Waring's 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 k^th 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) + 2^k - 2     for k ≥ 6.

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) and a simplified version of Hilbert's proof.