Abstract
Let Q(x) = Q(x(1), x(2),..., x(n)) be a quadratic form with integer coefficients and p be an odd prime. Let vertical bar vertical bar x vertical bar vertical bar = max vertical bar x(i)vertical bar. Then we show that for n >= 4, if Q(x) is nonsingular (mod p), then there exists a primitive solution of the congruence Q(x) = 0 (mod p(2)) (a solution x with gcd(x(1),..., x(n), p) = 1) with vertical bar vertical bar x vertical bar vertical bar<< p, which is best possible. Similar results are proven for rectangular boxes with sides of arbitrary lengths.