Editing Hilbert's problems

{{Short description|23 mathematical problems stated in 1900}}
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[[File:Hilbert.jpg|thumb|David Hilbert]]
'''Hilbert's problems''' are 23 problems in [[mathematics]] published by German mathematician [[David Hilbert]] in 1900. 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&nbsp;8 at the [[University of Paris|Sorbonne]]. The complete list of 23&nbsp;problems was published later, in English translation in 1902 by [[Mary Frances Winston Newson]] in the ''[[Bulletin of the American Mathematical Society]]''.<ref name="Hilbert_1902">{{cite journal |last=Hilbert |first=David |title=Mathematical Problems |url=https://www.ams.org/journals/bull/1902-08-10/home.html |journal=[[Bulletin of the American Mathematical Society]] |volume=8 |issue=10 |year=1902 |pages=437–479|doi=10.1090/S0002-9904-1902-00923-3 |doi-access=free }}
</ref> Earlier publications (in the original German) appeared in ''Archiv der Mathematik und Physik''.<ref name="Hilbert_1901">{{cite journal |last=Hilbert |first=David |title=Mathematische Probleme | url=https://www.digizeitschriften.de/id/252457811_1900%7Clog37?tify=%7B%22pages%22%3A%5B269%5D%2C%22pan%22%3A%7B%22x%22%3A0.427%2C%22y%22%3A0.819%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.525%7D#navi |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (News of the Society of Sciences at Göttingen, Mathematical-Physical Class) |year=1900 |pages=253–297 |lang=de}} and {{cite journal |last=Hilbert |first=David |title=Mathematische Probleme |journal=Archiv der Mathematik und Physik |series=3rd series |volume=1 |year=1901 |pages=44–63, 213–237 |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015049309860&seq=62 |lang=de}}</ref>

Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, 21, and 20 have resolutions that are accepted by consensus of the mathematical community. Problems 1, 2, 5, 6,{{refn|group=lower-alpha|Number&nbsp;6 is now considered a problem in physics rather than in mathematics.}} 9, 11, 12, 15, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems. That leaves 8 (the [[Riemann hypothesis]]), 13 and 16{{refn|group=lower-alpha|Some authors consider this problem as too vague to ever be described as solved, although there is still active research on it.}} unresolved. 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 in the [[Bulletin of the American Mathematical Society]].<ref name=Hilbert_1902/>

:1. [[Continuum hypothesis|Cantor's problem]] of the cardinal number of the continuum.
:2. The compatibility of the arithmetical axioms.
:3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
: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 (The "[[Riemann hypothesis|Riemann Hypothesis]]").
:9. Proof of the most general law of reciprocity in any number field.
:10. Determination of the solvability of a [[Diophantine equation]].
:11. [[Quadratic form]]s 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 calculus|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 (Boundary value problems in PD)
:21. Proof of the existence of linear differential equations having a prescribed [[monodromy]] group.
:22. Uniformization of analytic relations by means of [[Automorphic function|automorphic functions]].
:23. Further development of the methods of the calculus of variations.

== The 24th problem ==
{{Main article|Hilbert's twenty-fourth problem}}

Hilbert originally included 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.<ref name="Thiele">{{cite journal |last=Thiele |first=Rüdiger |date=January 2003 |title=Hilbert's twenty-fourth problem |url=http://www.maa.org/news/Thiele.pdf |journal=American Mathematical Monthly |volume=110 |pages=1–24 |doi=10.1080/00029890.2003.11919933 |s2cid=123061382}}</ref>

== 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 [[Conjecture|conjectural]] [[Langlands program|Langlands correspondence]] on representations of the absolute [[Galois group]] of a [[number field]].<ref name="Weinstein">{{cite journal | last=Weinstein | first=Jared | title=Reciprocity laws and Galois representations: recent breakthroughs | journal=Bulletin of the American Mathematical Society | publisher=American Mathematical Society (AMS) | volume=53 | issue=1 | date=2015-08-25 | issn=0273-0979 | doi=10.1090/bull/1515 | pages=1–39| doi-access=free }}</ref>  Still other problems, such as the 11th and the 16th, concern what are now flourishing mathematical subdisciplines, like the theories of [[quadratic form]]s and [[real algebraic curve]]s.

There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the [[Axiomatic system|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 now 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:

{{Blockquote
|text="So far, I have generally mentioned problems as definite and special as possible, in the opinion that it is just such definite and special problems that attract us the most and from which the most lasting influence is often exerted upon science. Nevertheless, I should like to close with a general problem, namely with the indication of a branch of mathematics repeatedly mentioned in this lecture—which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, is its due—I mean the calculus of variations."
}}

The other 21 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 (mathematician)|Martin Davis]]) generated similar acclaim. Aspects of these problems are still of great interest today.

==Knowability==
Following [[Gottlob Frege]] and [[Bertrand Russell]], Hilbert sought to define mathematics logically using the method of [[formal system]]s, i.e., [[finitism|finitistic]] [[Mathematical proof|proofs]] from an agreed-upon set of [[axiom]]s.<ref name="Frege_Gödel">{{cite book |pages=464ff |editor-first=Jean |editor-last=van Heijenoort |year=1976 |orig-year=1966 |title=From Frege to Gödel: A source book in mathematical logic, 1879–1931 |publisher=Harvard University Press |location=Cambridge MA |isbn=978-0-674-32449-7 |edition=(pbk.) |quote=A reliable source of Hilbert's axiomatic system, his comments on them and on the foundational 'crisis' that was on-going at the time (translated into English), appears as Hilbert's 'The Foundations of Mathematics' (1927).}}</ref> 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.{{refn|group=lower-alpha|See Nagel and Newman revised by Hofstadter (2001, p.&nbsp;107),<ref name="Hofstadter_2001">{{Cite book |last1=Nagel |first1=Ernest |title=Gödel's proof |last2=Newman |first2=James R. |last3=Hofstadter |first3=Douglas R. |date=2001 |publisher=New York University Press |isbn=978-0-8147-5816-8 |editor-last=Hofstadter |editor-first=Douglas R. |editor-link=Douglas Hofstadter |edition=Rev. |location=New York}}</ref> footnote&nbsp;37: "Moreover, although most specialists in mathematical logic do not question the cogency of [Gentzen's] proof, it is not finitistic in the sense of Hilbert's original stipulations for an absolute proof of consistency." Also see next page: "But these proofs [Gentzen's et al.] cannot be mirrored inside the systems that they concern, and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program." Hofstadter rewrote the original (1958) footnote slightly, changing the word "students" to "specialists in mathematical logic". And this point is discussed again on page&nbsp;109<ref name=Hofstadter_2001/> and was not modified there by Hofstadter (p.&nbsp;108).<ref name=Hofstadter_2001/>}}

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&nbsp;years after [[Kurt Gödel]] published his theorem, but does not seem to have written any formal response to Gödel's work.{{refn|group=lower-alpha|Reid reports that upon hearing about "Gödel's work from Bernays, he was 'somewhat angry'. ... At first he was only angry and frustrated, but then he began to try to deal constructively with the problem. ... It was not yet clear just what influence Gödel's work would ultimately have" (p.&nbsp;198–199).<ref name="Reid_1996">{{cite book |first=Constance |last=Reid |year=1996 |title=Hilbert |publisher=Springer-Verlag |location=New York, NY |isbn=978-0387946740 |url-access=registration |url=https://archive.org/details/hilbert0000reid }}</ref> Reid notes that in two papers in 1931 Hilbert proposed a different form of induction called "unendliche Induktion" (p.&nbsp;199).<ref name="Reid_1996"/>}}{{refn|group=lower-alpha|Reid's biography of Hilbert, written during the 1960s from interviews and letters, reports that "Godel (who never had any correspondence with Hilbert) feels that Hilbert's scheme for the foundations of mathematics 'remains highly interesting and important in spite of my negative results' (p.&nbsp;217). Observe the use of present tense – she reports that Gödel and Bernays among others "answered my questions about Hilbert's work in logic and foundations" (p.&nbsp;vii).<ref name=Reid_1996/>}}

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 integer]]s". That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.

In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.<ref group=lower-alpha>This issue that finds its beginnings in the "foundational crisis" of the early 20th&nbsp;century, in particular the controversy about under what circumstances could the [[Law of Excluded Middle]] be employed in proofs. See much more at [[Brouwer–Hilbert controversy]].</ref> He stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any "[[Ignoramus et ignorabimus|ignorabimus]]" (statement whose truth can never be known).<ref group=lower-alpha>"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ''ignorabimus''." (Hilbert, 1902, p.&nbsp;445)</ref>  It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus.

On the other hand, the status of the first and second problems is even more complicated: there is no clear mathematical consensus as to whether the results of Gödel (in the case of the second problem), or Gödel and Cohen (in the case of the first problem) give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one.{{refn|group=lower-alpha|Nagel, Newman and Hofstadter discuss this issue: "The possibility of constructing a finitistic absolute proof of consistency for a formal system such as ''Principia Mathematica'' is not excluded by Gödel's results. ... His argument does not eliminate the possibility ... But no one today appears to have a clear idea of what a finitistic proof would be like that is ''not'' capable of being mirrored inside ''Principia Mathematica'' (footnote&nbsp;39, page&nbsp;109). The authors conclude that the prospect "is most unlikely".<ref name=Hofstadter_2001/>}}

==Follow-ups==
Since 1900, mathematicians and mathematical organizations have announced problem lists but, with few exceptions, these have not had nearly as much influence nor generated as much work as Hilbert's problems.

One exception consists of three conjectures made by [[André Weil]] in the late 1940s (the [[Weil conjectures]]). In the fields of [[algebraic geometry]], number theory and the links between the two, the Weil conjectures were very important.<ref>{{Cite journal| last1=Weil | first1=André | author1-link=André Weil | title=Numbers of solutions of equations in finite fields | url=http://www.ams.org/bull/1949-55-05/S0002-9904-1949-09219-4/home.html | doi=10.1090/S0002-9904-1949-09219-4  | mr=0029393 | year=1949 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=55 | pages=497–508 | issue=5| doi-access=free }}</ref><ref name="Browder American Mathematical Society">{{cite book | last=Browder | first=Felix E. | title=Mathematical developments arising from Hilbert problems. | publisher=American Mathematical Society | publication-place=Providence | date=1976 | isbn=0-8218-1428-1 | oclc=2331329 | page=}}</ref> The first of these was proved by [[Bernard Dwork]]; a completely different proof of the first two, via [[Étale cohomology|ℓ-adic cohomology]], was given by [[Alexander Grothendieck]]. The last and deepest of the Weil conjectures (an analogue of the Riemann hypothesis) was proved by [[Pierre Deligne]]. Both Grothendieck and Deligne were awarded the [[Fields medal]]. However, the Weil conjectures were, in their scope, more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics. This is somewhat ironic, since arguably Weil was the mathematician of the 1940s and 1950s who best played the Hilbert role, being conversant with nearly all areas of (theoretical) mathematics and having figured importantly in the development of many of them.

[[Paul Erdős]] posed hundreds, if not thousands, of mathematical [[Paul Erdős#Erdős's problems|problems]], many of them profound. Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.<ref name="Chung Graham 1999 Erdös ">{{cite book | last1=Chung | first1=Fan R. K. | last2=Graham | first2=Ronald L. | title=Erdös on Graphs: his legacy of unsolved problems | publisher=A K Peters/CRC Press | publication-place=Natick, Mass | date=1999-06-01 | isbn=978-1-56881-111-6| oclc=42809520 | page=}}</ref>

The end of the millennium, which was also the centennial of Hilbert's announcement of his problems, provided a natural occasion to propose "a new set of Hilbert problems". Several mathematicians accepted the challenge, notably Fields Medalist [[Steve Smale]], who responded to a request by [[Vladimir Arnold]] to propose a list of 18 problems ([[Smale's problems]]).

At least in the mainstream media, the ''de facto'' 21st&nbsp;century analogue of Hilbert's problems is the list of seven [[Millennium Prize Problems]] chosen during 2000 by the [[Clay Mathematics Institute]]. Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and mathematicians in general, each prize problem includes a million-dollar bounty. As with the Hilbert problems, one of the prize problems (the [[Poincaré conjecture]]) was solved relatively soon after the problems were announced.

The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise. Although it has been attacked by major mathematicians of our day, many experts believe that it will still be part of unsolved problems lists for many centuries. Hilbert himself declared: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proved?"<ref name="Clawson">{{cite book |title=Mathematical Mysteries: The beauty and magic of numbers |first=Calvin C. |last=Clawson |date=8 December 1999 |page=258 |isbn=9780738202594 |lccn=99-066854 |publisher=Basic Books}}</ref>

In 2008, [[DARPA]] announced its own list of 23&nbsp;problems that it hoped could lead to major mathematical breakthroughs, "thereby strengthening the scientific and technological capabilities of the [[United States Department of Defense|DoD]]".<ref name="Cooney 2008 DARPA 23 questions">{{cite web | last=Cooney | first=Michael | title=The world's 23 toughest math questions | website=Network World | date=30 September 2008 | url=https://www.networkworld.com/article/905224/security-the-world-s-23-toughest-math-questions.html | access-date=7 April 2024}}</ref><ref name="DARPA_math_2019">{{Cite web |date=2008-09-26 |title=DARPA Mathematical Challenges |url=https://www.fbo.gov/?s=opportunity&mode=form&id=c120bc7171c203aa5f4b3903aa08e558&tab=core&_cview=0 |url-status=dead |archive-url=https://web.archive.org/web/20190112150040/https://www.fbo.gov/?s=opportunity&mode=form&id=c120bc7171c203aa5f4b3903aa08e558&tab=core&_cview=0 |archive-date=2019-01-12 |access-date=2021-03-31}}</ref> The DARPA list also includes a few problems from Hilbert's list, e.g. the Riemann hypothesis.

==Table of problems==
Hilbert's 23 problems are (for details on the solutions and references, see the articles that are linked to in the first column):
{| class="wikitable sortable"
|-
! Problem
!class="unsortable"| Brief explanation
! Status
! Year solved
|-
|style="text-align:center"| [[Hilbert's first problem|1st]]
| The [[continuum hypothesis]] (that is, there is no [[Set (mathematics)|set]] whose [[cardinality]] is strictly between that of the [[integer]]s and that of the [[real number]]s)
|{{partial|align=left|{{sort|2|}} 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 [[consistency|consistent]], i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem.}}
|style="text-align:center"| 1940, 1963
|-
|style="text-align:center"| [[Hilbert's second problem|2nd]]
| Prove that the [[axiom]]s of [[arithmetic]] are [[consistency|consistent]].
|{{partial|align=left|{{sort|2|}} There is no consensus on whether results of [[Kurt Gödel|Gödel]] and [[Gerhard Gentzen|Gentzen]] give a solution to the problem as stated by Hilbert. Gödel's [[Gödel's incompleteness theorems|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-founded relation|well-foundedness]] of the [[epsilon numbers (mathematics)|ordinal&nbsp;''ε''<sub>0</sub>]].}}
|style="text-align:center"| 1931, 1936
|-
|style="text-align:center"| [[Hilbert's third problem|3rd]]
| Given any two [[polyhedron|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?
|{{yes|align=left|{{sort|1|}} Resolved. Result: No, proved using [[Dehn invariant]]s.}}
|style="text-align:center"| 1900
|-
|style="text-align:center"| [[Hilbert's fourth problem|4th]]
| Construct all [[metric space|metrics]] where lines are [[geodesic]]s.
|{{unknown|align=left|style=font-size:inherit|{{sort|4|}} Too vague to be stated resolved or not.{{refn|According to Gray, most of the problems have been solved. Some were not defined completely, but enough progress has been made to consider them "solved"; Gray lists the fourth problem as too vague to say whether it has been solved.|group=lower-alpha}} }}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's fifth problem|5th]]
| Are continuous [[group (mathematics)|groups]] automatically [[Lie group|differential groups]]?
|{{partial|align=left|{{sort|2|}} Resolved by [[Andrew Gleason]], assuming one interpretation of the original statement. If, however, it is understood as an equivalent of the [[Hilbert–Smith conjecture]], it is still unsolved.}}
|style="text-align:center"| 1953?
|-
|style="text-align:center"| [[Hilbert's sixth problem|6th]]
| Mathematical treatment of the [[axiom]]s 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"
|{{partial|align=left|{{sort|2|}} Unresolved, or partially resolved, depending on how the original statement is interpreted.<ref>{{cite journal |last1=Corry |first1=L. |year=1997 |title=David Hilbert and the axiomatization of physics (1894–1905) |journal=Arch. Hist. Exact Sci. |volume=51 |issue=2 |pages=83–198 |doi=10.1007/BF00375141 |s2cid=122709777 }}</ref> Items&nbsp;(a) and (b) were two specific problems given by Hilbert in a later explanation.<ref name=Hilbert_1902/> [[probability axioms|Kolmogorov's axiomatics]] (1933) is now accepted as standard for the foundations of probability theory. There is some success on the way from the "atomistic view to the laws of motion of continua",<ref>{{cite journal |last1=Gorban |first1=A. N. |author-link=Alexander Nikolaevich Gorban |last2=Karlin |first2=I. |year=2014 |title=Hilbert's 6th Problem: Exact and approximate hydrodynamic manifolds for kinetic equations |journal=Bulletin of the American Mathematical Society |volume=51 |issue=2 |pages=186–246 |arxiv=1310.0406 |doi=10.1090/S0273-0979-2013-01439-3| doi-access= free}}</ref>}} but the transition from classical to quantum physics means that there would have to be two axiomatic formulations, with a clear link between them. [[John von Neumann]] made an early attempt to place [[quantum mechanics]] on a rigorous mathematical basis in his book ''[[Mathematical Foundations of Quantum Mechanics]]'',<ref>{{Cite book |last=Von Neumann |first=John |author-link=John von Neumann |url=https://press.princeton.edu/titles/11352.html |title=Mathematical foundations of quantum mechanics |date=2018 |publisher=Princeton University Press |isbn=978-0-691-17856-1 |editor-last=Wheeler |editor-first=Nicholas A. |location=Princeton Oxford |translator-last=Beyer |translator-first=Robert T.}}</ref> but subsequent developments have occurred, further challenging the axiomatic foundations of quantum physics. 
|style="text-align:center"| 1933–2002?
|-
|style="text-align:center"| [[Hilbert's seventh problem|7th]]
| Is ''a<sup>b</sup>'' [[transcendental number|transcendental]], for [[algebraic number|algebraic]] ''a'' ≠ 0,1 and [[irrational number|irrational]] algebraic ''b'' ?
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes, illustrated by the [[Gelfond–Schneider theorem]].}}
|style="text-align:center"| 1934
|-
|style="text-align:center"| [[Hilbert's eighth problem|8th]]
| The [[Riemann hypothesis]] ("the real part of any non-[[Triviality (mathematics)|trivial]] [[Zero of a function|zero]] of the [[Riemann zeta function]] is 1/2") and other prime-number problems, among them [[Goldbach's conjecture]] and the [[twin prime conjecture]]
|{{no|align=left|{{sort|3|}} Unresolved.}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's ninth problem|9th]]
| Find the most general law of the [[Quadratic reciprocity|reciprocity theorem]] in any [[algebra]]ic [[number field]].
|{{partial|align=left|{{sort|2|}} Partially resolved. Solved by [[Emil Artin]] in 1927 for [[abelian extension]]s of the [[rational number]]s during the development of [[class field theory]]. The non-abelian case remains unsolved, if one interprets that as meaning [[non-abelian class field theory]].|group=lower-alpha}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's tenth problem|10th]]
| Find an algorithm to determine whether a given polynomial [[Diophantine equation]] with integer coefficients has an integer solution.
|{{yes|align=left|{{sort|1|}} Resolved. Result: Impossible; [[Matiyasevich's theorem]] implies that there is no such algorithm.}}
|style="text-align:center"| 1970
|-
|style="text-align:center"| [[Hilbert's eleventh problem|11th]]
| Solving [[quadratic form]]s with algebraic numerical [[coefficient]]s.
|{{partial|align=left|{{sort|2|}} Partially resolved.<ref name="Hazewinkel">{{cite book |first=Michiel |last=Hazewinkel |date=2009 |title=Handbook of Algebra |publisher=Elsevier |page=69 |isbn=978-0080932811 |volume=6}}</ref>}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's twelfth problem|12th]]
| Extend the [[Kronecker–Weber theorem]] on abelian extensions of the [[rational number]]s to any base number field.
|{{partial|align=left|{{sort|2|}} Partially resolved.<ref>{{cite web |url=https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525/ |first=Kelsey |last=Houston-Edwards |title=Mathematicians Find Long-Sought Building Blocks for Special Polynomials |date=25 May 2021 }}</ref>}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's thirteenth problem|13th]]
| Solve [[septic equation|7th-degree equation]] using algebraic (variant: continuous) [[mathematical function|functions]] of two [[parameter]]s.
|{{no|align=left|{{sort|3|}} Unresolved. The continuous variant of this problem was solved by [[Vladimir Arnold]] in 1957 based on work by [[Andrei Kolmogorov]] (see [[Kolmogorov–Arnold representation theorem]]), but the algebraic variant is unresolved.{{refn|1=It is not difficult to show that the problem has a partial solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the [[Galois theory]] (see, for example, Abhyankar<ref name="Abyankar">{{cite book |url=http://www.emis.de/journals/SC/1997/2/pdf/smf_sem-cong_2_1-11.pdf |first=Shreeram S. |last=Abhyankar |title=Hilbert's Thirteenth Problem |series=Séminaires et Congrès |volume=2 |publisher=Société Mathématique de France |date=1997}}</ref> Vitushkin,<ref>{{cite journal |last1=Vitushkin |first1=Anatoliy G. |title=On Hilbert's thirteenth problem and related questions |journal=Russian Mathematical Surveys |date=2004 |volume=59 |issue=1 |pages=11–25 |doi=10.1070/RM2004v059n01ABEH000698 |publisher=Russian Academy of Sciences|bibcode=2004RuMaS..59...11V |s2cid=250837749 }}</ref> Chebotarev,<ref>{{cite journal |last1=Morozov |first1=Vladimir V. |title=О некоторых вопросах проблемы резольвент |journal=Proceedings of Kazan University |date=1954 |volume=114 |issue=2 |pages=173–187 |url=http://www.mathnet.ru/php/getFT.phtml?jrnid=uzku&paperid=406&what=fullt&option_lang=eng |publisher=Kazan University |language=ru |trans-title=On certain questions of the problem of resolvents}}</ref> and others). It appears from one of Hilbert's papers<ref>{{cite journal |first=David |last=Hilbert |title=Über die Gleichung neunten Grades |journal=Math. Ann. |volume=97 |year=1927 |pages=243–250 |doi=10.1007/BF01447867 |s2cid=179178089 }}</ref> that this was his original intention for the problem.
The language of Hilbert there is "{{lang|de|Existenz von ''algebraischen'' Funktionen|italic=unset}}" ("existence of ''algebraic'' functions").
As such, the problem is still unresolved.|group=lower-alpha}} }}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's fourteenth problem|14th]]
| Is the [[invariant theory|ring of invariants]] of an [[algebraic group]] acting on a [[polynomial ring]] always [[Finitely generated algebra|finitely generated]]?
|{{yes|align=left|{{sort|1|}} Resolved. Result: No, a counterexample was constructed by [[Masayoshi Nagata]].}}
|style="text-align:center"| 1959
|-
|style="text-align:center"| [[Hilbert's fifteenth problem|15th]]
| Rigorous foundation of [[Schubert's enumerative calculus]].
|{{partial|align=left|{{sort|2|}} Partially resolved.<ref name="KL">{{cite journal | last1=Kleiman | first1=S.L.| last2=Laksov | first2=Dan |author1-link=Steven Kleiman |author2-link=Dan Laksov|title= Schubert Calculus | publisher=American Mathematical Society|journal = American Mathematical Monthly | volume=79| issue=10 | year=1972 | issn=0377-9017 | doi=10.1080/00029890.1972.11993188 | pages=1061–1082 }}</ref> Haibao Duan and Xuezhi Zhao claimed that this problem is actually resolved.}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's sixteenth problem|16th]]
| Describe relative positions of ovals originating from a [[real number|real]] [[algebraic curve]] and as [[limit cycle]]s of a polynomial [[vector field]] on the plane.
|{{no|align=left|{{sort|3|}} Unresolved, even for algebraic curves of degree&nbsp;8.}}
|style="text-align:center"| —
|-
|style="text-align:center"| [[Hilbert's seventeenth problem|17th]]
| Express a nonnegative [[rational function]] as [[quotient]] of sums of [[Square (algebra)|squares]].
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes, due to [[Emil Artin]]. Moreover, an upper limit was established for the number of square terms necessary.}}
|style="text-align:center"| 1927
|-
|rowspan=3 style="text-align:center"| [[Hilbert's eighteenth problem|18th]]
| (a) Are there only finitely many essentially different [[space group]]s in ''n''-dimensional Euclidean space?
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes (by [[Ludwig Bieberbach]])}}
|style="text-align:center"| 1910
|-
| (b) Is there a polyhedron that admits only an [[anisohedral tiling]] in three dimensions?
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes (by [[Karl Reinhardt (mathematician)|Karl Reinhardt]]).}}
|style="text-align:center"| 1928
|-
| (c) What is the densest [[sphere packing]]?
|{{yes|align=left|{{sort|1|}} Widely believed to be resolved, by [[computer-assisted proof]] (by [[Thomas Callister Hales]]). Result: Highest density achieved by [[Close-packing of equal spheres|close packings]], each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.{{refn|Gray also lists the 18th problem as "open" in his 2000 book, because the sphere-packing problem (also known as the [[Kepler conjecture]]) was unsolved, but a solution to it has now been claimed.|group=lower-alpha}}}}
|style="text-align:center"| 1998
|-
|style="text-align:center"| [[Hilbert's nineteenth problem|19th]]
| Are the solutions of regular problems in the [[calculus of variations]] always necessarily [[Analytic function|analytic]]?
|{{yes|align=left|{{sort|1|}} Resolved. Result: Yes, proven by [[Ennio De Giorgi]] and, independently and using different methods, by [[John Forbes Nash]].}}
|style="text-align:center"| 1957
|-
|style="text-align:center"| [[Hilbert's twentieth problem|20th]]
| Do all [[calculus of variations|variational problems]] with certain [[boundary condition]]s have solutions?
|{{partial|align=left|{{sort|2|}} Partially resolved. A significant topic of research throughout the 20th century, resulting in solutions for some cases.<ref name="Gilbarg Trudinger 2001 Elliptic PDEs ">{{cite book | last1=Gilbarg | first1=David | last2=Trudinger | first2=Neil S. | title=Elliptic Partial Differential Equations of Second Order | publisher=Springer Science & Business Media | publication-place=Berlin New York | date=2001-01-12 | isbn=978-3-540-41160-4 | page=}}</ref><ref name="Serrin 1969 Dirichlet">{{cite journal | last= Serrin | first=James | title=The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables | journal=Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences | volume=264 | issue=1153 | date=1969-05-08 | issn=0080-4614 | doi=10.1098/rsta.1969.0033 | pages=413–496| bibcode=1969RSPTA.264..413S }}</ref><ref name="Mawhin 1999 pp. 195–228">{{cite journal | last=Mawhin | first=Jean | title=Leray-Schauder degree: a half century of extensions and applications | journal=Topological Methods in Nonlinear Analysis | publisher=Nicolaus Copernicus University in Toruń, Juliusz Schauder Center for Nonlinear Studies | volume=14 | issue=2 | date=1 January 1999 | issn=1230-3429 | doi=10.12775/TMNA.1999.029 | pages=195–228 | url=https://projecteuclid.org/journals/topological-methods-in-nonlinear-analysis/volume-14/issue-2/Leray-Schauder-degree--a-half-century-of-extensions-and/tmna/1475179840.full | access-date=8 April 2024}}</ref>}}
|style="text-align:center"| ?
|-
|style="text-align:center"| [[Hilbert's twenty-first problem|21st]]
| Proof of the existence of [[Fuchsian theory|Fuchsian]] [[linear differential equation]]s having a prescribed [[monodromy group]]
|{{Yes|{{sort|2|}} Resolved. Result: No, shown by A. Bolibrukh. <ref name="Plemelj">{{Citation | last1=Plemelj | first1=Josip | editor1-last=Radok. | editor1-first=J. R. M. | title=Problems in the sense of Riemann and Klein | url=https://books.google.com/books?id=f0urAAAAIAAJ | publisher=Interscience Publishers John Wiley & Sons Inc.|location= New York-London-Sydney | series= Interscience Tracts in Pure and Applied Mathematics | mr=0174815 | year=1964 | volume=16| isbn=9780470691250 }}</ref><ref name ="Anasov_Bolibruch">{{Citation | last1=Anosov | first1=D. V. | last2=Bolibruch | first2=A. A. | authorlink2=Andrei Bolibrukh | title=The Riemann-Hilbert problem | publisher=Friedr. Vieweg & Sohn | location=Braunschweig | series=Aspects of Mathematics, E22 | isbn=978-3-528-06496-9 | mr=1276272 | year=1994 | doi=10.1007/978-3-322-92909-9}}</ref><ref name ="Bolibruch1">{{Citation | last1=Bolibrukh | first1=A. A. | title=The Riemann-Hilbert problem | doi=10.1070/RM1990v045n02ABEH002350 | mr=1069347 | year=1990 | journal=Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | issn=0042-1316 | volume=45 | issue=2 | pages=3–47| bibcode=1990RuMaS..45Q...1B | s2cid=250853546 |language=Russian}}</ref><ref name ="Bolibruch2">{{Citation | last1=Bolibrukh | first1=A.A. | title=Sufficient conditions for the positive solvability of the Riemann-Hilbert problem | journal=Matematicheskie Zametki |pages=110–117 | language=Russian | year=1992| volume=51 | issue=2 |mr=1165460 | doi=10.1007/BF02102113| s2cid = 121743184}}</ref>|align=left}}
|style="text-align:center"| 1989
|-
|style="text-align:center"| [[Hilbert's twenty-second problem|22nd]]
| Uniformization of analytic relations by means of [[automorphic function]]s
|{{partial|align=left|{{sort|2|}} Partially resolved. [[Uniformization theorem]]}}
|style="text-align:center"| ?
|-
|style="text-align:center"| [[Hilbert's twenty-third problem|23rd]]
| Further development of the [[calculus of variations]]
|{{unknown|align=left|style=font-size:inherit|{{sort|4|}} Too vague to be stated resolved or not.}}
|style="text-align:center"| —
|}

==See also==
* [[Landau's problems]]
* [[Millennium Prize Problems]]
* [[Smale's problems]]
* [[Taniyama's problems]]
* [[Thurston's 24 questions]]

==Notes==
{{notelist|30em}}

==References==
{{Reflist|30em}}

== Further reading ==
{{refbegin|30em}}
* {{Cite book |last=Gray |first=Jeremy |author-link=Jeremy Gray |url=https://www.worldcat.org/title/ocm44153228 |title=The Hilbert challenge |publisher=Oxford University Press |year=2000 |isbn=978-0-19-850651-5 |location=Oxford; New York |oclc=ocm44153228}}
* {{Cite book |last=Yandell |first=Ben |author-link=Benjamin Yandell |url=https://archive.org/details/honorsclasshilbe0000yand |title=The honors class: Hilbert's problems and their solvers |publisher=A.K. Peters |year=2002 |isbn=978-1-56881-141-3 |location=Natick, Mass |url-access=registration}}
* {{Cite book |last=Thiele |first=Rüdiger |title=Mathematics and the historian's craft: the Kenneth O. May lectures; [presented at CSHPM meetings since 1990] |title-link=Kenneth May |publisher=Springer |year=2005 |isbn=978-0-387-25284-1 |editor-last=Brummelen |editor-first=Glen Van |series=[[Canadian Mathematical Society|CMS]] Books in Mathematics |volume=21 |location=New York, NY [Heidelberg] |pages=243–295 |chapter=On Hilbert and his twenty-four problems |editor-last2=Kinyon |editor-first2=Michael |editor-last3=Van Brummelen |editor-first3=Glen |editor-last4=Canadian Society for History and Philosophy of Mathematics}}
* {{Cite book |last1=Dawson |first1=John W. |title=Logical dilemmas: the life and work of Kurt Gödel |last2=Gödel |first2=Kurt |publisher=Peters |year=1997 |isbn=978-1-56881-256-4 |edition=Reprint |location=Wellesley, Mass}}<br />''A wealth of information relevant to Hilbert's "program" and [[Gödel]]'s impact on the Second Question, the impact of [[Arend Heyting]]'s and [[L. E. J. Brouwer|Brouwer]]'s [[Intuitionism]] on Hilbert's philosophy.''
*  {{Cite book |last=Browder |first=Felix Earl |title=Proceedings of Symposia in Pure Mathematics XXVIII |publisher=American Mathematical Society |year=1976 |isbn=978-0-8218-1428-4 |editor-last=Browder |editor-first=Felix E. |editor-link=Felix Browder |location=Providence (R.I) |chapter=Mathematical Developments Arising from Hilbert Problems}}<br />''A collection of survey essays by experts devoted to each of the 23&nbsp;problems emphasizing current developments.''
* {{Cite book |last1=Matijasevič |first1=Jurij V. |title=Hilbert's tenth problem |last2=Matijasevič |first2=Jurij V. |publisher=MIT Press |year=1993 |isbn=978-0-262-13295-4 |edition=3. |series=Foundations of computing |location=Cambridge, Mass.}}<br />''An account at the undergraduate level by the mathematician who completed the solution of the problem.''
{{refend}}

== External links ==
{{Wikisource|Mathematical Problems}}
* {{springer|title=Hilbert problems|id=p/h120080}}
* {{cite web |url=http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/rede.html |title=Original text of Hilbert's talk, in German. |access-date=2005-02-05 |archive-url=https://web.archive.org/web/20120205025851/http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/rede.html |archive-date=2012-02-05 |url-status=dead }}
* {{cite web |url=https://www.ams.org/journals/bull/2000-37-04/S0273-0979-00-00881-8/S0273-0979-00-00881-8.pdf |title=David Hilbert's "Mathematical Problems": A lecture delivered before the International Congress of Mathematicians at Paris in 1900.}}
* {{librivox book | title=Mathematical Problems| author=Hilbert}}

{{Hilbert's problems}}
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{{DEFAULTSORT:Hilbert's Problems}}
[[Category:Hilbert's problems| ]]
[[Category:Unsolved problems in mathematics]]