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Pairs of additive congruences: cubic congruences

Published online by Cambridge University Press:  26 February 2010

R. J. Cook
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH
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Abstract

We consider two additive cubic equations

in p-adic fields. Davenport and Lewis showed that the equations have a non-trivial solution in every p-adic field, if n ≥ 16, and need not have a solution in the 7-adic field, if n = 15. Here we prove that if p ≠ 7 the equations have a non-trivial solution in p-adic fields if n ≥ 13. When n = 12 such a result fails for every prime p ≡ 1 (mod 3).

Type
Research Article
Copyright
Copyright © University College London 1985

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References

Chowla, S., Mann, H. B. and Straus, E. G.. Some applications of the Cauchy-Davenport theorem. Kon. Norske Vidensk. Selsk. Fork, 32 (1959), 7480.Google Scholar
Cook, R. J.. Pairs of additive equations. Michigan Math. J., 19 (1972), 325331.CrossRefGoogle Scholar
Davenport, H. and Lewis, D. J.. Cubic equations of additive type. Phil. Trans. Roy. Soc. London, A, 261 (1966), 97136.Google Scholar
Davenport, H. and Lewis, D. J.. Two additive equations. Proc. Symp. in Pure Math., 12 (1967), 7498.CrossRefGoogle Scholar
Davenport, H. and Lewis, D. J.. Simultaneous equations of additive type. Phil Trans. Roy. Soc London, A, 264 (1969), 557595.Google Scholar
Dodson, M. M.. Homogeneous additive congruences. Phil. Trans. Roy. Soc. London, A, 261 (1966), 163210.Google Scholar
Lewis, D. J.. Cubic congruences. Michigan Math. J., 4 (1957), 8595.CrossRefGoogle Scholar
Vaughan, R. C.. On pairs of additive cubic equations. Proc. London Math. Soc, 34 (1977), 354364.CrossRefGoogle Scholar
Vaughan, R. C.. The Hardy-Littlewood Method (Cambridge University Press, 1981).Google Scholar