On the Representations of a Number as the Sum of Three Cubes and a Fourth or Fifth Power
Tóm tắt
Let R
k
(n) 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 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 kth powers, and we also use various exponential sum estimates.
Tài liệu tham khảo
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