Editing Waring's problem (section)

==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)&nbsp;=&nbsp;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.&nbsp;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}}.