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

Abstract

Let R k (n) denote the number of representations of a natural number n as the sum of three cubes and a k th power. In this paper, we show that R 3 (n) ≪ n 5/9+ε , and that R 4 (n) ≪ n 47/90+ε , where ε > 0 is arbitrary. This extends work of 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 k th powers, and we also use various exponential sum estimates.