Waring's problem

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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 kth powers of natural numbers (for example, every number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc.). The affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909.[1] Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants."

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[edit] 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 was conjectured in Bachet's 1621 edition of Diophantus; Fermat claimed to have a proof, but did not publish it.[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. Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers.

That g(3) = 9 was established from 1909 to 1912 by Wieferich[3] and A. 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].

Euler conjectured that, with [x] and {x} denoting the integral and fractional part of x respectively, g(k)=2k+[(3/2)k]-2.[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)=2k+[(3/2)k]-2   if   2k{(3/2)k}+[(3/2)k]≤ 2k
g(k)=2k+[(3/2)k]+[(4/3)k]-2   if   2k{(3/2)k}+[(3/2)k]>2k   and   [(4/3)k][(3/2)k]+[(4/3)k]+[(3/2)k]=2k
g(k)=2k+[(3/2)k]+[(4/3)k]-3   if   2k{(3/2)k}+[(3/2)k]>2k   and   [(4/3)k][(3/2)k]+[(4/3)k]+[(3/2)k]>2k.

(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 2k{(3/2)k}+[(3/2)k]>2k, which has been shown to happen for at most finitely many k by Mahler[10], in fact never occurs. If the conjecture holds, then indeed   g(k)=2k+[(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 ... are listed in Sloane's A002804.

[edit] 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.e. every integer greater than some constant) can be represented as a sum of at most s kth 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). The exact value of G(k) is unknown for any other k, but there exist bounds.

[edit] Lower bounds for G(k)

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

2r+2 if k=2r with r ≥ 2, or k=2r3;
pr+1 if p is a prime greater than 2 and k=pr(p-1);
(pr+1-1)/2 if p is a prime greater than 2 and k=pr(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.

[edit] 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.)

[edit] Further reading

  • W. J. Ellison: Waring's problem. American Mathematical Monthly, volume 78 (1971), pp. 10-76. Survey, contains the precise formula for g(k), a simplified version of Hilbert's proof and a wealth of references.
  • 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.

[edit] Notes

  1. ^ 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)
  2. ^ Dickson, Leonard Eugene (1920). "Chapter VIII", History of the Theory of Numbers, Volume II: Diophantine Analysis. Carnegie Institute of Washington. 
  3. ^ Wieferich, Arthur (1909). "Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt". Mathematische Annalen 66: 95-101. 
  4. ^ Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen 72: 387-399. 
  5. ^ 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
  6. ^ 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
  7. ^ Pillai, S. S. On Waring's problem g(6)=73, Proc. Indian Acad. Sci. 12A, pp. 30-40
  8. ^ Euler's Conjecture - from Wolfram MathWorld
  9. ^ Niven, Ivan M. (1944). "An unsolved case of the Waring problem". American Journal of Mathematics 66: 137–143. 
  10. ^ Mahler, K. On the fractional parts of the powers of a rational number II, 1957, Mathematika, 4, pages 122-124
  11. ^ 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
  12. ^ 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
  13. ^ 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)