Hilbert's 23 problems (1900)
David Hilbert, ICM Paris, 8 August 1900 (ten spoken; all 23 in the written list). English: Hilbert, Bull. Amer. Math. Soc. 8 (1902), 437–479.
Status cutoff 2026-08-17. "Resolved" means a community-accepted yes/no or impossibility proof for a clean reading. Several items are lists, not one question. Sources: Wikipedia; Simons / ICM 2026.
| # | Header (1902) | Status | Note |
|---|---|---|---|
| 1 | Continuum hypothesis | Controversial | Gödel–Cohen: independent of ZFC (1940, 1963). ICM 2026: Asperó–Schindler on extra axioms. |
| 2 | Consistency of arithmetic | Controversial | Gödel 1931; Gentzen 1936 via . Not Hilbert-finitistic. |
| 3 | Scissor congruence of equal-volume polyhedra | Resolved (no) | Dehn 1900. |
| 4 | Straight line as shortest distance | Too vague | |
| 5 | Lie groups without differentiability | Partial | Gleason 1953 for manifolds. Hilbert–Smith still open. |
| 6 | Axioms of physics | Split | (a) Kolmogorov 1933. (b) Deng–Hani–Ma arXiv:2503.01800 (2025), peer review. |
| 7 | Transcendence of | Resolved (yes) | Gelfond–Schneider 1934. |
| 8 | Problems of prime numbers | Open | (a) RH. (b) Goldbach / twins (see landau.md). (c) Dedekind zeta / GRH. |
| 9 | Most general reciprocity law | Open | Artin for abelian; non-abelian still conjectural. |
| 10 | Algorithm for Diophantine solvability | Resolved (impossible) | Matiyasevich 1970 (Davis–Putnam–Robinson). |
| 11 | Quadratic forms over number fields | Resolved | Hasse 1924; Witt 1937. |
| 12 | Kronecker–Weber for any base field | Open | CM / totally-real special cases (Shimura; Dasgupta–Kakde). |
| 13 | Seventh-degree equation, two arguments | Split | Continuous: Kolmogorov–Arnold 1957 (no). Algebraic: open. |
| 14 | Finite generation of invariant rings | Resolved (no) | Nagata 1959. |
| 15 | Schubert enumerative calculus | Controversial | Duan–Zhao claim a resolution; no consensus. |
| 16 | Topology of real algebraic curves and surfaces | Open | Ovals even in degree 8; limit cycles finite (Ilyashenko/Écalle) but no effective bound. Also Smale 13. |
| 17 | Nonnegative forms as sums of squares | Resolved (yes) | Artin 1927. |
| 18 | Building space from congruent polyhedra | Resolved | Bieberbach; Reinhardt; Hales sphere packing (1998 / Flyspeck). |
| 19 | Regular variational problems analytic? | Resolved (yes) | De Giorgi; Nash 1957. |
| 20 | General boundary-value problems | Open | Many cases solved; no complete answer. |
| 21 | Fuchsian equations with given monodromy | Resolved (no in general) | Bolibrukh 1989. |
| 22 | Uniformization by automorphic functions | Open | Riemann-surface case known. |
| 23 | Further development of the calculus of variations | Too vague | Hilbert's closing programme, not a yes/no. |
| 24 | Proof simplicity (unpublished) | Too vague | Recovered by Thiele 2000. |
Hilbert 8 overlaps Landau. Hilbert 16 is Smale 13.
See also the Clay list in millennium.md (Riemann is 8(a)).