Famous math problems
The hardest questions in mathematics, stated exactly and explained in plain words. 9 of the 14 problems here are still unsolved: nobody, human or computer, has a proof. Each page says honestly what is known, who proved what, and where to read more.
The seven Millennium Prize Problems
Announced by the Clay Mathematics Institute in Paris on 24 May 2000, each with a prize of one million US dollars. Only the Poincaré conjecture has been solved.
Every non-trivial zero of the Riemann zeta function has real part 1/2.
If a solution can be checked quickly, can it also be found quickly?
Do smooth solutions of the 3D incompressible Navier–Stokes equations exist for all time?
Construct quantum Yang–Mills theory rigorously and prove its lightest particle has positive mass.
On projective algebraic varieties, every Hodge class is a rational combination of algebraic cycles.
The rank of an elliptic curve equals the order of vanishing of its L-function at s = 1.
Every simply connected, closed 3-manifold is a 3-sphere. Proved by Perelman.
Other famous problems and equations
The equations of general relativity: not an unsolved problem, but highly nonlinear PDEs with famous exact solutions.
Halve if even, triple and add one if odd: every start eventually reaches 1?
Every even integer greater than 2 is the sum of two primes.
There are infinitely many primes p with p + 2 also prime.
For coprime a + b = c, c is rarely much larger than the product of the distinct primes of abc.
No positive integers satisfy xⁿ + yⁿ = zⁿ for n > 2. Proved by Andrew Wiles.
Every map in the plane can be coloured with four colours so that neighbouring regions differ.
Can a math solver solve these?
No. A solver, a calculator or an AI chatbot cannot solve an open problem by typing it in; any site that shows a “solution” to the Riemann hypothesis or P versus NP is wrong. When you paste one of these into the MathGPT solver, it recognises the problem and shows the exact statement, its status and the established facts, clearly labelled as reference information.
What a solver can do is check special cases: try the solver on a Collatz step, a sum of two primes or a derivative from the Navier–Stokes equations.