The continuum hypothesis
Independent of the standard axioms of set theory (Gödel 1940, Cohen 1963); new axiomatic viewpoints keep the question alive.
Hilbert's 23 problems
Presented by David Hilbert at the 1900 International Congress of Mathematicians. Many are solved in one setting and open in another, so “partially solved” is common.
7Solved12Partially solved3Open1Too vague
Partially solved
Consistency of the axioms of arithmetic
Gödel's incompleteness theorems show arithmetic cannot prove its own consistency; Gentzen (1936) gave a proof by other means. Whether this settles Hilbert's question is debated.
Equidecomposability of polyhedra
Max Dehn, 1900: two polyhedra of equal volume need not be scissors-congruent.
The straight line as the shortest distance
Considered too vague to have a definitive resolution, though it inspired work on exotic geometries.
Lie groups without differentiability
Gleason, Montgomery and Zippin (1950s) showed differentiability is not needed; a more general interpretation is still open.
Axiomatization of physics
Fields are axiomatized one at a time, from classical mechanics to fluid dynamics (Deng, Hani and Ma, 2025).
Transcendence of certain numbers
Gelfond and Schneider, 1934: a^b is transcendental for algebraic a ≠ 0, 1 and irrational algebraic b.
Problems of prime numbers (Riemann hypothesis)
The Riemann hypothesis remains one of the most important open problems in mathematics.
The most general reciprocity law
Major progress (including Artin reciprocity for abelian extensions), but not completely resolved.
Solvability of Diophantine equations
Matiyasevich, 1970, building on Davis, Putnam and Robinson: no general algorithm exists.
Quadratic forms with algebraic coefficients
Solved by Hasse over number fields; work continues in other settings.
Kronecker's theorem for arbitrary number fields
Gave rise to class field theory, but only a few such extensions are completely understood.
Solving seventh-degree equations
Kolmogorov and Arnold (1957) answered the continuous version; the algebraic version is open again.
Finiteness of systems of invariants
Nagata, 1959: a counterexample.
Rigorous foundation of Schubert calculus
No consensus on whether Duan and Zhao (2020) fully resolve it.
Topology of algebraic curves and surfaces
Open even for real plane curves of degree 8.
Definite functions as sums of squares
Emil Artin, 1927.
Polyhedra, tilings and sphere packing
Bieberbach (1910), Reinhardt (1928), and Hales (1998) for the Kepler conjecture, formally verified two decades later.
Regularity in the calculus of variations
Grouped with problems 20 and 23 by the Simons Foundation: deep progress, no single endpoint.
Boundary value problems
Existence of solutions for classes of variational problems; continuing development.
Linear differential equations with prescribed monodromy
Bolibrukh (1990) found a counterexample to Plemelj's 1908 'solution': such systems need not exist.
Uniformization of analytic relations
Poincaré and Koebe (1907) resolved one setting; work continues.
Further development of the calculus of variations
Open-ended by design; the field keeps developing.
Sources
Status labels: Evelyn Lamb, “Solved, Unsolved and Unsolvable: The Status of Hilbert’s 23 Problems in Mathematics”, Simons Foundation, July 2026.
Background: M. Hazewinkel, “Hilbert’s 1900 ICM lecture. The 23 problems”, CWI.
We map “Resolved” to Solved, “Unresolved” to Open, and “Resolved for some cases” and “It’s complicated” to Partially solved.