Sums and differences of three $k$th powers

Journal article

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.

Authors: Heath-Brown, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: JOURNAL OF NUMBER THEORY
• Volume: 129
• Issue: 6
• Pages: 1579-1594
• Publication date: 2009-06-01
• DOI: 10.1016/j.jnt.2009.01.012
• ISSN: 0022-314X
• Language: English