Sums and Differences of Three k-th Powers

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(BθN1/10) ways, providing that N << B3/13. The significance of this is that we may take θ strictly less than 1. We also prove the estimate O(B10/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 xk-1y-zk allows us to establish an asymptotic formula for (k-1)-free values of pk+c, when p runs over primes, answering a problem raised by Hooley.