The Hilbert-Waring theorem

Abstract

Waring’s problem for exponent k is to prove that the set of nonnegative integers is a basis of finite order, that is, to prove that every nonnegative integer can be written as the sum of a bounded number of kth powers. We denote by g(k) the smallest number s such that every nonnegative integer is the sum of exactly s kth powers of nonnegative integers. Waring’s problem is to show that g(k) is finite; Hilbert proved this in 1909. The goal of this chapter is to prove the Hilbert-Waring theorem: the kth powers are a basis of finite order for every positive integer k.

Nous ne devons pas douter que ces considérations, qui permettent ainsi d’obtenir des relations arithmétiques en les faisant sortir d’identités où figurent des intégrales définies, ne puissent un jour, quand on en aura bien compris de sens, être appliquées à des problèmes bien plus étendus que celui de Waring. ¹ H. Poincaré [96]