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Reference: Heath-Brown, DR, (2009). Sums and differences of three $k$th powers. JOURNAL OF NUMBER THEORY, 129 (6), 1579-1594.Citable link to this page:


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 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.

Peer Review status:Peer reviewedPublication status:PublishedVersion:Publisher versionNotes:Copyright 2009 Elsevier B.V. All rights reserved. Re-use of this article is permitted in accordance with the Terms and Conditions set out at

Bibliographic Details

Publisher: Elsevier B.V.

Publisher Website:

Journal: JOURNAL OF NUMBER THEORYsee more from them

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Issue Date: 2009-6


Urn: uuid:22939eab-37d8-411d-96a6-92fd3457c552

Source identifier: 21162


Issn: 0022-314X Item Description

Type: Journal article;

Language: eng

Version: Publisher version Tiny URL: pubs:21162


Autor: Heath-Brown, DR - institutionUniversity of Oxford Oxford, MPLS, Mathematical Inst - - - - Bibliographic Details Publisher: Elsevi



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