On the Representations of a Number as the Sum of Three Cubes and a Fourth or Fifth Power |
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Authors: | Joel M Wisdom |
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Institution: | (1) Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1109, U.S.A. |
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Abstract: | 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+, and that R
4
(n) n
47/90+, where > 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. |
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Keywords: | Cubes exponential sums fourth power fifth power sieve methods Waring's problem |
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