Hilbert's twenty-three problems are:
| Problem | Brief explanation | Status | Year Solved |
|---|---|---|---|
| 1st | The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers) | Proven to be impossible to prove or disprove within the Zermelo–Fraenkel set theory with or without the Axiom of Choice. There is no consensus on whether this is a solution to the problem. | 1963 |
| 2nd | Prove that the axioms of arithmetic are consistent. | There is no consensus on whether results of Gödel and Gentzen give a solution to the problem as stated by Hilbert. Gödel's second incompleteness theorem, proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0. | 1936? |
| 3rd | Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? | Resolved. Result: no, proved using Dehn invariants. | 1900 |
| 4th | Construct all metrics where lines are geodesics. | Too vague to be stated resolved or not.[n 1] | – |
| 5th | Are continuous groups automatically differential groups? | Resolved by Andrew Gleason, depending on how the original statement is interpreted. If, however, it is understood as an equivalent of the Hilbert–Smith conjecture, it is still unsolved. | 1953? |
| 6th | Axiomatize all of physics | Unresolved. [n 2] | – |
| 7th | Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? | Resolved. Result: yes, illustrated by Gelfond's theorem or the Gelfond–Schneider theorem. | 1935 |
| 8th | The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is ½") and other prime number problems, among them Goldbach's conjecture and the twin prime conjecture | Unresolved. | – |
| 9th | Find most general law of the reciprocity theorem in any algebraic number field | Partially resolved.[n 3] | – |
| 10th | Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. | Resolved. Result: impossible, Matiyasevich's theorem implies that there is no such algorithm. | 1970 |
| 11th | Solving quadratic forms with algebraic numerical coefficients. | Partially resolved.[citation needed] | – |
| 12th | Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field. | Unresolved. | – |
| 13th | Solve all 7-th degree equations using continuous functions of two parameters. | Resolved. The problem was solved affirmatively by Vladimir Arnold based on work by Andrei Kolmogorov. [n 5] | 1957 |
| 14th | Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? | Resolved. Result: no, counterexample was constructed by Masayoshi Nagata. | 1959 |
| 15th | Rigorous foundation of Schubert's enumerative calculus. | Partially resolved.[citation needed] | – |
| 16th | Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. | Unresolved. | – |
| 17th | Expression of definite rational function as quotient of sums of squares | Resolved. Result: An upper limit was established for the number of square terms necessary.[citation needed] | 1927 |
| 18th | (a) Is there a polyhedron which admits only an anisohedral tiling in three dimensions? (b) What is the densest sphere packing? | (a) Resolved. Result: yes (by Karl Reinhardt). (b) Resolved by computer-assisted proof. Result: cubic close packing and hexagonal close packing, both of which have a density of approximately 74%.[n 6] | (a) 1928 (b) 1998 |
| 19th | Are the solutions of Lagrangians always analytic? | Resolved. Result: yes, proven by Ennio de Giorgi and, independently and using different methods, by John Forbes Nash. | 1957 |
| 20th | Do all variational problems with certain boundary conditions have solutions? | Resolved. A significant topic of research throughout the 20th century, culminating in solutions[citation needed] for the non-linear case. | – |
| 21st | Proof of the existence of linear differential equations having a prescribed monodromic group | Resolved. Result: Yes or no, depending on more exact formulations of the problem.[citation needed] | – |
| 22nd | Uniformization of analytic relations by means of automorphic functions | Resolved.[citation needed] | – |
| 23rd | Further development of the calculus of variations | Unresolved. | – |
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