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Bounded representations of integers by 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 ƒ = ƒ(x1, …, xk) be a quadratic form in k variables, which has integral coefficients and is not degenerate. Let n ≠ 0 be any integer representable by ƒ, that is, such that the equation

is soluble in integers x1, …, xk. We shall call a solution of (1) a bounded representation of n by ƒ if it satisfies

Type
Research Article
Information
Mathematika , Volume 4 , Issue 1 , June 1957 , pp. 17 - 24
Copyright
Copyright © University College London 1957

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References

page 17 note * See, e.g., Landau, , Vorlesungen über Zahlentheorie I, 141, Satz 203.Google Scholar

page 17 note † Meyer, A., Vierteljahrsschrift der Naturforschenden Ges. in Zürich, 29 (1884), 209–222.Google Scholar

page 19 note * The determinant of g itself may be fractional, if g has product terms with odd coefficients.

page 19 note † For a simple proof, see Pall, Gordon, "On the order invariants of integral quadratic forms", Quart. J. of Math. (Oxford), 6 (1935), 30–51 (35, Lemma 2).CrossRefGoogle Scholar

page 20 note * An appeal to this theory can be avoided by remarking that Lemma 1 (ii) continues t o hold if q is any square-free positive integer which is relatively prime to the fixed number 2|Δ|. The existence of such an integer satisfying (16) and (17), and also relatively prime to d (as we require later) is easily proved.

page 22 note * See, for example, Hermite, Oeuvres I, 122–135.