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Cubic Diophantine equations: a supplementary congruence condition

Published online by Cambridge University Press:  26 February 2010

G. L. Watson
G. L. Watson
Affiliation:
Department of Mathematics, University College, London
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For the solubility of an inhomogeneous polynomial Diophantine equation, there is one well-known necessary, but not sufficient condition; namely the necessary congruence condition (NCC) explained in §2, below. Till recently, no progress had been made with the general cubic equation, because no one knew what else to assume. Examples given here, see (4.3), (5.4), indicate that some rather subtle hypothesis is needed. The first such hypothesis, see Davenport and Lewis [1], was very far from being necessary for the solubility of the equation. It would seem that any supplementary hypothesis which (loosely) is somewhere near necessary and also (together with the NCC) somewhere near sufficient deserves separate detailed investigation before one proceeds to use it.

Type
Research Article
Information
Mathematika , Volume 12 , Issue 1 , June 1965 , pp. 39 - 48
Copyright
Copyright © University College London 1965

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References

1. Davenport, H. and Lewis, D. J., “Non-homogeneous cubic equations”, Journal London Math. Soc, 39 (1964), 657–672.CrossRefGoogle Scholar
2. Watson, G. L., “Indefinite quadratic Diophantine equations”, Malhematika, 8 (1961), 32–38.CrossRefGoogle Scholar
3. Watson, G. L., “Cubic congruences”, Mathematika, 11 (1964), 142–150.CrossRefGoogle Scholar
4. Watson, G. L., “Cubic Diophantine equations: the necessary congruence condition”, Mathematika, 12 (1965), 30–38.CrossRefGoogle Scholar