Sums and differences of three kth powers

Author: 

Heath-Brown, D

Publication Date: 

1 June 2009

Journal: 

Journal of Number Theory

Last Updated: 

2020-02-02T09:18:22.987+00:00

Issue: 

6

Volume: 

129

DOI: 

10.1016/j.jnt.2009.01.012

page: 

1579-1594

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 3kth powers, using integers of size at most B, in O (Bθ N1 / 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 non-singular ternary from. However "non-trivial" must then be suitably defined. Consideration of the singular form xk - 1 y - 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. © 2009 Elsevier Inc. All rights reserved.

Symplectic id: 

21162

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article