Waring\'s problem - Citizendia

In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s k^th powers of natural numbers (for example, every number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Year 1770 ( MDCCLXX) was a Common year starting on Monday (see link for calendar of the Gregorian calendar (or a Common year starting on Friday Edward Waring (1736 – August 15, 1798) was an English Mathematician who was born in Old Heath (near Shrewsbury) In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French ). The affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Year 1909 ( MCMIX) was a Common year starting on Friday (link will display full calendar of the Gregorian calendar (or a Common year starting ^[1] Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants. The Mathematics Subject Classification (MSC is an alphanumerical Classification scheme formulated by the American Mathematical Society based on the coverage of two "

1 The number g(k)
2 The number G(k)
- 2.1 Lower bounds for G(k)
- 2.2 Upper bounds for G(k)
3 Further reading
4 Notes

The number g(k)

For every k, we denote by g(k) the minimum number s of kth powers needed to represent all integers. Note we have g(1) = 1. Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 fourth-powers; these examples show that g(2) ≥ 4, g(3) ≥ 9, and g(4) ≥ 19. Waring conjectured that these values were in fact the best possible.

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 g(2) = 4. Lagrange's four-square theorem, also known as Bachet's conjecture, was proven in 1770 by Joseph Louis Lagrange. Lagrange's four-square theorem was conjectured in Bachet's 1621 edition of Diophantus; Fermat claimed to have a proof, but did not publish it. Claude Gaspard Bachet de Méziriac ( October 9, 1581 - February 26, 1638) was a French Mathematician born in Bourg-en-Bresse Arithmetica is an ancient Greek text on Mathematics written by the Mathematician Diophantus in the 3rd century CE. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the ^[2]

Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example, Liouville showed that g(4) is at most 53. Joseph Liouville ( March 24 1809 &ndash September 8 1882) was a French Mathematician. Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers. Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 John Edensor Littlewood ( 9 June 1885 &ndash 6 September 1977) was a British Mathematician, best known for his long collaboration

That g(3) = 9 was established from 1909 to 1912 by Wieferich^[3] and A. Arthur Josef Alwin Wieferich ( April 27, 1884 – September 15, 1954) was a German Mathematician and teacher remembered J. Kempner^[4], g(4) = 19 in 1986 by R. Balasubramanian, F. Dress, and J. -M. Deshouillers^[5]^[6], g(5) = 37 in 1964 by Chen Jingrun, and g(6) = 73 in 1940 by Pillai^[7]. Chen Jingrun ( May 22 1933 – March 19 1996) was a Chinese Mathematician who made significant contributions to Number Subbayya Sivasankaranarayana Pillai (1901-1950 was an Indian Mathematician, well known for his work in Number theory.

Euler conjectured that, with [x] and {x} denoting the integral and fractional part of x respectively, g(k)=2^k+[(3/2)^k]-2. In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The ^[8] Later work by Dickson, Pillai, Rubugunday and Niven^[9] expanded on this idea, and now, apart from a certain ambiguity, all the other values of g are also known:

g(k)=2^k+[(3/2)^k]-2 if 2^k{(3/2)^k}+[(3/2)^k]≤ 2^k

g(k)=2^k+[(3/2)^k]+[(4/3)^k]-2 if 2^k{(3/2)^k}+[(3/2)^k]>2^k and [(4/3)^k][(3/2)^k]+[(4/3)^k]+[(3/2)^k]=2^k

g(k)=2^k+[(3/2)^k]+[(4/3)^k]-3 if 2^k{(3/2)^k}+[(3/2)^k]>2^k and [(4/3)^k][(3/2)^k]+[(4/3)^k]+[(3/2)^k]>2^k. Leonard Eugene Dickson ( 22 January[[ 874]] Independence Iowa – 17 January[[ 954]] Harlingen Texas) (often called L Ivan Morton Niven ( October 25 1915 &ndash May 9 1999) was a Canadian - American Mathematician, specializing in

(Here [(3/2)^k] is the usual shorthand for "the integer part of (3/2)^k", and {(3/2)^k} = (3/2)^k - [(3/2)^k]. )

It is conjectured that 2^k{(3/2)^k}+[(3/2)^k]>2^k, which has been shown to happen for at most finitely many k by Mahler^[10], in fact never occurs. Kurt Mahler ( 26 July 1903, Krefeld, Germany – 25 February 1988, Canberra, Australia) was a mathematician If the conjecture holds, then indeed g(k)=2^k+[(3/2)^k]-2 for each positive integer k. The conjecture indeed is verified for all reasonably small k'. The first proven or conjectured values 1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055 . Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In mathematics Four is the smallest Composite number, its proper Divisors being and. In mathematics Nine is a Composite number, its proper Divisors being 1 and 3. 19 ( nineteen) is the Natural number following 18 and preceding 20. 37 ( thirty-seven) is the Natural number following 36 and preceding 38. 73 ( seventy-three) is the Natural number following 72 and preceding 74. . . are listed in Sloane's A002804. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

The number G(k)

From the work of Hardy and Littlewood, more fundamental than g(k) turned out to be G(k), which is defined to be the least positive integer s such that every sufficiently large integer (i. Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 John Edensor Littlewood ( 9 June 1885 &ndash 6 September 1977) was a British Mathematician, best known for his long collaboration In Mathematics, the phrase sufficiently large is used in contexts such as P is true for sufficiently large x which is actually shorthand e. every integer greater than some constant) can be represented as a sum of at most s k^th powers of positive integers. It is easy to see that G(2)≥ 4 since every integer congruent to 7 modulo 8 cannot be represented as a sum of three squares. Since G(k) ≤ g(k) for all k, this shows that G(2) = 4. Davenport showed that 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 1985 reduced this from 14 to 13). Harold Davenport ( 30 October 1907 – 9 June 1969) was an English mathematician known for his extensive work in Number theory The exact value of G(k) is unknown for any other k, but there exist bounds.

Lower bounds for G(k)

The number G(k) is greater than or equal to

2^r+2 if k=2^r with r ≥ 2, or k=2^r3;

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 k+1.

Upper bounds for G(k)

The following upper bounds for G(k) are known:

k          3   5   6   7   8   9  10  11  12  13  14   15   16   17   18   19   20
G(k) =<    7  17  21  33  42  50  59  67  76  84  92  100  109  117  125  134  142

G(3) is at least four (since cubes are congruent to 0, 1 or -1 mod 9); 1290740 is the last number less than 1. 3e9 to require six cubes, and the number of numbers between N and 2N requiring five cubes drops off with increasing N at sufficient speed to have people believe G(3)=4; the largest number now known not to be a sum of four cubes is 7373170279850 ^[11], and the authors give reasonable arguments there that this may be the largest possible.

13792 is the largest number to require seventeen fourth powers (Deshouillers, Hennecart and Landreau showed in 2000 ^[12] that every number between 13793 and 10²⁴⁵ required at most sixteen, and Kawada, Wooley and Deshouillers extended Davenport's 1939 result to show that every number above 10²²⁰ required no more than sixteen). Sixteen fourth powers are always needed to write a number of the form 16^n*31.

617597724 is the last number less than 1. 3e9 which requires ten fifth powers, and 51033617 the last number less than 1. 3e9 which requires eleven.

Using his improved Hardy-Littlewood method, I. M. Vinogradov has shown that

$G(k)\le k(3\log k +11).$

T. D. Wooley has established the bound, in big O notation,

$G(k)\le k\log k+k\log\log k+O(k).$ (See ^[13] for a proof. In Mathematics, the Hardy–Littlewood circle method is one of the most frequently used techniques of Analytic number theory. Ivan Matveevich Vinogradov (Иван Матвеевич Виноградов September 14, 1891 &ndash March 20, 1983) was a Russian Trevor D Wooley FRS is a British Mathematician and currently Professor of Mathematics at the University of Bristol. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments )

Notes

^ D. Hilbert, Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem), Mathematische Annalen, 67, pages 281-300 (1909)
^ Dickson, Leonard Eugene (1920). Leonard Eugene Dickson ( 22 January[[ 874]] Independence Iowa – 17 January[[ 954]] Harlingen Texas) (often called L "Chapter VIII", History of the Theory of Numbers, Volume II: Diophantine Analysis. Carnegie Institute of Washington. The Carnegie Institution for Science (also called the Carnegie Institution of Washington (CIW) is a organization in the United States established to support Scientific
^ Wieferich, Arthur (1909). Arthur Josef Alwin Wieferich ( April 27, 1884 – September 15, 1954) was a German Mathematician and teacher remembered "Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt". Mathematische Annalen 66: 95-101.
^ Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen 72: 387-399.
^ Balasubramanian, Ramachandran; Deshouillers, Jean-Marc; Dress, François, Problème de Waring pour les bicarrés. I. Schéma de la solution. (French. English summary) [Waring's problem for biquadrates. I. Sketch of the solution] C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 4, pp. 85-88
^ Balasubramanian, Ramachandran; Deshouillers, Jean-Marc; Dress, François, Problème de Waring pour les bicarrés. II. Résultats auxiliaires pour le théorème asymptotique. (French. English summary) [Waring's problem for biquadrates. II. Auxiliary results for the asymptotic theorem] C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 5, pp. 161-163
^ Pillai, S. S. On Waring's problem g(6)=73, Proc. Indian Acad. Sci. 12A, pp. 30-40
^ Euler's Conjecture - from Wolfram MathWorld
^ Niven, Ivan M. (1944). Ivan Morton Niven ( October 25 1915 &ndash May 9 1999) was a Canadian - American Mathematician, specializing in "An unsolved case of the Waring problem". American Journal of Mathematics 66: 137–143. American Journal of Mathematics is a Bimonthly mathematics journal published by the Johns Hopkins University Press, founded in 1878 by James Joseph Sylvester
^ Mahler, K. On the fractional parts of the powers of a rational number II, 1957, Mathematika, 4, pages 122-124
^ Jean-Marc Deshouillers, François Hennecart, Bernard Landreau, 7373170279850, Mathematics of Computation 69 (2000) 421--439, available at http://www.ams.org/mcom/2000-69-229/S0025-5718-99-01116-3/S0025-5718-99-01116-3.pdf
^ Deshouillers, Hennecart, Landreau, Waring's Problem for sixteen biquadrates - numerical results, Journal de Théorie des Nombers de Bordeaux 12 (2000), 411-422; http://www.math.ethz.ch/EMIS/journals/JTNB/2000-2/Dhl.ps
^ The Hardy-Littlewood method, R. C. Vaughan, 2nd ed. , Cambridge Tracts in Mathematics, CUP, 1997

Yu. V. Linnik, "An elementary solution of the problem of Waring by Schnirelman's method". Mat. Sb. , N. Ser. 12 (54), 225–230 (1943)

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Contents

The number g(k)

The number G(k)

Lower bounds for G(k)

Upper bounds for G(k)

Further reading

Notes