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

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The iconic 12

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

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Browse by prominence

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

Iconic

Iconic12 problems

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

Landmark

Landmark99 problems

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

Major

Major147 problems

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

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Browse by field

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

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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 12

Riemann Hypothesis

Birch–Swinnerton-Dyer Conjecture

Strong Goldbach Conjecture

Twin Prime Conjecture

Collatz Conjecture

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