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1 results

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

  • Joel M. Wisdom
  • Journal:
    Compositio Mathematica / Volume 121 / Issue 1 / March 2000
    Published online by Cambridge University Press:
    04 December 2007, pp. 55-78
    Print publication:
    March 2000
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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.