The generalized waring problem: A new property of positive integers

Abstract

ABSTRACT. The paper deals with the problem of whether a positive integer n _> I can be written as the sum of s summands that are rth powers of integers > m, where m _> 0 is a chosen integer (for m = 0 we have the classical Waring problem). For this problem, we define in a natural way arithmetic functions G(m, r) and g(m, r) that are the analogs of the Hilbert functions G(r) and g(r) for the classical Waring problem.It is proved that every positive integer n exceeding some threshold value can be written as the above sum, simultaneously for all s, 1 < s <n, with a finite number of exceptions~ which are determined explicitly.Consider the problem of representing positive integers n > 1 as sums of the form