Sums and Differences of Three k-th Powers

TL;DR

This paper investigates the representation of integers as sums or differences of three k-th powers, providing bounds on the number of such representations and extending results to prime-related forms, with implications for number theory.

Contribution

It introduces new bounds for representations of integers as sums or differences of three k-th powers, including cases involving prime numbers and singular forms.

Findings

Bound of O(B^{ heta}N^{1/10}) for representations, with ta<1

Estimate of O(B^{10/k}) for large k

Asymptotic formula for (k-1)-free values of p^k+c over primes

Abstract

Let k>2 be a fixed integer exponent and let \theta > 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^{\theta}N^{1/10}) ways, providing that N << B^{3/13}. The significance of this is that we may take \theta strictly 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.