Riemann hypothesis
There are 25 primes up to 100 and 168 up to 1,000. Their uneven counts follow a broad trend; a conjecture about the zeta function predicts how far they can stray from a refined estimate.
Millennium Prize Problems
Five open problems from Clay’s famous prize program, linking mathematics with computation and mathematical physics.
The Clay Mathematics Institute announced seven Millennium Prize Problems in 2000, with a million-dollar prize for each. This view brings together the five it lists as unsolved, as checked on 1 October 2026. The original seven also include the Poincaré conjecture, which has been solved, and Navier–Stokes existence and smoothness. Clay’s 11 September 2026 announcement describes Navier–Stokes as apparently settled while its evaluation process continues; that announcement is not a final prize decision. Follow Clay’s pages for the full program and updates.
Subjects: Mathematics · Computational complexity · Particle physics
Problems in this collection
5 problemsP versus NP
A proposed guest list can be checked against the rules. Does easy checking always come with an efficient way to find a list that works?
Birch and Swinnerton-Dyer conjecture
An elliptic curve has both rational solutions and an analytic function. The conjecture predicts that these two ways of studying it encode the same rank.
Hodge conjecture
Some topological features of a complex algebraic shape look as though they should come from smaller algebraic shapes inside it. Must they?
Yang–Mills existence and mass gap
A successful framework for particle physics still needs a rigorous four-dimensional construction with a positive gap above its vacuum energy.
About this collection
- The Millennium Prize Problems
The program’s official roster distinguishes five unsolved problems, the active Navier–Stokes problem, and the solved Poincaré conjecture.
- Navier-Stokes Announcement
Clay reports an apparent resolution and explains that its deliberate evaluation process will continue with updates.