Math Conjectures

As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development.

— David Hilbert, Mathematical Problems, 1900

Start with the icons

The iconic 13

The questions that define modern mathematics — most have stood for a century or more.

Browse iconic problems →

Choose your depth

Browse by prominence

Every record carries one of three prominence tiers, so you can start famous and go deep.

Iconic

Iconic13 problems

The most famous open questions in mathematics — the problems whole subjects are organized around.

Landmark

Landmark137 problems

Field-shaping problems that every specialist knows, many with decades of partial progress.

Major

Major156 problems

Substantial open questions under active pursuit inside their own research communities.

Choose your terrain

Browse by field

Each problem has one primary home; cross-field links are preserved on its page.

Full field index →

Proof-claim watch

Publicized proof or refutation claims, audited and graded rather than silently accepted — only claims backed by substantive evidence are listed.

The enduring conjectures

The iconic 13

Riemann Hypothesis

Birch–Swinnerton-Dyer Conjecture

Strong Goldbach Conjecture

Twin Prime Conjecture

Collatz Conjecture

Hilbert's Twelfth Problem

Hodge Conjecture

Smooth four-dimensional Poincaré conjecture

Three-dimensional Navier–Stokes existence and smoothness

Weak cosmic censorship (future-null-infinity form)

Uniform finiteness in Hilbert's sixteenth problem

PPP versus NPNPNP