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The representation of integers by positive ternary quadratic forms

Published online by Cambridge University Press:  26 February 2010

G. L. Watson
G. L. Watson
Affiliation:
University College, London
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Extract

Let f = f(x, y, z) be a positive definite form of the type

where x, y, z are integral valued variables, and the coefficients a, …, t are integers whose highest common factor is 1. As the determinant of such a form may be fractional, I define

and

thus — C is the discriminant of the binary form f(x, y, 0), and the necessary and sufficient condition for f to be positive definite is that a > 0, C > 0, and d > 0.

Type
Research Article
Information
Mathematika , Volume 1 , Issue 2 , June 1954 , pp. 104 - 110
Copyright
Copyright © University College London 1954

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References

page note 104 * Izvestia Akad. Nauk S.S.S.R. 4, (1940) 363–402.Google Scholar

page note 104 † Annals of Math., 28 (1927), 333–341.Google Scholar

page note 104 ‡ Acta Math., 70 (1939), 165–191.CrossRefGoogle Scholar

page note 106 * See e.g., Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers Oxford, 1938), 96, Theorem 123.Google Scholar

page note 107 * Nachrichten K. Ges. Wiss. Göttingen, Math.-Phya. Klasse, 1918, 21–29. Pólya states the result for a proper character, but this restriction is easily removed.