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. Six of the problems remain unsolved. 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, 2026b).
| problem | status |
| Birch and Swinnerton-Dyer conjecture | unsolved |
| Hodge conjecture | unsolved |
| Navier-Stokes existence and smoothness | unsolved |
| P versus NP problem | unsolved |
| Poincaré conjecture | solved |
| Riemann hypothesis | unsolved |
| Yang-Mills existence and mass gap | unsolved |
The names in the table follow CMI's official formulations. In particular, the fourth asks the central question of complexity theory,
whether 
and 
are equal; the Navier-Stokes problem asks for existence and smoothness of solutions
in three dimensions, and the Yang-Mills equation
problem asks for a rigorous construction of quantum Yang-Mills theory with a positive
mass gap.
