Warings problem

Waring's Problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated number 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.

For every k, we denote the least such s by g(k).

Lagranges Theorem states that every natural number is the sum of at least four squares; since three squares are not enough, this theorem establishes g(2)=4. g(3)=9 was established around 1912 and g(4) = 19 in 1986. These values had already been conjectured by Waring.

Using the Hardy Littlewood Method, g(k) can now readily be computed for all other values of k as well.