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

Published online by Cambridge University Press:  04 December 2007

Joel M. Wisdom
Joel M. Wisdom
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, U.S.A. E-mail: wisdom@math.lsa.umich.edu
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Abstract

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Let Rk(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 R3(n) [Lt ] n5/9+ε, and that R4(n) [Lt ] n47/90+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.

Keywords

Cubesexponential sumsfourth powerfifth powersieve methodsWaring's problem
Type
Research Article
Information
Compositio Mathematica , Volume 121 , Issue 1 , March 2000 , pp. 55 - 78
Copyright
© 2000 Kluwer Academic Publishers