Hilbert's problems
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 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 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 ε ₀ 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 , proved using Dehn invariants 1900
4th Construct all metrics where lines are geodesics Too vague to be stated resolved or not -
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 Mathematical treatment of the axioms of physics ( 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 Partially resolved depending on how the original statement is interpreted . Items ( a ) and ( b ) were two specific problems given by Hilbert in a later explanation . 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 1933-2002 ?
7th Is a 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 -
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 -
12th Extend the Kronecker-Weber theorem on Abelian extensions of the rational numbers to any base number field Unresolved -
13th Solve 7th degree equation using algebraic ( variant : continuous ) functions of two parameters Unresolved . The continuous variant of this problem was solved by Vladimir Arnold in 1957 based on work by Andrei Kolmogorov , but the algebraic variant is unresolved -
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 ( 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 ?
Table of problems
Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were 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 in the Sorbonne. The complete list of 23 problems was published later, most notably in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.