The generalized waring problem: A new property of positive integers

Abstract

The paper deals with the problem of whether a positive integern>1 can be written as the sum ofs summands that arerth powers of integers ≥m, wherem≥0 is a chosen integer (form=0 we have the classical Waring problem). For this problem, we define in a natural way arithmetic functionsG(m, r) andg(m, r) that are the analogs of the Hilbert functionsG(r) andg(r) for the classical Waring problem. It is proved that every positive integern exceeding some threshold value can be written as the above sum, simultaneously for alls, 1≤s≤n, with a finite number of exceptions, which are determined explicitly.