Millennium Prize Problems
Five open problems from Clay’s famous prize program, linking mathematics with computation and mathematical physics.
5 problemsCollections
Selected problems that share a subject or a research tradition. Each leads to its own overview and sources.
Within Mathematics · All collections
Primes and integers
Patterns in prime numbers, and equations whose solutions must be whole numbers.
14 problemsPrime patterns
Prime pairs, prime sums, and the distribution of primes through the lens of the zeta function.
7 problemsDiophantine equations
Equations over integers and rational numbers, from elliptic curves to the limits of decision algorithms.
7 problemsGraphs and coloring
Colors, subgraphs, labels and paths that visit every vertex.
20 problemsGeometric extremes
Packing, covering, distances and other limits set by geometry.
11 problemsProofs and certificates
Which statements have short proofs, and how do proof methods compare?
3 problemsFields and rational points
Field symmetries and rational points on curves and higher-dimensional varieties.
7 problemsAlgebras and modules
Representations, extensions and resolutions that reveal algebraic structure.
4 problemsDynamics
Repeated maps, billiard trajectories and periodic motion.
5 problemsBases and transversals
Rearranging bases, choosing distinct symbols and listing minimal hitting sets.
4 problemsAlgebraic geometry
Polynomial equations define spaces whose shapes, singularities and algebraic cycles pose different open problems.
8 problemsGroups and operators
Structure in groups, representations and infinite-dimensional spaces.
7 problemsRandom geometry
Fastest paths and self-avoiding walks ask for different geometric limits in random models.
2 problemsLinear recurrences
Two different decisions about recurrence sequences: whether a zero occurs and whether all terms are nonnegative.
2 problemsWaves and spectra
Vibration frequencies, high-frequency wave patterns and reconstruction from frequency information.
3 problemsKnots and manifolds
Smooth structures, curved spaces and the manifolds produced by knot surgery.
8 problemsGames and strategies
Perfect play in chess, winners in graph games, and pursuit strategies.
3 problems