Number theory
Integers, prime numbers, and arithmetic structures.
36 problemsMathematics
Mathematical structures, relationships, and proofs.
Explore Mathematics
Algebra and geometry
Shapes, spaces, and algebraic structures.
64 problemsCombinatorics
Finite structures, arrangements, and counting.
46 problemsAnalysis and dynamics
Continuous change, differential equations, and limiting behavior.
34 problemsLogic and computation
What can be decided, proved, or computed.
13 problemsProbability
Random models and the laws governing their fluctuations and limits.
3 problemsCollections
Browse collectionsMillennium Prize Problems
Five open problems from Clay’s famous prize program, linking mathematics with computation and mathematical physics.
5 problemsPrimes 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 problemsA few problems to explore
146 totalCollatz conjecture
Choose a positive whole number. Halve it if it is even; otherwise triple it and add one. Repeat. Does every starting number reach one?
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.
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.
Goldbach conjecture
Every even whole number greater than two appears to be the sum of two primes. A proof covering all of them is still missing.