Millennium Prize Problems

The Millennium Prize Problems are seven mathematical problems designated by the Clay Mathematics Institute (CMI). They were announced in Paris on May 24, 2000, with a prize of $1 million allocated to a solution of each problem. As of Sep. 11, 2026, CMI lists five problems as unsolved and labels the Navier-Stokes equations problem "Active." The Poincaré conjecture was proved by Perelman in preprints posted in 2002 and 2003, and CMI announced the award of the Millennium Prize for the proof in 2010 (Clay Mathematics Institute 2026a, 2026d).

OpenAI (2026) announced a proposed resolution of the Navier-Stokes equations existence and smoothness problem on Sep. 8, 2026, by claiming breakdown alternatives (C) and (D) in the official formulation. The construction uses a nonzero smooth external force. On Sep. 11, 2026, CMI described the problem as "apparently" settled and stated that evaluation and assignment of credit would follow its prize rules (Clay Mathematics Institute 2026b). The announcement did not award a prize, and CMI continues to label the problem "Active." CMI's rules require a proposed solution to be published in a qualifying outlet, to receive general acceptance in the global mathematical community, and to have been published for at least two years before CMI will consider it (Clay Mathematics Institute 2026a, 2026c, 2026e).

The table summarizes CMI's problem descriptions and status labels. In particular, the fourth asks the central question of complexity theory, whether and are equal. The Navier-Stokes equations problem asks for existence and smoothness of solutions in three dimensions, and the Yang-Mills existence and mass gap problem asks for a rigorous construction of quantum Yang-Mills theory with a positive mass gap.