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以下列出希尔伯特的23个问题。欲了解更多信息，请阅读表格第一排条目链接。
| 问题 | 简述 | 状态 | 解决年份 |
|---|---|---|---|
| 第1题 | 连续统假设，即不存在一个基数绝对大于可列集而绝对小于实数集的集合。 | 美国数学家保罗·柯恩证明了连续统假设无法从带选择公理的策梅洛-弗兰克尔集合论（ZFC）中得到证明或反证，也就是说，连续统假设成立与否无法由ZFC确定。对于该结论是否可以算作对希尔伯特原始问题的答复，不存在共识。 | 1940, 1963 |
| 第2题 | 算术公理之相容性 | 哥德尔于1931年发表第二不完备定理，证明了算术的相容性不可能从算术本身中得到。而根岑在1936年则证明了算术的相容性取决于序数 ε₀ 的良基性。对于该结论在何种意义上可以算作对该问题的回答，不存在共识。一些数学家认为哥德尔给出了否定的回答，但另一些则认为根岑给出了肯定的回答。 | 1931, 1936 |
| 第3题 | 已知两个多面体有相同体积，能否把其中一个多面体分割成有限块再将之结合成另一个？ | 已解决，答案是不能。希尔伯特的学生马克斯·德恩以一反例给出了证明，后将其使用的证明方法称为德恩不变量。 | 1900 |
| 第4题 | 建立所有度量空间使得所有线段为测地线。 | 英国数学家Jeremy John Gray认为，大部分相关问题都已经得到解决，没解决的部分也取得了相当大的进展。然而希尔伯特对于这个问题的定义过于含糊，故不能确定是否解决。 | – |
| 第5题 | 所有连续群是否皆为可微群？ | 1953年日本数学家山边英彦已得到完全肯定的结果。但是，该问题还可以解读为希尔伯特－史密斯猜想的表述，而这个问题目前仍未解决。 | 1953? |
| 6th | Mathematical treatment of the axioms of physics | Partially resolved depending on how the original statement is interpreted.[1] In particular, in a further explanation Hilbert proposed two specific problems: (i) axiomatic treatment of probability with limit theorems for foundation of statistical physics and (ii) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua." Kolmogorov's axiomatics (1933) is now accepted as standard. There is some success on the way from the "atomistic view to the laws of motion of continua."[2] | 1933–2002? |
| 7th | Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? | Resolved. Result: yes, illustrated by Gelfond's theorem or the Gelfond–Schneider theorem. | 1934 |
| 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 the most general law of the reciprocity theorem in any algebraic number field. | Partially resolved.[n 1] | – |
| 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.[3] | – |
| 12th | Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field. | Unresolved. | – |
| 13th | Solve 7-th degree equation using algebraic (variant: continuous) functions of two parameters. | The problem was partially solved by Vladimir Arnold based on work by Andrei Kolmogorov.[n 2] | 1957 |
| 14th | Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? | Resolved. Result: no, a counterexample was constructed by Masayoshi Nagata. | 1959 |
| 15th | Rigorous foundation of Schubert's enumerative calculus. | Partially resolved. | – |
| 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, even for algebraic curves of degree 8. | – |
| 17th | Express a nonnegative rational function as quotient of sums of squares. | Resolved. Result: yes, due to Emil Artin. Moreover, an upper limit was established for the number of square terms necessary. | 1927 |
| 18th | (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? (b) What is the densest sphere packing? |
(a) Resolved. Result: yes (by Karl Reinhardt). (b) Widely believed to be resolved, by computer-assisted proof (by Thomas Callister Hales). Result: Highest density achieved by close packings, each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.[n 3] |
(a) 1928 (b) 1998 |
| 19th | Are the solutions of regular problems in the calculus of variations always necessarily 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 for the non-linear case. | ? |
| 21st | Proof of the existence of linear differential equations having a prescribed monodromic group | Partially resolved. Result: Yes, no, open depending on more exact formulations of the problem. | ? |
| 22nd | Uniformization of analytic relations by means of automorphic functions | Resolved. | ? |
| 23rd | Further development of the calculus of variations | Too vague to be stated resolved or not. | – |
- ^ Corry, L. David Hilbert and the axiomatization of physics (1894–1905). Arch. Hist. Exact Sci. 1997, 51 (2): 83–198. doi:10.1007/BF00375141.
- ^ Gorban, A.N.; Karlin, I. Hilbert's 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations (PDF). Bull. Amer. Math. Soc. 2014, 51 (2): 186–246. doi:10.1090/S0273-0979-2013-01439-3.
- ^ Hazewinkel, Michiel. Handbook of Algebra 6. Elsevier. 2009: 69. ISBN 0080932819.
- ^ D. Hilbert, "¨Uber die Gleichung neunten Grades", Math. Ann. 97 (1927), 243–250
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