Author
Heath-Brown, D
Journal title
Journal of Number Theory
DOI
10.1016/j.jnt.2009.01.012
Issue
6
Volume
129
Last updated
2023-12-21T20:13:34.87+00:00
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
Favourite
On
Publication type
Journal Article
Publication date
01 Jun 2009
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