Open problems
Unsolved. The questions still waiting for answers.
From the Riemann hypothesis to the Collatz conjecture — the Millennium Prize Problems and classical open questions that have resisted the best minds in mathematics for decades or centuries.
Twin Prime Conjecture
Are there infinitely many prime pairs like (3,5), (11,13), (17,19)? One of the oldest open problems in number theory, with dramatic recent progress.
Proposed -300 by Euclid (implicitly)
Read → OpenGoldbach's Conjecture
One of the oldest and simplest-to-state unsolved problems in number theory. Almost three centuries old and still open.
Proposed 1742 by Christian Goldbach
Read → OpenNavier–Stokes Existence and Smoothness
Millennium Prize — $1,000,000
The Navier-Stokes equations describe fluid motion. Whether their solutions always remain well-behaved is one of mathematics' great open problems.
Proposed 1845 by Claude-Louis Navier, George Gabriel Stokes
Read → OpenRiemann Hypothesis
Millennium Prize — $1,000,000
The most famous unsolved problem in mathematics. A proof would unlock dozens of other theorems in number theory.
Proposed 1859 by Bernhard Riemann
Read → OpenCollatz Conjecture
A problem so simple a child can understand it, so hard that no mathematician can solve it. The 3n+1 problem.
Proposed 1937 by Lothar Collatz
Read → OpenHodge Conjecture
Millennium Prize — $1,000,000
A deep conjecture connecting topology, complex analysis, and algebraic geometry — one of the most technical of the Millennium problems.
Proposed 1950 by W.V.D. Hodge
Read → OpenYang–Mills Existence and Mass Gap
Millennium Prize — $1,000,000
The mathematical foundations of a theory that physicists use daily — rigorously placed on secure footing.
Proposed 1954 by Chen Ning Yang, Robert Mills
Read → OpenBirch and Swinnerton-Dyer Conjecture
Millennium Prize — $1,000,000
A deep conjecture connecting elliptic curves, L-functions, and the distribution of rational solutions to cubic equations.
Proposed 1965 by Bryan Birch, Peter Swinnerton-Dyer
Read → OpenP vs NP
Millennium Prize — $1,000,000
The central problem of theoretical computer science: are the problems we can verify quickly the same as the problems we can solve quickly?
Proposed 1971 by Stephen Cook (independently Leonid Levin)
Read → OpenABC Conjecture
A deep conjecture about the additive and multiplicative structure of integers. Said to imply many other theorems, including Fermat's Last Theorem as a corollary.
Proposed 1985 by Joseph Oesterlé, David Masser
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