Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. He intended to rival the master of French mathematics, Henri Poincaré, and to prove that he was cut from the same cloth. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, and translated into English in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society. Earlier publications (in the original German) appeared in Archiv der Mathematik und Physik. and
Of the cleanly formulated Hilbert problems: 3, 6a, 7, 10, 11, 14, 17, 18, 19, and 21 have resolutions that are accepted by consensus of the mathematical community. The status of problems 1, 2, 5, 6b, 8c, 13, and 15 is controversial: there are some results, but there exists some controversy as to whether they resolve the problems. Problems 8a, 8b, 9, 12, 16, 20 and 22 are unresolved or widely agreed as unresolved despite some partial results. Problems 4 and 23 are considered as too vague to ever be described as solved; the withdrawn 24 would also be in this class.
List of Hilbert's problems The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation of Hilbert's presentation, published in the Bulletin of the American Mathematical Society.
Cantor's problem of the cardinal number of the continuum The compatibility of the arithmetical axioms Scissor congruence of polyhedra of equal volumes Problem of the straight line as the shortest distance between two points. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group Mathematical treatment of the axioms of physics Irrationality and transcendence of certain numbers Problems of prime numbers Proof of the most general law of reciprocity in any number field Determination of the solvability of a Diophantine equation Quadratic forms with any algebraic numerical coefficients Extensions of the Kronecker–Weber theorem on Abelian fields to any algebraic realm of rationality Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments Proof of the finiteness of certain complete systems of functions Rigorous foundation of Schubert's enumerative calculus Problem of the topology of algebraic curves and surfaces Expression of definite forms by squares Building up of space from congruent polyhedra Are the solutions of regular problems in the calculus of variations always necessarily analytic? The general problem of boundary values Proof of the existence of linear differential equations having a prescribed monodromy group Uniformization of analytic relations by means of automorphic functions Further development of the methods of the calculus of variations
The 24th problem
Hilbert originally had 24 problems on his list, but decided against including one of them in the published list. The "24th problem" (in proof theory, on a criterion for simplicity and general methods) was rediscovered in Hilbert's original manuscript notes by German historian Rüdiger Thiele in 2000.
Table of problems Hilbert's 23 problems, and the unpublished 24th problem, are listed below. For details on the solutions and references, see the articles that are linked to in the first column. Problem Brief explanation Status Year solved 1st The continuum hypothesis: there is no set whose cardinality is strictly between that of the integers and that of the real numbers. 1940, 1963? 2nd Prove that the axioms of arithmetic are consistent. 1931, 1936? 3rd Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second? 1900 4th Construction of geometries satisfying axioms of classical geometry, where lines are geodesics. 5th Are continuous groups automatically differential groups? 1953? 6th Mathematical treatment of the axioms of physics. In a later explanation given by Hilbert:
(a) axiomatic treatment of probability with limit theorems for foundation of statistical physics
(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"(a) Resolved. The Kolmogorov axioms are accepted as the foundation of probability theory. 1933(b) Depends on the interpretation of the problem.
If treated as a physical problem: since the publication of Hilbert's list, new discoveries challenged classical mechanics and led to the formulation of quantum field theory, which holds an "atomistic view" of physical laws; and general relativity, which describes "motion of continua" at large scales. Despite many attempts to unify them into a theory of everything, it is still not obvious how to make clear link between them.
Some authors tried to solve this as a mathematical problem in a classical mechanics framework, which was the dominant physical theory during the publication of the list. In March 2025, Deng, Hani, and Ma published a paper claiming to have solved this problem by deriving continuous fluid equations and Boltzmann equations from Newton's laws applied for particles. The paper is currently in peer review. 2025? 7th Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? 1934 8th (a) The Riemann hypothesis: the real part of any non-trivial zero of the Riemann zeta function is . (b) For pairwise coprime integers: determine solvability of diophantine equation: for x and y being prime numbers. Goldbach's conjecture and the twin prime conjecture are special cases of this problem. (c) Generalize results using Riemann zeta function for distribution of prime numbers in integers, to apply them to Dedekind zeta functions for distribution of prime ideals in ring of integers for any number field. 1917? 9th Find the most general reciprocity law in any number field. 10th Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. 1970 11th Develop the theory of quadratic forms with any number of variables and coefficients over any number field, especially resolving quadratic equations by elements of this field and its ring of integers. 1924, 1937 12th Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to abelian extensions of any base number field. 13th Prove that the general case of septic equation cannot be solved using finite composition of continuous functions (variant: algebraic functions) of two parameters. For continuous variant: at least construct analytic function of three variables that cannot be represented as such composition. 1957? 14th If is a field and is a subfield of containing , is always finitely generated over ? 1954, 1958-2005 15th Rigorous foundation of Schubert's enumerative calculus. 1987–2020? 16th Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. 17th Express a nonnegative rational function as quotient of sums of squares. 1927 18th (a) Are there only finitely many essentially different space groups in n-dimensional Euclidean space? 1910 (b) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? 1928 (c) What is the densest sphere packing? 1998 19th Are the solutions of regular problems in the calculus of variations always necessarily analytic? 1957 20th Do all variational problems with certain boundary conditions have solutions? 21st Proof of the existence of Fuchsian linear differential equations with given singular points and a prescribed monodromy group 1908, 1989 22nd Uniformization of analytic relations by means of automorphic functions 23rd Further development of the calculus of variations Unpublished 24th problem 24th Development of a theory of proof simplicity
Nature and influence of the problems Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer. For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist. Some of Hilbert's statements were not precise enough to specify a particular problem, but were suggestive enough that certain problems of contemporary nature seem to apply; for example, most modern number theorists would probably see the 9th problem as referring to the conjectural Langlands correspondence on representations of the absolute Galois group of a number field. Still other problems, such as the 11th and the 16th, concern flourishing mathematical subdisciplines, like the theories of quadratic forms and real algebraic curves.
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that 20th-century developments seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner that is generally judged to be too vague to enable a definitive answer.
The 23rd problem was purposefully set as a general indication by Hilbert to highlight the calculus of variations as an underappreciated and understudied field. In the lecture introducing these problems, Hilbert made the following introductory remark to the 23rd problem:
The other 20 problems have all received significant attention, and late into the 20th century work on these problems was still considered to be of the greatest importance. Paul Cohen received the Fields Medal in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by Yuri Matiyasevich (completing work by Julia Robinson, Hilary Putnam, and Martin Davis) generated similar acclaim. Aspects of these problems remain of great interest.
Knowability Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms. One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.
However, Gödel's second incompleteness theorem gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.
Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.