  ## Encyclopedia > Warings problem

Article Content

# Waring's problem

Redirected from Warings problem

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. The affirmative answer was provided by David Hilbert in 1909. Sometimes this topic is described as Hilbert-Waring's theorem.

For every k, we denote the least such s by g(k). Note we have g(1) = 1.

Lagrange's Four Square Theorem from 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 by Fermat in 1640 and was first stated in 1621.

g(3) = 9 was established from 1909 to 1912 by Wieferich[?] and A. J. Kempner, g(4) = 19 in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers, g(5) = 37 in 1964 by Jing-run Chen and g(6) = 73 in 1940 by Pillai[?]. The values g(3) = 9 and g(4) = 19 had already been conjectured in 1778 by Waring.

All the other values of g are now also known, as a result of work by Dickson, Pillai, Rubugunday and Niven. Their formula contains two cases, and it is conjectured that the second case never occurs; in the first case, the formula reads

g(k) = floor((3/2)k) + 2k - 2     for k ≥ 6.

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Autocracy ... Wikipedia <  