Waring's problem

In 1770 E. Waring asserted without proof in his Meditationes Algebraicae that every natural number is a sum of at most nine positive integral cubes, also a sum of at most 19 biquadrates, and so on. By this it is usually assumed that he believed that for every natural number k ≥ 2 there exists a number s such that every natural number is a sum of at most s kth powers of natural numbers, and that the least such s, say g(k), satisfies g(3) = 9, g(4) = 19.

It was probably known to Diophantus, albeit in a different form, that every natural number is the sum of at most four squares. The four square theorem was first stated explicitly by Bachet in 1621, and a proof was claimed by Fermat but he died before disclosing it. It was not until 1770 that one was given, by Lagrange, who built on earlier work of Euler. For an account of this theorem see Chapter 20 of Hardy & Wright (1979).

In the 19th century the existence of g(k) was established for many values of k, but it was not until the present century that substantial progress was made. First of all Hilbert (1909a, b) demonstrated the existence of g(k) for every k by a difficult combinatorial argument based on algebraic identities (see Rieger, 1953a, b, c; Ellison, 1971). His method gives a very poor bound for g (k).