The Millennium Prize Problems · Clay Mathematics Institute, 2000
Seven questions mathematics has not answered.
In May 2000 the Clay Mathematics Institute named seven unsolved problems and put a million dollars on each. One has been solved, one has a claim against it as of September 2026, and five are wide open. Every page here starts from a picture you can move, then works up to what the problem actually says.
01
P versus NP
If a computer can check an answer quickly, must it also be able to find one quickly?
Open
02
The Riemann Hypothesis
The primes look scattered. One line in the complex plane may say exactly how scattered.
Open
03
Navier–Stokes Smoothness
The equations for moving fluid are written down. In September 2026 a preprint claimed the first flow that breaks them, given a push from outside.
Claimed 2026
04
The Poincaré Conjecture
Is every three-dimensional shape without holes a sphere in disguise? Yes. Grigori Perelman proved it, then turned down the money.
Solved 2003
05
Yang–Mills and the Mass Gap
The theory behind the strong nuclear force works in every experiment. It has never been shown to exist in mathematics.
Open
06
Birch and Swinnerton-Dyer
How many whole-number solutions does a cubic curve have? A seemingly unrelated function appears to know the answer.
Open
07
The Hodge Conjecture
Some shapes can be cut out by equations. The conjecture says their holes alone tell you which ones.
Open
7
problems named in 2000, chosen to mark the turn of the century.
1
solved and accepted, by a mathematician who declined both the prize and the medal.
$6M
still unpaid. Clay awards nothing until a proof is published and has stood for two years.