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== Table of problems ==
Tabel berikut memuat 23 masalah Hilbert. Untuk detail solusi dan referensi lebih lanjut, silahkan lihat artikel yang dipranala di kolom pertama.
Hilbert's twenty-three problems are (for details on the solutions and references, see the detailed articles that are linked to in the first column):
{| class="wikitable sortable" style="text-align:left"
! style="text-align:center;" width="6%" |Problem▼
! style="text-align:center;" width="44%" class="unsortable" |Brief explanation▼
! style="text-align:center;" width="44%" |Status▼
! style="text-align:center;" width="6%" |Year Solved▼
|-
| !style="text-align:center;" width=6% |[[Hilbert's first problem|1st]]Masalah
▲! style="text-align:center;" width="44%" class="unsortable" |Brief explanationPenjelasan secara singkat
|The [[continuum hypothesis]] (that is, there is no [[Set (mathematics)|set]] whose [[cardinality]] is strictly between that of the [[Integer|integers]] and that of the [[Real number|real numbers]])| {{partial|{{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;" width=44% |1940, 1963Status
▲! style="text-align:center;" width="6%" |Problem Terpecahkan pada tahun
|-
| style="text-align:center;" | [[Hilbert'sMasalah secondpertama problemHilbert|2ndMasalah ke-1]]
| [[Hipotesis kontinum]], suatu hipotesis yang mengatakan bahwa tidak ada [[Himpunan (matematika)|himpunan]] yang mempunyai [[kardinalitas]] antara kardinalitas [[bilangan bulat]] dan kardinalitas [[bilangan real]])
|Prove that the [[Axiom|axioms]] of [[arithmetic]] are [[Consistency|consistent]].| {{partial|{{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>]].}}
| {{partial|{{sort|2|}}Masalah ini mustahil untuk dibuktikan atau dibantahkan dalam [[teori himpunan Zermelo–Fraenkel]] dengan atau tanpa menggunakan [[aksioma pemilihan]]. Teorema himpunan Zermelo–Fraenkel adalah [[Konsistensi|konsisten]], dalam artian bahwa teorema ini tidak mengandung kontradiksi. Masih belum ada konsensus apakah ini adalah solusi untuk masalah tersebut.}}
| style="text-align:center;" |1931, 1936▼
▲! |style="text-align:center;"| width="44%"1940, |Status1963
|-
| style="text-align:center;" | [[Hilbert'sMasalah thirdkedua problemHilbert|3rdMasalah ke-2]]
| Buktikan bahwa [[aksioma]] [[aritmetika]] adalah [[konsistensi|konsisten]].
|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|{{sort|1|}}Resolved. Result: No, proved using [[Dehn invariant]]s.}}
| {{partial|{{sort|2|}}Belum ada konsensus mengenai apakah hasil [[Kurt Gödel|Gödel]] dan [[Gerhard Gentzen|Gentzen]] memberikan solusi untuk masalah yang dinyatakan terpecahkan atau tidak. [[Teorema ketaklengkapan Gödel|teorema ketaklengkapan kedua]] Gödel, yang dibuktikan di tahun 1931, menunjukkan bahwa tiada bukti konsistensinya yang dapat diselesaikan dalam aritmetika itu sendiri. Gentzen membuktikan di tahun 1936 bahwa konsistensi aritmetika yang diikuti dari ''[[Relasi well-founded|well-foundedness]]'' dari [[Bilangan epsilon (matematika)|ordinal &nbsp;''ε''<sub>0</sub>]].}}
| style="text-align:center;" |1900▼
▲| style="text-align:center;"| |1931, 1936
|-
| style="text-align:center;" | [[Hilbert'sMasalah fourthketiga problemHilbert|4thMasalah ke-3]]
| Diberikan sebarang dua [[polihedron]] yang mempunyai volume yang sama. Apakah polihedron pertama yang dipotong menjadi potongan yang berhingga banyaknya akan selalu dapat disatukan kembali agar menghasilkan polihedron kedua?
|Construct all [[Metric space|metrics]] where lines are [[Geodesic|geodesics]].| {{dunno|{{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}}}}▼
| {{yes|{{sort|1|}}Terpecahkan. Jawabannya adalah tidak. Ini terbukti menggunakan [[invarian Dehn]].}}
| style="text-align:center;" |—▼
▲! |style="text-align:center;" width="6%" |Year Solved1900
|-
| style="text-align:center;" | [[Hilbert'sMasalah fifthkeempat problemHilbert|5thMasalah ke-4]]
| Konstruksi semua [[ruang metrik]] dengan garis-garis adalah [[geodesik]].
|Are continuous [[Group (mathematics)|groups]] automatically [[Lie group|differential groups]]?| {{partial|{{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.}}▼
▲|Construct all [[Metric space|metrics]] where lines are [[Geodesic|geodesics]].| {{dunno|{{sort|4|}}TooBelum vaguejelas toapakah beterselesaikan statedatau resolved or nottidak.{{refn|According toMenurut Gray, mosthampir ofsemua themasalah problemstelah have been solvedterpecahkan. 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;" |1953?▼
▲| style="text-align:center;"| |1900—
|-
| style="text-align:center;" | [[Hilbert'sMasalah sixthkelima problemHilbert|6thMasalah ke-5]]
| Are continuous [[group (mathematics)|groups]] automatically [[Lie group|differential groups]]?
|Mathematical treatment of the [[Axiom|axioms]] of [[physics]]
▲|Are continuous [[Group (mathematics)|groups]] automatically [[Lie group|differential groups]]?| {{partial|{{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.}}
(a) axiomatic treatment of probability with limit theorems for foundation of [[statistical physics]]
▲| style="text-align:center;"| |1953?
 
(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"| {{partial|{{sort|2|}}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. 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>}}▼
| style="text-align:center;" |1933–2002?▼
|-
| style="text-align:center;" | [[Hilbert'sMasalah seventhkeenam problemHilbert|7thMasalah ke-6]]
| Mathematical treatment of the [[axiom]]s of [[physics]]<br /><br />(a) axiomatic treatment of probability with limit theorems for foundation of [[statistical physics]]<br /><br />(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"
|Is ''a<sup>b</sup>'' [[Transcendental number|transcendental]], for [[Algebraic number|algebraic]] ''a'' ≠ 0,1 and [[Irrational number|irrational]] algebraic ''b'' ?| {{yes|{{sort|1|}}Resolved. Result: Yes, illustrated by [[Gelfond's theorem]] or the [[Gelfond–Schneider theorem]].}}
▲(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"| {{partial|{{sort|2|}}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. 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>}}
| style="text-align:center;" |1934▼
▲| style="text-align:center;"| |1933–2002?
|-
| style="text-align:center;" | [[Hilbert'sMasalah eighthketujuh problemHilbert|8thMasalah ke-7]]
| Apakah ''a<sup>b</sup>'' [[bilangan transenden|transenden]], untuk [[bilangan aljabar]] ''a'' ≠ 0,1 dan [[bilangan irasional]] aljabar ''b''?
|The [[Riemann hypothesis]]
| {{yes|{{sort|1|}}Terpecahkan. Jawabannya adalah bisa. Ini dapat diilustrasikan dengan [[teorema Gelfond]] atau [[teorema Gelfond–Schneider]].}}
("the real part of any non-[[Trivial (mathematics)|trivial]] [[Root of a function|zero]] of the [[Riemann zeta function]] is {{frac|1|2}}")
▲| style="text-align:center;"| |1934
and other prime number problems, among them [[Goldbach's conjecture]] and the [[twin prime conjecture]]| {{no|{{sort|3|}}Unresolved.}}
| style="text-align:center;" |—▼
|-
| style="text-align:center;" | [[Hilbert'sMasalah ninthkedelapan problemHilbert|9thMasalah ke-8]]
| [[Hipotesis Riemann]] ("bagian real dari sebarang [[Akar fungsi|akar]] non-[[Trivialitas (matematika)|trivial]] dari [[fungsi zeta Riemann]] adalah <math display="inline"> \frac{1}{2} </math>"), dan masalah-masalah lain yang membahas tentang bilangan prima, seperti [[konjektur Goldbach]] dan [[konjektur bilangan prima kembar]]
|Find the most general law of the [[Quadratic reciprocity|reciprocity theorem]] in any [[Algebra|algebraic]] [[number field]].| {{partial|{{sort|2|}}Partially resolved.{{refn|Problem 9 has been solved by [[Emil Artin]] in 1927 for [[Abelian extension]]s of the [[rational numbers]] 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}}}}▼
| {{no|{{sort|3|}}Belum terpecahkan.}}
| style="text-align:center;" |—▼
▲| style="text-align:center;"| |—
|-
| style="text-align:center;" | [[Hilbert'sMasalah tenthkesembilan problemHilbert|10thMasalah ke-9]]
| Find the most general law of the [[Quadratic reciprocity|reciprocity theorem]] in any [[algebra]]ic [[number field]].
| style="text-align:left;" |Find an algorithm to determine whether a given polynomial [[Diophantine equation]] with integer coefficients has an integer solution.| {{yes|{{sort|1|}}Resolved. Result: Impossible; [[Matiyasevich's theorem]] implies that there is no such algorithm.}}
▲|Find the most general law of the [[Quadratic reciprocity|reciprocity theorem]] in any [[Algebra|algebraic]] [[number field]].| {{partial|{{sort|2|}}Partially resolved.{{refn|Problem 9 has been solved by [[Emil Artin]] in 1927 for [[Abelian extension]]s of the [[rational numbers]] 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;" |1970▼
▲| style="text-align:center;"| |—
|-
| style="text-align:center;" | [[Hilbert'sMasalah eleventhkesepuluh problemHilbert|11thMasalah ke-10]]
|style="text-align:left;"| Carilah algoritma untuk menentukan apakah suatu polinomial [[persamaan Diophantine]] yang diberikan dengan koefisien bialngan bulat memiliki penyelesaian berupa bilangan bulat.
|Solving [[Quadratic form|quadratic forms]] with algebraic numerical [[Coefficient|coefficients]].| {{partial|{{sort|2|}}Partially resolved.<ref>{{cite book|first=Michiel|last=Hazewinkel|date= 2009|title= Handbook of Algebra|publisher=Elsevier|page=69|isbn=978-0080932811|volume=6}}</ref>}}▼
| {{yes|{{sort|1|}}Terpecahkan. Jawabannya adalah mustahil. [[Teorema Matiyasevich]] mengimplikasikan tiada algoritma yang menentukan solusi bilangan bulat pada polinomial persamaan Diophantine.}}
| style="text-align:center;" |—▼
▲| style="text-align:center;"| |1970
|-
| style="text-align:center;" | [[Hilbert'sMasalah twelfthkesebelas problemHilbert|12thMasalah ke-11]]
| Solving [[quadratic form]]s with algebraic numerical [[coefficient]]s.
|Extend the [[Kronecker–Weber theorem]] on Abelian extensions of the [[Rational number|rational numbers]] to any base number field.| {{partial|{{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>}}▼
▲|Solving [[Quadratic form|quadratic forms]] with algebraic numerical [[Coefficient|coefficients]].| {{partial|{{sort|2|}}Partially resolved.<ref>{{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;"| |—
|-
| style="text-align:center;" | [[Hilbert'sMasalah thirteenthkeduabelas problemHilbert|13thMasalah ke-12]]
| Extend the [[Kronecker–Weber theorem]] on Abelian extensions of the [[rational number]]s to any base number field.
|Solve [[Septic equation|7th degree equation]] using algebraic (variant: continuous) [[Mathematical function|functions]] of two [[Parameter|parameters]].| {{no|{{sort|3|}}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.{{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>{{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}}</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.▼
▲|Extend the [[Kronecker–Weber theorem]] on Abelian extensions of the [[Rational number|rational numbers]] to any base number field.| {{partial|{{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;"| [[Masalah ketigabelas Hilbert|Masalah ke-13]]
| Solve [[septic equation|7th degree equation]] using algebraic (variant: continuous) [[mathematical function|functions]] of two [[parameter]]s.
▲|Solve [[Septic equation|7th degree equation]] using algebraic (variant: continuous) [[Mathematical function|functions]] of two [[Parameter|parameters]].| {{no|{{sort|3|}}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.{{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>{{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}}</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 "...&nbsp;Existenz von ''algebraischen'' Funktionen&nbsp;...", [existence of ''algebraic'' functions].
As such, the problem is still unresolved.|group=lower-alpha}} }}
| style="text-align:center;"| |—
|-
| style="text-align:center;" | [[Hilbert'sMasalah fourteenthkeempatbelas problemHilbert|14thMasalah ke-14]]
| Is the [[Invariantinvariant theory|ring of invariants]] of an [[algebraic group]] acting on a [[polynomial ring]] always [[Finitely generated algebra|finitely generated]]?
| {{yes|{{sort|1|}}Resolved. Result: No, a counterexample was constructed by [[Masayoshi Nagata]].}}
| style="text-align:center;"| |1959
|-
| style="text-align:center;" | [[Hilbert'sMasalah fifteenthkelimabelas problemHilbert|15thMasalah ke-15]]
| Rigorous foundation of [[Schubert's enumerative calculus]].
| {{partial|{{sort|2|}}Partially resolved.{{Citation needed|date=November 2019}}}}
| style="text-align:center;"| |—
|-
| style="text-align:center;" | [[Hilbert'sMasalah sixteenthkeenambelas problemHilbert|16thMasalah ke-16]]
| Describe relative positions of ovals originating from a [[Realreal numbers|real]] [[algebraic curve]] and as [[Limitlimit cycle|limit cycles]]s of a polynomial [[vector field]] on the plane.
| {{no|{{sort|3|}}Unresolved, even for algebraic curves of degree 8.}}
| style="text-align:center;"| |—
|-
| style="text-align:center;" | [[Hilbert'sMasalah seventeenthketujuhbelas problemHilbert|17thMasalah ke-17]]
| Express a nonnegative [[rational function]] as [[quotient]] of sums of [[Square (algebra)|squares]].
| {{yes|{{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
|-
| style="text-align:center;" | [[Hilbert'sMasalah eighteenthkedelapanbelas problemHilbert|18thMasalah ke-18]]
| (a) Are there only finitely many essentially different space groups in n-dimensional Euclidean space? <br/> <br/> (b) Is there a polyhedron that admits only an [[anisohedral tiling]] in three dimensions?<br /><br />(c) What is the densest [[sphere packing]]?
(c) What is the densest [[sphere packing]]?| {{yes|{{sort|1|}}(a) Resolved. Result Yes (by [[Ludwig Bieberbach]]) <br/> <br/> (b) Resolved. Result: Yes (by [[Karl Reinhardt (mathematician)|Karl Reinhardt]]).<br /><br />(c) 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}}}}▼
(b) Is there a polyhedron that admits only an [[anisohedral tiling]] in three dimensions?
| style="text-align:center;"| |{{sort|1928|(a) 1910 <br/> <br/> (b) 1928<br /><br />(c) 1998}}▼
 
▲(c) What is the densest [[sphere packing]]?| {{yes|{{sort|1|}}(a) Resolved. Result Yes (by [[Ludwig Bieberbach]]) <br/> <br/> (b) Resolved. Result: Yes (by [[Karl Reinhardt (mathematician)|Karl Reinhardt]]).<br /><br />(c) 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;" |{{sort|1928|(a) 1910 <br/> <br/> (b) 1928<br /><br />(c) 1998}}
|-
| style="text-align:center;" | [[Hilbert'sMasalah nineteenthkesembilanbelas problemHilbert|19thMasalah ke-19]]
| Are the solutions of regular problems in the [[calculus of variations]] always necessarily [[Analytic function|analytic]]?
| {{yes|{{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'sMasalah twentiethkeduapuluh problemHilbert|20thMasalah ke-20]]
| Do all [[Calculuscalculus of variations|variational problems]] with certain [[Boundaryboundary condition|boundary conditions]]s have solutions?
| {{yes|{{sort|1|}}Resolved. A significant topic of research throughout the 20th century, culminating in solutions for the non-linear case.}}
| style="text-align:center;" | ?
|-
| style="text-align:center;" | [[Hilbert'sMasalah twenty-firstkeduapuluhsatu problemHilbert|21stMasalah ke-21]]
| Proof of the existence of [[Linearlinear differential equation|linear differential equations]]s having a prescribed [[monodromic group]]
| {{partial|{{sort|2|}} Partially resolved. Result: Yes/No/Open depending on more exact formulations of the problem.}}
| style="text-align:center;" | ?
|-
| style="text-align:center;" | [[Hilbert'sMasalah twenty-secondkeduapuluhdua problemHilbert|22ndMasalah ke-22]]
| Uniformization of analytic relations by means of [[Automorphic function|automorphic functionsfunction]]s
| {{partial|{{sort|2|}}Partially resolved. [[Uniformization theorem]]}}
| style="text-align:center;" | ?
|-
| style="text-align:center;" | [[Hilbert'sMasalah twenty-thirdkeduapuluhtiga problemHilbert|23rdMasalah ke-23]]
| Further development of the [[calculus of variations]]
| {{dunno|{{sort|4|}}Too vague to be stated resolved or not.}}
| style="text-align:center;"| |—
|}
 
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