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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

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