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Small solutions of quadratic congruences

Published online by Cambridge University Press:  18 May 2009

D. R. Heath-Brown
D. R. Heath-Brown
Affiliation:
Magdalen College, Oxford OX1 4AU
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Let Q(x) = Q(x1, …, xn)∈ℤ[x1, …, xn] be a quadratic form. We investigate the size of the smallest non-zero solution of the congruence Q(x)≡0 (mod q). We seek a bound Bn(q), independent of Q, such that there is always a non-zero solution satisfying

The form gives the trivial lower bound Bn(q)≥(q/n)½ for all q and n, since if x≠0 and q∣ Q(x), then Q(x)≥q.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 27 , October 1985 , pp. 87 - 93
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Borevich, Z. I. and Shafarevich, I. R., Number theory (Academic Press, New York, 1966).Google Scholar
2.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (Oxford University Press, 1960).Google Scholar
3.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