On the Representations of a Number as the Sum of Three Cubes and a Fourth or Fifth Power

Abstract

Let RkOnU denote the number of representations of a natural number n as the sum of three cubes and a kth power. In this paper, we show that R3OnUn 5=9ae ,a nd that R4OnUn 47=90ae ,w heree > 0 is arbitrary.This extends workof Hooley concerning sums of four cubes, to the case of sums of mixed powers. To achieve these bounds, we use a variant of the Selberg sieve method introduced by Hooley to study sums of two kth powers, and we also use various exponential sum estimates.