RENDtCONT! DEL CIRCOLO MATEMATICO DI PALERMO Serie 11, Tomo L (2(~1), pp. 239-246 AN ADDITIVE PROBLEM ABOUT POWERS OF FIXED INTEGERS GIUSEPPE MELFI Soit A un ensemble fini d'entiers > 2. Nous 6tudions les propri6tfs de l'ensemble E(Pow(A)) des entiers positifs qui sont une somme de puissances distinctes d'616ments de A. Erd6s posa ie probl~me suivant: d6montrer que I:(Pow ({3, 4})) a densit6 asymtotique superieure positive. Nous d6montrons que la fonction qui les 6num~re v6rifie P{3,4} (x) >> x 0"9659. 1. Introduction. Let A be a set of distinct integers > 2 and s a nonnegative integer. Define : Y~ga,k ak, ga,k E {0, 1} E(Pow (A; s)) aEA k>_s Burr, Erd6s, Graham and Wen-Ching Li [1] proved several results providing sufficient conditions in order that E(Pow (A; s)) contains all sufficiently large integers. They also stated the following CONJECTURE 1. Let s > 1. The set E(Pow(A; s)) sufficiently large integers if and only if 1 y~-- > 1 and gcd{a 6 A}= 1. a-I - aEA contains all In particular the conjectured result is independent of s > 1. In this paper we fix our attention for finite sets A. In section 2 we prove some