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Pairs of additive congruences to a large prime modulus

Part of: Diophantine equations

Published online by Cambridge University Press:  09 April 2009

O. D. Atkinson  and
R. J. Cook
O. D. Atkinson
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, England
R. J. Cook
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, England
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Abstract

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This paper is concerned with non-trivial solvability in p–adic integers, for relatively large primes p, of a pair of additive equations of degree k > 1: where the coefficients a1,…, an, b1,…, bn are rational integers.

Our first theorem shows that the above equations have a non-trivial solution in p–adic integers if n > 4k and p > k6. The condition on n is best possible.

The later part of the paper obtains further information for the particular case k = 5. specifically we show that when k = 5 the above equations have a non-trivial solution in p–adic integers (a) for all p > 3061 if n ≥ 21; (b) for all p execpt p = 5, 11 if n ≥ 26.

MSC classification

Secondary: 11D88: $p$-adic and power series fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 3 , June 1989 , pp. 438 - 455
Copyright
Copyright © Australian Mathematical Society 1989

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