Sums and Differences of Three k-th Powers

2008/06/26 by Heath-Brown, D. R.
#11D45 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0806.4330

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-1y-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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