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Homogeneous quadratic equations

Published online by Cambridge University Press:  26 February 2010

H. Davenport
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Let

be a quadratic form with integral coefficients, and suppose the equation

has a solution in integers x1…, xn, not all 0. It was proved by Cassels [2] that there is such a solution, which satisfies the estimate

where F = max|fij|. It was later observed by Birch and Davenport [1] that the result can be stated in a slightly more general form. Let

be a quadratic form which assumes only integral values at the points (x1 …, x2) of an n-dimensional lattice Λ of determinant Δ. Suppose there is some point of Λ, other than the origin, at which ø = 0. Then there is such a point for which also

where Φ = max |øij|.

Type
Research Article
Information
Mathematika , Volume 18 , Issue 1 , June 1971 , pp. 1 - 4
Copyright
Copyright © University College London 1971

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References

1.Birch, B. J. and Davenport, H., Proc. Camb. Phil. Soc., 54 (1958), 135–138.CrossRefGoogle Scholar
2.Cassels, J. W. S., Proc. Camb. Phil. Soc., 51 (1955), 262–264 and 52 (1956), 604.CrossRefGoogle Scholar
3.Minkowski, H., Geometrie der Zahlen, Kap. 5.Google Scholar