∞ Open Problems

The hardest problems in mathematics

These problems have resisted proof for decades — sometimes centuries. Submit your work on any of them. Every submission requires you to explain what you tried and where it breaks down. Serious engagement only.

Number Theory

Research frontierMillennium Prize — $1,000,000since 1859

Riemann Hypothesis

The non-trivial zeros of the Riemann zeta function all have real part 1/2. A proof or disproof would transform number theory and our understanding of prime distribution — with direct consequences for cryptography and the structure of primes.

submissions

Accessiblesince 1742

Goldbach's Conjecture

Every even integer greater than 2 can be expressed as the sum of two primes. Stated in 1742 and verified computationally for all even numbers up to 4 × 10¹⁸, yet no general proof exists.

submissions

Accessible

Twin Prime Conjecture

There are infinitely many pairs of primes differing by 2, such as (11, 13) or (17, 19). Zhang's 2013 breakthrough proved infinitely many prime pairs within a bounded gap — since compressed to 246 — but the twin prime case remains open.

submissions

Accessiblesince 1937

Collatz Conjecture

Take any positive integer. If even, divide by 2. If odd, multiply by 3 and add 1. Repeat. The conjecture: you always eventually reach 1. Verified for all starting values up to 2⁶⁸, yet no proof is known. Erdős said of it: "Mathematics is not yet ready for such problems."

submissions

Research frontiersince 1985

ABC Conjecture

For coprime positive integers a, b, c with a + b = c, the radical of abc — the product of its distinct prime factors — is rarely much smaller than c. A claimed proof by Mochizuki (2012) remains contested by the mathematical community.

submissions

Complexity Theory

Research frontierMillennium Prize — $1,000,000since 1971

P versus NP

Does every problem whose solution can be quickly verified also have a solution that can be quickly found? One of the most consequential open questions in mathematics and computer science — its resolution would reshape cryptography, optimization, and artificial intelligence.

submissions

Analysis / PDEs

Research frontierMillennium Prize — $1,000,000since 1845

Navier-Stokes Existence and Smoothness

Do smooth, globally defined solutions to the Navier-Stokes equations always exist for all time, or can they develop singularities? These equations model fluid flow and are central to physics, meteorology, and engineering — yet their mathematical foundations remain incomplete.

submissions

Mathematical Physics

Research frontierMillennium Prize — $1,000,000since 1954

Yang-Mills Existence and Mass Gap

Prove rigorously that quantum Yang-Mills theory exists and exhibits a mass gap — a positive lower bound on the energy of any excitation above the vacuum. Foundational to the Standard Model of particle physics, yet the rigorous mathematics remains out of reach.

submissions

Algebraic Geometry

Research frontierMillennium Prize — $1,000,000since 1965

Birch and Swinnerton-Dyer Conjecture

The rank of an elliptic curve over the rationals — the number of independent rational points — is equal to the order of vanishing of its L-function at s = 1. One of the deepest open connections between algebraic geometry and analytic number theory.

submissions

Research frontierMillennium Prize — $1,000,000since 1950

Hodge Conjecture

For a smooth complex projective algebraic variety, every rational Hodge class is a rational linear combination of the cohomology classes of algebraic cycles. A profound statement about the relationship between topology and algebraic structure.

submissions

Algebra

Research frontiersince 1939

Jacobian Conjecture

If a polynomial map from ℂⁿ to ℂⁿ has a constant nonzero Jacobian determinant everywhere, must it have a polynomial inverse? Stated in 1939, it has been "proved" and retracted multiple times, and remains open for n ≥ 2.

submissions

Functional Analysis

Research frontier

Invariant Subspace Problem

Does every bounded linear operator on a separable infinite-dimensional Hilbert space over ℂ have a non-trivial closed invariant subspace? Solved for specific operator classes, but the general case remains open — one of the oldest unsolved problems in functional analysis.

submissions