Glasgow Mathematical Journal

Research Article

Small solutions of quadratic congruences

D. R. Heath-Browna1

a1 Magdalen College, Oxford OX1 4AU

Let Q(x) = Q(x1, …, xn)xs2208xs2124[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 S0017089500006091_inline1gives the trivial lower bound Bn(q)≥(q/n)½ for all q and n, since if x≠0 and qQ(x), then Q(x)≥q.