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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. ...

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Publication status:
Published
Peer review status:
Peer reviewed

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Publisher copy:
10.1016/j.jnt.2009.01.012

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Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Role:
Author
Publisher:
Elsevier
Journal:
JOURNAL OF NUMBER THEORY More from this journal
Volume:
129
Issue:
6
Pages:
1579-1594
Publication date:
2009-06-01
DOI:
ISSN:
0022-314X
Language:
English
UUID:
uuid:22939eab-37d8-411d-96a6-92fd3457c552
Local pid:
pubs:21162
Source identifiers:
21162
Deposit date:
2012-12-19

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