Mathematika

Research Article

Small zeros of quadratic congruences modulo pq

Todd Cochranea1

a1 Department of Mathematics, Kansas State University, Cardwell Hall, Manhattan, Kansas 66506-2602, U.S.A.

Let Q(x) = Q(x1, x2,…, xn) be a quadratic form with integer coefficients. Schinzel, Schickewei and Schmidt [9, Theorem 1] have shown that for any modulus m there exists a nonzero such that

S0025579300012985_eqn1

and ║x║≤m(1/2)+(1/2(n-1)), where ║x║ = max |xi|. When m is a prime Heath-Brown [8] has obtained a nonzero solution of (1) with ║x║≤m1/2 log m. Yuan [10] has extended Heath-Brown's work to all finite fields. We have proved related results in [5] and [6]. In this paper we extend Heath-Brown's work to moduli which are a product of two primes. Throughout the paper we shall assume that n is even and n>2. For any odd prime p let

S0025579300012985_eqnU1

where det Q is the determinant of the integer matrix representing Q and is the Legendre symbol.

(Received December 12 1989)

Key Words:

  • 10B05: NUMBER THEORY; Diophantine equations; Quadratic equations.