Millennium Prize Problems

The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. Each of the problems has a prize of one million US dollars for the first correct solution, pledged by the Clay Mathematics Institute. The Clay Mathematics Institute officially designated the title Millennium Prize 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 Prize 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. The Clay Institute only considers proposed solutions to Millennium Prize problems after at least two years since publication. The claimed result has yet to be verified by the independent mathematical community and is the subject of a priority dispute.