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# Slim Exceptional Sets for Sums of Cubes

Published:2002-04-01
Printed: Apr 2002
• Trevor D. Wooley
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## Abstract

We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by $9$, and not exceeding $X$, that fail to have a representation as the sum of $7$ cubes of prime numbers, is $O(X^{23/36+\eps})$. For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is $O(X^{11/36+\eps})$.
 Keywords: Waring's problem, exceptional sets
 MSC Classifications: 11P32 - Goldbach-type theorems; other additive questions involving primes 11P05 - Waring's problem and variants 11P55 - Applications of the Hardy-Littlewood method [See also 11D85]

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