Hilbert's problems

Hilbert's problems are 23 big questions in mathematics that a German mathematician named David Hilbert shared in 1900. At that time, none of these problems had been solved, and they ended up shaping a lot of what mathematicians worked on during the 1900s. Hilbert talked about ten of these problems at a big meeting in Paris, and later, all 23 problems were written down and translated into English.

Many of these problems have now been solved, and they helped guide mathematicians to new ideas and discoveries. Some problems still remain unsolved, which means mathematicians today are still working on them. Even the ones that are considered too vague to solve have inspired lots of interesting research and new ways of thinking about math.

List of Hilbert's problems

In 1900, a mathematician named David Hilbert shared 23 big questions in math that nobody had solved yet. These questions became known as Hilbert's problems and helped guide math research for many years. He talked about ten of these problems at a big meeting in Paris.

These are the titles of the 23 problems as they were written in a book in 1902:

1. Cantor's problem of the cardinal number of the continuum.
2. The compatibility of the arithmetical axioms.
3. Scissor congruence of polyhedra of equal volumes.
4. Problem of the straight line as the shortest distance between two points.
5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.
6. Mathematical treatment of the axioms of physics.
7. Irrationality and transcendence of certain numbers.
8. Problems of prime numbers.
9. Proof of the most general law of reciprocity in any number field.
10. Determination of the solvability of a Diophantine equation.
11. Quadratic forms with any algebraic numerical coefficients.
12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality.
13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.
14. Proof of the finiteness of certain complete systems of functions.
15. Rigorous foundation of Schubert's enumerative calculus.
16. Problem of the topology of algebraic curves and surfaces.
17. Expression of definite forms by squares.
18. Building up of space from congruent polyhedra.
19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?
20. The general problem of boundary values.
21. Proof of the existence of linear differential equations having a prescribed monodromy group.
22. Uniformization of analytic relations by means of automorphic functions.
23. Further development of the methods of the calculus of variations.

The 24th problem

Main article: Hilbert's twenty-fourth problem

Hilbert had 24 problems on his list, but he chose not to include one in the final version. This "24th problem" was found in his original notes by a German historian named Rüdiger Thiele in the year 2000. It was about finding simple and general methods to solve mathematical problems.

Nature and influence of the problems

Hilbert's problems covered many different areas of mathematics, some with clear goals and others more open-ended. For example, the 3rd problem was solved early on, while the 8th problem, known as the Riemann hypothesis, is still not solved today. Some problems, like the 5th, have accepted answers though related questions remain unanswered. Others, such as the 11th and 16th, relate to active areas of study like quadratic forms and real algebraic curves.

Two of the problems, the 4th and 6th, are considered either too vague or less important by today’s standards. The 23rd problem was meant to draw attention to the calculus of variations, a part of mathematics Hilbert felt was not appreciated enough. The other 20 problems have been widely studied, with mathematicians like Paul Cohen earning top honors for their work on these challenges. Even today, these problems inspire important research.

Main article: the Riemann hypothesis

Main articles: number theorists, conjectural, Langlands correspondence, Galois group, number field, quadratic forms, real algebraic curves, axiomatization, physics, foundations of geometry

Further information: Paul Cohen, Fields Medal, Yuri Matiyasevich, Julia Robinson, Hilary Putnam, Martin Davis

Knowability

David Hilbert wanted to use logic to clearly define mathematics. He believed that important math rules could be checked using a special method. One of his big goals was to prove that these rules were consistent, meaning they wouldn’t lead to any mistakes.

Later, a mathematician named Kurt Gödel showed that some of Hilbert’s ideas could not actually work. This was surprising and changed how people thought about math. Hilbert lived many years after Gödel’s discovery but didn’t write much about it. Even with these discoveries, Hilbert believed that every math problem should have an answer, one way or another.

Table of problems

Hilbert's 23 problems, along with an unpublished 24th problem, are shown below. To learn more about how these problems were solved and where to find extra information, click on the links in the first column.

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)	Paul Cohen demonstrated that the hypothesis is impossible to prove or disprove within Zermelo–Fraenkel set theory with or without the axiom of choice (provided Zermelo–Fraenkel set theory is consistent, i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem.	1940, 1963?
2nd	Prove that the axioms of arithmetic are consistent.	There is no consensus on whether the results of Gödel and Gentzen give a solution to the problem as stated by Hilbert. Gödel's second incompleteness theorem, proven in 1931, shows that no such proof 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.	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?	Resolved. Result: No, proven by Max Dehn using Dehn invariants. Even different Platonic solids of equal volume cannot be obtained this way from each other.	1900
4th	Construction of geometries satisfying axioms of classical geometry, where lines are geodesics.	Too vague to be stated resolved or not.	—
5th	Are continuous groups automatically differential groups?	Depends on the interpretation of "continuous group". If the term is understood as a topological group that is also a topological manifold: yes, proven by Andrew Gleason. If "continuous group" is understood as a topological group acting on a manifold, the problem becomes the Hilbert–Smith conjecture, which is still unresolved.	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. Kolmogorov's axiomatics 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 ?	Resolved. Result: Yes, illustrated by the Gelfond–Schneider theorem.	1934
8th	(a) The Riemann hypothesis: the real part of any non-trivial zero of the Riemann zeta function is 1⁄2.	Unresolved. Partial results involve much weaker estimations that almost all nontrivial zeroes have real part arbitrarily close to 1⁄2, and at least 5⁄12 of non-trivial zeros have real part equal to 1⁄2.	—
	(b) For pairwise coprime integers: a , b , c {\displaystyle a,b,c} determine solvability of diophantine equation: a x + b y + c = 0 {\displaystyle ax+by+c=0} for x and y being prime numbers. Goldbach's conjecture and the twin prime conjecture are special cases of this problem.	Unresolved, even the special cases of this equation are hard open problems. Partial results include Yitang Zhang's proof of bounded gaps between primes, later improved by the Polymath Project.	—
	(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.	Depends on the interpretation of expected results. In 1917, Erich Hecke constructed an analytic continuation for Dedekind zeta functions and proved functional equation, which allowed for obtaining results similar to that currently accessible using Riemann zeta function. However, if understood as proving an extended Riemann hypothesis, then the problem is still unresolved.	1917?
9th	Find the most general law of the reciprocity theorem in any algebraic number field.	Unresolved. Partial results involve the Artin reciprocity law for abelian extensions of number fields, key result in class field theory. Development of non-abelian class field theory that would work for the general case of number fields is still a largely conjectural area.	—
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 any number of variables and coefficients over any number field.	Resolved. Helmut Hasse in 1924 created a general theory of classification and deciding solvability of quadratic forms over number fields using the local-global principle. His methodology was later simplified by Ernst Witt using Witt rings.	1924
12th	Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to abelian extensions of any base number field.	Unresolved. Partial results involve construction using Hilbert modular forms for CM-fields by Goro Shimura and special cases of totally real fields using Brumer-Stark units by Dasgupta and Kadke.	—
13th	Prove that the general case of 7th-degree 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.	Depends on the variant of the problem. For the continuous variant: No; the Kolmogorov–Arnold representation theorem shows that every multivariate continuous function can be obtained through such composition. Some authors argue that Hilbert intended for a solution within the space of algebraic functions and possible extension of the Galois theory, thus continuing their own work on the algebraic case. It appears from one of later Hilbert's papers that this was his original intention for the problem. For the algebraic variant, the problem is unresolved.	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.	Significant developments for resolving this problem have been made since the publication of the list: Major enumerative examples of Schubert have been verified by Aluffi, Harris, Kleiman, Xambó, et al. Special presentations of the Chow rings of flag manifolds have been worked out by Borel, Marlin, Billey-Haiman and Duan-Zhao, et al.; Schubert's characteristic problem has been solved by Haibao Duan and Xuezhi Zhao. Duan and Zhao claimed that their result actually resolved this problem. Currently there is no consensus whether the problem is resolved completely or partially.	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.	Unresolved. Exact description of position of components for real algebraic curves is an open problem, even for small degrees like 8. For polynomial vector fields, partial results include proof that they have finitely many limit cycles, but no effective bound is known.	—
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) Are there only finitely many essentially different space groups in n-dimensional Euclidean space?	Resolved. Result: Yes (by Ludwig Bieberbach)	1910
	(b) Is there a polyhedron that admits only an anisohedral tiling in three dimensions?	Resolved. Result: Yes (by Karl Reinhardt).	1928
	(c) What is the densest sphere packing?	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.	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?	Unresolved. A significant topic of research throughout the 20th century, resulting in solutions for some cases.	—
21st	Proof of the existence of Fuchsian linear differential equations having a prescribed monodromy group	Resolved. Result: No, a counterexample was shown by Andrei Bolibrukh. Despite a negative answer in the most general case, Fuchsian equations may exist in special cases under some additional assumptions.	1989
22nd	Uniformization of analytic relations by means of automorphic functions	Unresolved. Partial results involve the uniformization theorem for Riemann surfaces.	—
23rd	Further development of the calculus of variations	Too vague to be stated resolved or not. Since the list was proposed, Hilbert and many other mathematicians have made numerous contributions to the calculus of variations. The dynamic programming of Richard Bellman is considered an alternative to the calculus of variations.	—
Unpublished 24th problem
24th	Development of a theory of proof simplicity	Recovered from Hilbert's unpublished notes. Too vague to be stated resolved or not.	—

Follow-ups

Since 1900, many mathematicians and groups have created their own lists of big math questions. But most of these have not been as important or popular as Hilbert's problems.

One famous set of questions came from a mathematician named André Weil in the 1940s. These were very important for the study of shapes and numbers. Different mathematicians found answers to these questions over time.

Another mathematician, Paul Erdős, shared hundreds of tricky math problems and sometimes offered money to anyone who could solve them.

In 1982, William Thurston shared a list of 24 problems, but these were all about the shapes of space. Most of these were solved within 30 years, which is faster than Hilbert's problems.

At the end of the 1900s, some mathematicians tried to make a new list like Hilbert's. Steve Smale made a list of 18 problems in response to a request from Vladimir Arnold.

Today, the big math challenges are often called the Millennium Prize Problems. These seven problems each have a million-dollar prize for whoever solves them. One of these, the Poincaré conjecture, was solved not long after the list was made.

The Riemann hypothesis is one question that appears in many of these lists. It is still unsolved today, and many think it will remain a big mystery for many years to come.