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. Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 fourth-powers. Waring conjected that these values were in fact the best possible.

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.

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 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.

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.
• 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.