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Quadratic equations in several variables

Published online by Cambridge University Press:  24 October 2008

B. J. Birch  and
H. Davenport
B. J. Birch
Affiliation:
Trinity College, Cambridge
H. Davenport
Affiliation:
University College, London
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Extract

Let

be a quadratic form in n variables (n ≥ 3) with integral coefficients. One of us gave recently (2) a simple proof of a result of Cassels (l), in the following form: if the equation f = 0 is properly soluble in integers, then it has a solution satisfying

where γn–1 denotes Hermite's constant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 2 , April 1958 , pp. 135 - 138
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

REFERENCES

(1)Cassels, J. W. S.Bounds for the least solutions of homogeneous quadratic equations. Proc. Camb. Phil. Soc. 51 (1955), 262–4 and 52 (1956), 604.CrossRefGoogle Scholar
(2)Davenport, H.Note on a theorem of Cassels. Proc. Camb. Phil. Soc. 53 (1957), 539–40.CrossRefGoogle Scholar
(3)Mordell, L. J.On the equation ax 2 + by 2 – cz 2 = 0. Mh. Math. Phys. 55 (1951), 323–7.CrossRefGoogle Scholar