J Austral Math Soc Ser A 46 pp438--455, 1989.

Pairs of Additive Congruences to a Large Prime Modulus

O. D. Atkinson and R. J. Cook

Abstract

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:

f(x) = a1x1k + ···+ anxnk = 0,

g(x) = b1x1k + ···+ bnxnk = 0,

where the coefficients a₁, ..., a_n, b₁, ..., b_n 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 chapter obtains further information for the particular case k = 5. Specifically we show that when k = 5 the above equations have a non-trivial solutin in p-adic integers (a) for all p > 3061 if n ³ 21; (b) for all p except p = 5, 11 if n ³ 26.

1980 AMS Subject Classification (1985 Revision): 11D88

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Authors

O. D. Atkinson

R. J. Cook

Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, England.