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arXiv · 0806.4330

Sums and Differences of Three k-th Powers

D. R. Heath-Brown
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Abstract

Let k>2 be a fixed integer exponent and let θ> 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^θN^{1/10}) ways, providing that N << B^{3/13}. The significance of this is that we may take θstrictly less than 1. We also prove the estimate O(B^{10/k}), (subject to N << B) which is better for large k. The results extend to representations by an arbitrary fixed nonsingular ternary from. However ``non-trivial'' must then be suitably defined. Consideration of the singular form x^{k-1}y-z^k allows us to establish an asymptotic formula for (k-1)-free values of p^k+c, when p runs over primes, answering a problem raised by Hooley.

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D. R. Heath-Brown. 2008-06-26. Sums and Differences of Three k-th Powers. https://arxiv.org/abs/0806.4330

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