Editing Waring's problem (section)

===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}}</ref> that every number between {{val|13793}} and 10<sup>245</sup> 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 10<sup>220</sup> required at most 16). Numbers of the form 31·16<sup>''n''</sup> 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.&nbsp;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/>