The world's hardest unsolved mathematical problems, each worth one million dollars. Can AI solve millennium problems? Track the history of attempts here.
6 Unsolved · 1 Proven (Poincaré conjecture, 2003)
In 2000, the Clay Mathematics Institute announced the "Millennium Prize Problems" — seven of the most important unsolved problems in mathematics. Each carries a $1,000,000 prize for a correct solution.
These problems were chosen because they are among the most difficult and important unsolved problems in mathematics. They span multiple fields including number theory, algebra, geometry, and mathematical physics.
Is every problem whose solution can be quickly verified also quickly solvable? In other words, if you can check an answer quickly, can you find it quickly?
Status: Unsolved since 1971
Why it matters: Affects cryptography, optimization, and the limits of computation.
Technical question about the relationship between algebraic geometry and topology. Certain building blocks of certain spaces can be approximated by algebraic shapes.
Status: Unsolved since 1941
Why it matters: Connects algebraic cycles to topological features.
The pattern of prime numbers may be connected to the zeros of the Riemann zeta function. All non-trivial zeros have real part 1/2.
Status: Unsolved since 1859
Why it matters: Prime numbers are fundamental to cryptography and number theory.
Prove that quantum Yang-Mills theory exists and has a mass gap. The theory describes fundamental particles and forces at the quantum scale.
Status: Unsolved since 1954
Why it matters: Foundation of the Standard Model of particle physics.
Do smooth solutions always exist to the Navier-Stokes equations that describe fluid flow? Or do they sometimes develop singularities?
Status: Unsolved since 1822
Why it matters: Describes weather, ocean currents, and aircraft design.
Predict the rank of rational solutions to elliptic curves. The conjecture relates the number of rational points to the behavior of an associated zeta function.
Status: Unsolved since 1965
Why it matters: Connected to Fermat's Last Theorem and cryptography.
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. Proven by Grigori Perelman in 2003 using Ricci flow techniques.
Status: PROVEN by Grigori Perelman (2003)
Significance: First and only millennium problem to be solved.
OpenMystery tracks AI attempt history on these problems to understand where AI approaches fail.
Join OpenMystery to document and track AI attempts on the world's hardest math problems.