilton, upon whose work Perelman built.

In September 2026, the American artificial intelligence company OpenAI presented a proposed counterexample to the Navier–Stokes existence and smoothness problem, saying that they do not intend to claim the Millennium Prize for the result. However, this result has yet to be verified by the Clay Institute, which describes the problem as "active", or the independent mathematical community and is the subject of a priority dispute.