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{{Short description|Mathematical problem in number theory}}
In [[number theory]], '''Waring's problem''' asks whether each [[natural number]] ''k'' has an associated [[positive integer]] ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by [[Edward Waring]], after whom it is named. Its affirmative answer, known as the '''Hilbert–Waring theorem''', was provided by [[David Hilbert|Hilbert]] in 1909.<ref>{{cite journal| first = David | last = Hilbert | author-link = David Hilbert | title=Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem) |language=de |journal=[[Mathematische Annalen]] |volume=67 | pages=281–300 |year=1909 | issue=3 | doi=10.1007/bf01450405 | mr=1511530| s2cid = 179177986 | url = https://zenodo.org/record/1428266 }}</ref> Waring's problem has its own [[Mathematics Subject Classification]], 11P05, "Waring's problem and variants".

==Relationship with Lagrange's four-square theorem==
Long before Waring posed his problem, [[Diophantus]] had asked whether every positive integer could be represented as the [[Lagrange's four-square theorem|sum of four perfect squares]] greater than or equal to zero. This question later became known as Bachet's conjecture, after the 1621 translation of Diophantus by [[Claude Gaspard Bachet de Méziriac]], and it was solved by [[Joseph-Louis Lagrange]] in his [[Lagrange's four-square theorem|four-square theorem]] in 1770, the same year Waring made his conjecture. Waring sought to generalize this problem by trying to represent all positive integers as the sum of cubes, integers to the [[fourth power]], and so forth, to show that any positive integer may be represented as the sum of other integers raised to a specific exponent, and that there was always a maximum number of integers raised to a certain exponent required to represent all positive integers in this way.

==The number ''g''(''k'')==
For every <math>k</math>, let <math>g(k)</math> denote the minimum number <math>s</math> of <math>k</math>th powers of naturals needed to represent all positive integers. Every positive integer is the sum of one first power, itself, so <math>g(1) = 1</math>. Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes,<ref>Remember we restrict ourselves to ''positive'' natural numbers. With general integers, it is not hard to write 23 as the sum of 4 cubes, e.g. <math>2^3 + 2^3 + 2^3 + (-1)^3</math> or <math>29^3 + 17^3 + 8^3 + (-31)^3</math>.</ref> and 79 requires 19 fourth powers; these examples show that <math>g(2) \ge 4</math>, <math>g(3) \ge 9</math>, and <math>g(4) \ge 19</math>. Waring conjectured that these lower bounds were in fact exact values.

[[Lagrange's four-square theorem]] of 1770 states that every natural number is the sum of at most four squares. Since three squares are not enough, this theorem establishes <math>g(2) = 4</math>. Lagrange's four-square theorem was conjectured in [[Claude Gaspard Bachet de Méziriac|Bachet]]'s 1621 edition of [[Diophantus]]'s ''[[Arithmetica]]''; [[Pierre de Fermat|Fermat]] claimed to have a proof, but did not publish it.<ref>{{cite book | last = Dickson | first = Leonard Eugene | author-link = Leonard Eugene Dickson | title = History of the Theory of Numbers |volume = II: Diophantine Analysis | publisher = [[Carnegie Institution of Washington|Carnegie Institute of Washington]] | year = 1920 | chapter = Chapter VIII}}</ref>

Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example, [[Joseph Liouville|Liouville]] showed that <math>g(4)</math> is at most 53. [[G. H. Hardy|Hardy]] and [[John Edensor Littlewood|Littlewood]] showed that all sufficiently large numbers are the sum of at most 19 fourth powers.

That <math>g(3) = 9</math> was established from 1909 to 1912 by [[Arthur Wieferich|Wieferich]]<ref>{{cite journal | last = Wieferich | first = Arthur | author-link = Arthur Wieferich | title = Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt |language = de | journal = Mathematische Annalen | volume = 66 | issue = 1 | pages = 95–101 | year = 1909 | url = http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D38240 | doi = 10.1007/BF01450913| s2cid = 121386035 }}</ref> and [[Aubrey J. Kempner|A. J. Kempner]],<ref>{{cite journal | last = Kempner | first = Aubrey | title = Bemerkungen zum Waringschen Problem | language = de | journal = Mathematische Annalen | volume = 72 | issue = 3 | pages = 387–399 | year=1912 | url = http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?did=D28751 | doi = 10.1007/BF01456723| s2cid = 120101223 }}</ref> <math>g(4) = 19</math> in 1986 by [[Ramachandran Balasubramanian|R. Balasubramanian]], F. Dress, and [[Jean-Marc Deshouillers|J.-M. Deshouillers]],<ref>{{cite journal | last1=Balasubramanian | first1=Ramachandran | last2=Deshouillers | first2=Jean-Marc | last3=Dress | first3=François | title=Problème de Waring pour les bicarrés. I. Schéma de la solution | language=fr | trans-title=Waring's problem for biquadrates. I. Sketch of the solution | journal=Comptes Rendus de l'Académie des Sciences, Série I | volume=303 | year=1986 | issue=4 | pages=85–88 | mr=0853592}}</ref><ref>{{cite journal | last1=Balasubramanian | first1=Ramachandran | last2=Deshouillers | first2=Jean-Marc | last3=Dress | first3=François | title=Problème de Waring pour les bicarrés. II. Résultats auxiliaires pour le théorème asymptotique | language=fr | trans-title=Waring's problem for biquadrates. II. Auxiliary results for the asymptotic theorem | journal=Comptes Rendus de l'Académie des Sciences, Série I | volume=303 |year=1986 | issue= 5 | pages=161–163 | mr=0854724}}</ref> <math>g(5) = 37</math> in 1964 by [[Chen Jingrun]],<ref name="ChenWaring5">{{cite journal |last=Chen |first= Jing-run|date=1964 |title=Waring's problem for g(5)=37 |journal=Scientia Sinica |volume=13 |pages=1547–1568 |language=Chinese}}</ref> and <math>g(6) = 73</math> in 1940 by [[Subbayya Sivasankaranarayana Pillai|Pillai]].<ref>{{cite journal | last1 = Pillai | first1 = S. S. | title = On Waring's problem ''g''(6) = 73 | journal = Proc. Indian Acad. Sci. | volume = 12 | year=1940 | pages = 30–40 | mr=0002993| doi = 10.1007/BF03170721 | s2cid = 185097940 }}</ref>

Let <math>\lfloor x\rfloor</math> and <math>\{x\}</math> respectively denote the [[integral part|integral]] and [[fractional part]] of a positive real number <math>x</math>. Given the number <math>c = 2^k \lfloor(3/2)^k\rfloor - 1 < 3^k</math>, only <math>2^k</math> and <math>1^k</math> can be used to represent <math>c</math>; the most economical representation requires
<math>\lfloor(3/2)^k\rfloor - 1</math> terms of <math>2^k</math> and <math>2^k - 1</math> terms of <math>1^k</math>. It follows that <math>g(k)</math> is at least as large as <math>2^k + \lfloor(3/2)^k\rfloor - 2</math>. This was noted by [[Johann Euler|J. A. Euler]], the son of [[Leonhard Euler]], in about 1772.<ref>[[Euler|L. Euler]], [https://archive.org/stream/leonhardieuleri00petrgoog#page/n219/mode/2up"Opera posthuma"] (1), 203–204 (1862).</ref>

The Ideal Waring Theorem would be an unconditional strengthening of Euler's observation:

: Define: g*(k) <math> = 2^k + \lfloor(3/2)^k\rfloor - 2</math>. Then g(k) = g*(k).

Work by [[Leonard Eugene Dickson|Dickson]] and [[Subbayya Sivasankaranarayana Pillai|Pillai]] in 1936, [[R. K. Rubugunday|Rubugunday]]<ref name="Rubugunday">{{cite journal | last=Rubugunday | first=R.K. | title=On g(k) in Waring's Problem | journal= Journal of the Indian Mathematical Society | volume=6 | date=1942 | pages=192–198}}</ref> in 1942, [[Ivan M. Niven|Niven]] in 1944<ref>{{cite journal | last = Niven | first = Ivan M. |author-link = Ivan M. Niven |title = An unsolved case of the Waring problem |journal = [[American Journal of Mathematics]] |volume = 66 |pages = 137–143 |year = 1944 |issue = 1 |doi = 10.2307/2371901 |publisher = The Johns Hopkins University Press |jstor = 2371901 | mr=0009386}}</ref> and many others has proved that

: <math>
g(k) = \begin{cases}
2^k + \lfloor(3/2)^k\rfloor - 2
&\text{if}\quad
2^k \{(3/2)^k\} + \lfloor(3/2)^k\rfloor \le 2^k, \\
2^k + \lfloor(3/2)^k\rfloor + \lfloor(4/3)^k\rfloor - 2
&\text{if}\quad
2^k \{(3/2)^k\} + \lfloor(3/2)^k\rfloor > 2^k
\text{ and }
\lfloor(4/3)^k\rfloor \lfloor(3/2)^k\rfloor + \lfloor(4/3)^k\rfloor + \lfloor(3/2)^k\rfloor = 2^k, \\
2^k + \lfloor(3/2)^k\rfloor + \lfloor(4/3)^k\rfloor - 3
&\text{if}\quad
2^k \{(3/2)^k\} + \lfloor(3/2)^k\rfloor > 2^k
\text{ and }
\lfloor(4/3)^k\rfloor \lfloor(3/2)^k\rfloor + \lfloor(4/3)^k\rfloor + \lfloor(3/2)^k\rfloor > 2^k.
\end{cases}
</math>

Dickson's 1936 proof<ref name="Dickson1936">{{cite journal | last=Dickson | first=L. E. | title=Solution of Waring's Problem | journal=American Journal of Mathematics | volume=58 | issue=3 | date=1936 | doi=10.2307/2370970 | pages=530–535| jstor=2370970 }}</ref> applies when k > 6, and Pillai's<ref name="Pillai1936">{{cite journal | last=Pillai | first=S.S. | title=On Waring's Problem | journal= Journal of the Indian Mathematical Society | volume=2 | date=1936 | pages=16–44}}</ref> when k > 7, leaving g(4), g(5), and g(6) to be resolved as documented above.

No value of <math>k</math> is known for which <math>2^k\{(3/2)^k\} + \lfloor(3/2)^k\rfloor > 2^k</math>. [[Kurt Mahler|Mahler]]<ref>{{cite journal | last1 = Mahler | first1 = Kurt | year =1957 | title = On the fractional parts of the powers of a rational number II | journal = [[Mathematika]] | volume = 4 | issue = 2 | pages = 122–124 | doi=10.1112/s0025579300001170 |mr=0093509}}</ref> proved that there can only be a finite number of such <math>k</math>, and Kubina and Wunderlich<ref>{{cite journal | last1=Kubina | first1=Jeffrey M. | last2=Wunderlich | first2=Marvin C. | title=Extending Waring's conjecture to 471,600,000 | journal=[[Math. Comp.]] | volume=55 | pages=815–820 | year=1990 | issue=192 | mr=1035936 | doi=10.2307/2008448 | jstor=2008448 | bibcode=1990MaCom..55..815K }}</ref> have shown that any such <math>k</math> must satisfy <math>k > 471\,600\,000</math>, extending work of Stemmler.<ref name="Stemmler 1964">{{cite journal | last=Stemmler | first=Rosemarie M. | title=The ideal Waring theorem for exponents 401-200,000 | journal=Mathematics of Computation | volume=18 | issue=85 | date=1964 | issn=0025-5718 | doi=10.1090/S0025-5718-1964-0159803-X | doi-access=free | pages=144–146 | url=https://www.ams.org/mcom/1964-18-085/S0025-5718-1964-0159803-X/S0025-5718-1964-0159803-X.pdf | access-date=4 February 2025}}</ref> Thus it is conjectured that this never happens, that is, <math>g(k) = 2^k + \lfloor(3/2)^k\rfloor - 2</math> for every positive integer <math>k</math>.

The first few values of <math>g(k)</math> are:
: 1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, 263619, 526502, 1051899, ... {{OEIS|A002804}}.

==The number ''G''(''k'')==
From the work of [[G. H. Hardy|Hardy]] and [[John Edensor Littlewood|Littlewood]],<ref name="Hardy Littlewood 1922 pp. 161–188">{{cite journal | last1=Hardy | first1=G. H. | last2=Littlewood | first2=J. E. | title=Some problems of ''Partitio Numerorum'': IV. The singular series in Waring's Problem and the value of the number G(k) | journal=Mathematische Zeitschrift | volume=12 | issue=1 | date=1922 | issn=0025-5874 | doi=10.1007/BF01482074 | pages=161–188}}</ref> the related quantity ''G''(''k'') was studied with ''g''(''k''). ''G''(''k'') is defined to be the least positive integer ''s'' such that every [[sufficiently large]] integer (i.e. every integer greater than some constant) can be represented as a sum of at most ''s'' positive integers to the power of ''k''. Clearly, ''G''(1) = 1. Since squares are congruent to 0, 1, or 4 (mod 8), no integer congruent to 7 (mod 8) can be represented as a sum of three squares, implying that {{nowrap|''G''(2) ≥ 4}}. Since {{nowrap|''G''(''k'') ≤ ''g''(''k'')}} for all ''k'', this shows that {{nowrap|1=''G''(2) = 4}}. [[Harold Davenport|Davenport]] showed<ref>{{cite journal| first = H. | last = Davenport | author-link = Harold Davenport | title=On Waring's Problem for Fourth Powers |language=en |journal=[[Annals of Mathematics]] |volume=40 | pages=731–747 |year=1939 | issue=4 | doi = 10.2307/1968889 | jstor = 1968889 | bibcode = 1939AnMat..40..731D }}</ref> that {{nowrap|1=''G''(4) = 16}} in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers (Vaughan in 1986<ref name="Vaughan 1986 pp. 445–463">{{cite journal | last=Vaughan | first=R. C. | title=On Waring's Problem for Smaller Exponents | journal=Proceedings of the London Mathematical Society | volume=s3-52 | issue=3 | date=1986 | doi=10.1112/plms/s3-52.3.445 | pages=445–463}}</ref> and 1989<ref name="Vaughan 1989 pp. 1–71">{{cite journal | last=Vaughan | first=R. C. | title=A new iterative method in Waring's problem | journal=Acta Mathematica | volume=162 | date=1989 | issn=0001-5962 | doi=10.1007/BF02392834 | pages=1–71}}</ref> reduced the 14 biquadrates successively to 13 and 12). The exact value of ''G''(''k'') is unknown for any other ''k'', but there exist bounds.

===Lower bounds for ''G''(''k'')===

{| class="wikitable" style="float:right; text-align: center"
! Bounds
|-
| 1 = ''G''(1) = 1
|-
| 4 = ''G''(2) = 4
|-
| 4 ≤ ''G''(3) ≤ 7
|-
| 16 = ''G''(4) = 16
|-
| 6 ≤ ''G''(5) ≤ 17
|-
| 9 ≤ ''G''(6) ≤ 24
|-
| 8 ≤ ''G''(7) ≤ 33
|-
| 32 ≤ ''G''(8) ≤ 42
|-
| 13 ≤ ''G''(9) ≤ 50
|-
| 12 ≤ ''G''(10) ≤ 59
|-
| 12 ≤ ''G''(11) ≤ 67
|-
| 16 ≤ ''G''(12) ≤ 76
|-
| 14 ≤ ''G''(13) ≤ 84
|-
| 15 ≤ ''G''(14) ≤ 92
|-
| 16 ≤ ''G''(15) ≤ 100
|-
| 64 ≤ ''G''(16) ≤ 109
|-
| 18 ≤ ''G''(17) ≤ 117
|-
| 27 ≤ ''G''(18) ≤ 125
|-
| 20 ≤ ''G''(19) ≤ 134
|-
| 25 ≤ ''G''(20) ≤ 142
|}
The number ''G''(''k'') is greater than or equal to
: {|
| 2''r''+2 || if ''k'' = 2''r'' with ''r'' ≥ 2, or ''k'' = 3 × 2''r'';
|-
| ''p''''r''+1 || if ''p'' is a prime greater than 2 and ''k'' = ''p''''r''(''p'' − 1);
|-
| (''p''''r''+1 − 1)/2   || if ''p'' is a prime greater than 2 and ''k'' = p''r''(p − 1)/2;
|-
| ''k'' + 1 || for all integers ''k'' greater than 1.
|}

In the absence of congruence restrictions, a density argument suggests that ''G''(''k'') should equal {{nowrap|''k'' + 1}}.

===Upper bounds for ''G''(''k'')===

''G''(3) is at least 4 (since cubes are congruent to 0, 1 or −1 mod 9); for numbers less than 1.3{{e|9}}, {{val|1290740}} is the last to require 6 cubes, and the number of numbers between ''N'' and 2''N'' requiring 5 cubes drops off with increasing ''N'' at sufficient speed to have people believe that {{nowrap|1=''G''(3) = 4}};<ref>{{harvtxt|Nathanson|1996|p=71}}.</ref> the largest number now known not to be a sum of 4 cubes is {{val|7373170279850}},<ref name="x7373170279850">{{cite journal |last1=Deshouillers |first1=Jean-Marc |last2=Hennecart |first2= François |last3=Landreau |first3=Bernard |last4=I. Gusti Putu Purnaba |first4=Appendix by |title=7373170279850 |journal=Mathematics of Computation |volume=69 |issue=229 |year=2000 |pages=421–439 |doi=10.1090/S0025-5718-99-01116-3 |doi-access=free}}</ref> and the authors give reasonable arguments there that this may be the largest possible. The upper bound {{nowrap|''G''(3) ≤ 7}} is due to Linnik in 1943.<ref>U. V. Linnik. "On the representation of large numbers as sums of seven cubes". Mat. Sb. N.S. 12(54), 218–224 (1943).</ref> (All nonnegative integers require at most 9 cubes, and the largest integers requiring 9, 8, 7, 6 and 5 cubes are conjectured to be 239, 454, 8042, {{val|1290740}} and {{val|7373170279850}}, respectively.)

{{val|13792}} is the largest number to require 17 fourth powers (Deshouillers, Hennecart and Landreau showed in 2000<ref name="sixteen-biquadrates">{{cite journal |last1=Deshouillers |first1=Jean-Marc |last2=Hennecart |first2=François |last3=Landreau |first3=Bernard |title=Waring's Problem for sixteen biquadrates – numerical results |journal=[[Journal de théorie des nombres de Bordeaux]] |volume=12 |issue=2 |year=2000 |pages=411–422 |url= http://www.math.ethz.ch/EMIS/journals/JTNB/2000-2/Dhl.ps |doi=10.5802/jtnb.287 |doi-access=free|url-access=subscription }}</ref> that every number between {{val|13793}} and 10245 required at most 16, and Kawada, Wooley and Deshouillers extended<ref name="Deshouillers Kawada Wooley 2005 pp. 1–120">{{cite journal | last1=Deshouillers | first1=Jean-Marc | last2=Kawada | first2=Koichi | last3=Wooley | first3=Trevor D. | title=On Sums of Sixteen Biquadrates | journal=Mémoires de la Société Mathématique de France | volume=1 | date=2005 | issn=0249-633X | doi=10.24033/msmf.413 | pages=1–120}}</ref> Davenport's 1939 result to show that every number above 10220 required at most 16). Numbers of the form 31·16''n'' always require 16 fourth powers.

{{val|68578904422}} is the last known number that requires 9 fifth powers ([[Integer sequence]] S001057, Tony D. Noe, Jul 04 2017), {{val|617597724}} is the last number less than 1.3{{e|9}} that requires 10 fifth powers, and {{val|51033617}} is the last number less than 1.3{{e|9}} that requires 11.

The upper bounds on the right with {{nowrap|1=''k'' = 5, 6, ..., 20}} are due to [[R. C. Vaughan|Vaughan]] and [[Trevor Wooley|Wooley]].<ref name=Vaughan-Wooley>{{cite book | first1 = R. C. | last1 = Vaughan | author-link2 = Trevor Wooley | first2 = Trevor | last2 = Wooley | chapter = Waring's Problem: A Survey |title=Number Theory for the Millennium |volume=III |publisher=A. K. Peters |pages=301–340 |year=2002 |isbn=978-1-56881-152-9 | mr=1956283 | location=Natick, MA | editor1-last=Bennet | editor1-first=Michael A. | editor2-last=Berndt | editor2-first=Bruce C. | editor3-last=Boston | editor3-first=Nigel | editor4-last=Diamond | editor4-first=Harold G. | editor5-last=Hildebrand | editor5-first=Adolf J. | editor6-first=Walter | editor6-last=Philipp}}</ref>

Using his improved [[Hardy–Ramanujan–Littlewood circle method|Hardy–Ramanujan–Littlewood method]], [[Ivan Matveyevich Vinogradov|I. M. Vinogradov]] published numerous refinements leading to
: <math>G(k) \le k(3\log k + 11)</math>
in 1947<ref name="Vinogradov 1947 ">{{cite book | last=Vinogradov | first=Ivan Matveevich |translator-last1=Roth |translator-first1=K.F. |translator-last2=Davenport |translator-first2=Anne | title=The Method of Trigonometrical Sums in the Theory of Numbers | publisher=Dover Publications | publication-place=Mineola, NY | date=1 Sep 2004 |orig-date=1947 | isbn=978-0-486-43878-8}}</ref> and, ultimately,
: <math>G(k) \le k(2\log k + 2\log\log k + C\log\log\log k)</math>
for an unspecified constant ''C'' and sufficiently large ''k'' in 1959.<ref name="Math-Net.Ru z658">{{cite journal |last1=Vinogradov |first1=I. M. |title=On an upper bound for $G(n)$ |journal=Izv. Akad. Nauk SSSR Ser. Mat. |date=1959 |volume=23 |issue=5 |pages=637–642 |url=https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3799&option_lang=eng | language=Russian}}</ref>

Applying his [[p-adic|''p''-adic]] form of the Hardy–Ramanujan–Littlewood–Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors, [[Anatolii Alexeevitch Karatsuba]] obtained<ref>{{cite journal |first=A. A. |last=Karatsuba |title=On the function ''G''(''n'') in Waring's problem | journal=Izv. Akad. Nauk SSSR Ser. Mat. |volume=27 |issue=4 |pages=935–947 |year=1985 |bibcode=1986IzMat..27..239K |doi=10.1070/IM1986v027n02ABEH001176}}</ref> in 1985 a new estimate, for <math>k \ge 400</math>:

: <math>G(k) \le k(2\log k + 2\log\log k + 12).</math>

Further refinements were obtained by Vaughan in 1989.<ref name="Vaughan 1989 pp. 1–71">{{cite journal | last=Vaughan | first=R. C. | title=A new iterative method in Waring's problem | journal=Acta Mathematica | volume=162 | date=1989 | issn=0001-5962 | doi=10.1007/BF02392834 | pages=1–71}}</ref>

Wooley then established that for some constant ''C'',<ref name=Vaughan>{{cite book | zbl=0868.11046 | last=Vaughan | first=R. C. | title=The Hardy–Littlewood method | edition=2nd | series=Cambridge Tracts in Mathematics | volume=125 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1997 | isbn=0-521-57347-5 }}</ref>
: <math>G(k) \le k(\log k + \log\log k + C).</math>

Vaughan and Wooley's survey article from 2002 was comprehensive at the time.<ref name=Vaughan-Wooley/>

==See also==
* [[Centered polygonal number theorem]]
* [[Fermat polygonal number theorem]], that every positive integer is a sum of at most ''n'' of the ''n''-gonal numbers
* [[Waring–Goldbach problem]], the problem of representing numbers as sums of powers of primes
* [[Subset sum problem]], an algorithmic problem that can be used to find the shortest representation of a given number as a sum of powers
* [[Pollock's conjectures]]
* [[Sums of three cubes]], discusses what numbers are the sum of three ''not necessarily positive'' cubes
* [[Sums of four cubes problem]], discusses whether every integer is the sum of four cubes of integers
* [[Jacobi's four-square theorem]], provides the number of ways a positive integer can be represented as the sum of 4 squares

==Notes==
{{clear}}
{{Reflist|30em}}

==References==
* G. I. Arkhipov, V. N. Chubarikov, [[Anatolii Alexeevitch Karatsuba|A. A. Karatsuba]], "Trigonometric sums in number theory and analysis". Berlin–New-York: Walter de Gruyter, (2004).
* G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, "Theory of multiple trigonometric sums". Moscow: Nauka, (1987).
* [[Yu. V. Linnik]], "An elementary solution of the problem of Waring by Schnirelman's method". ''Mat. Sb., N. Ser.'' '''12''' (54), 225–230 (1943).
* [[R. C. Vaughan]], "A new iterative method in Waring's problem". ''Acta Mathematica'' (162), 1–71 (1989).
* [[Ivan Matveyevich Vinogradov|I. M. Vinogradov]], "The method of trigonometrical sums in the theory of numbers". ''Trav. Inst. Math. Stekloff'' (23), 109 pp. (1947).
* I. M. Vinogradov, "On an upper bound for ''G''(''n'')". ''Izv. Akad. Nauk SSSR Ser. Mat.'' (23), 637–642 (1959).
* I. M. Vinogradov, A. A. Karatsuba, "The method of trigonometric sums in number theory", ''Proc. Steklov Inst. Math.'', 168, 3–30 (1986); translation from Trudy Mat. Inst. Steklova, 168, 4–30 (1984).
* {{cite journal | last1 = Ellison | first1 = W. J. | year = 1971 | title = Waring's problem | url = http://www.maa.org/programs/maa-awards/writing-awards/warings-problem| journal = American Mathematical Monthly | volume = 78 | issue = 1| pages = 10–36 | doi=10.2307/2317482| jstor = 2317482 }} Survey, contains the precise formula for ''G''(''k''), a simplified version of Hilbert's proof and a wealth of references.
* {{Cite book | author-link = Aleksandr Khinchin | last = Khinchin | first = A. Ya. | title = Three Pearls of Number Theory | publisher = Dover | location = Mineola, NY | year = 1998 | isbn = 978-0-486-40026-6 }} Has an elementary proof of the existence of ''G''(''k'') using [[Schnirelmann density]].
* {{cite book | first=Melvyn B. | last=Nathanson | title=Additive Number Theory: The Classical Bases | volume=164 | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | year=1996 | isbn=0-387-94656-X | zbl=0859.11002 }} Has proofs of Lagrange's theorem, the [[polygonal number theorem]], Hilbert's proof of Waring's conjecture and the Hardy–Littlewood proof of the asymptotic formula for the number of ways to represent ''N'' as the sum of ''s'' ''k''th powers.
* [[Hans Rademacher]] and [[Otto Toeplitz]], ''The Enjoyment of Mathematics'' (1933) ({{isbn|0-691-02351-4}}). Has a proof of the Lagrange theorem, accessible to high-school students.

==External links==
{{wikisource|de:David Hilbert Gesammelte Abhandlungen Erster Band – Zahlentheorie/Kapitel 11|Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)}}
* {{springer|title=Waring problem|id=p/w097100}}

{{DEFAULTSORT:Waring's Problem}}
[[Category:Additive number theory]]
[[Category:Mathematical problems]]
[[Category:Unsolved problems in number theory]]
[[Category:Squares in number theory]]

Waring's problem - Revision history

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