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Small solutions of quadratic congruences, II

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

D. R. Heath-Brown
D. R. Heath-Brown
Affiliation:
Magdalen College, Oxford, OX1 4AU
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Let Q(x) = Q(x1,…, xn) є ęZ x1, …, xn] be a quadratic form. The primary purpose of this paper is to bound the smallest non-zero solution of the congruence Q(x) = 0 (mod q). The problem may be formulated as follows. We ask for the least bound Bn(q) such that, for any Ki > 0 satisfying

and any Q, the congruence has a non-zero solution satisfying

MSC classification

Secondary: 11D79: Congruences in many variables
Type
Research Article
Information
Mathematika , Volume 38 , Issue 2 , December 1991 , pp. 264 - 284
Copyright
Copyright © University College London 1991

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References

1.Baker, R. C.. Diophantine Inequalities, London Math. Soc. Monographs New Series, 1 (Oxford, 1986).Google Scholar
2.Baker, R. C. and Harman, G.. Small fractional parts of quadratic forms. Proc. Edinburgh Math. Soc, 25 (1982), 269–277.CrossRefGoogle Scholar
3.Borevich, Z. I. and Shafarevich, I. R.. Number Theory (Academic Press, New York, 1966).Google Scholar
4.Cochrane, T.. Small zeros of quadratic congruences modulo pq. Mathematika, 37 (1990), 261–272.Google Scholar
5.Cochrane, T.. Small zeros of quadratic congruences modulo p, III. J. Number Theory, 37 (1991), 92–99.Google Scholar
6.Danicic, I.. An extension of a theorem of Heilbronn. Mathematika, 5 (1958), 30–37.Google Scholar
7.Gérardin, P. and Lin, C.-Y.. An identity involving quadratic characters. J. Number Theory, 33 (1989), 356–361.CrossRefGoogle Scholar
8.Heath-Brown, D. R.. Small solutions of quadratic congruences. Glasgow Math. J., 27 (1985), 87–93.CrossRefGoogle Scholar
9.Sander, J. W.. A reciprocity formula for quadratic forms. Monatsh. Math., 104 (1987), 125–132.Google Scholar
10.Schinzel, A., Schlickewei, H.-P. and Schmidt, W. M.. Small solutions of quadratic congruences and small fractional parts of quadratic forms. Acta Arith., 37 (1980), 241–248.CrossRefGoogle Scholar