Primitive zeros of quadratic forms mod $p^2$

Abstract

Abstract. Let Q(x) = Q(x 1 , x 2 , . . . , x n ) be a quadratic form with integer coefficients, p be an odd prime and ∥x∥ = max i |x i | . A solution of the congruence Q(x) ≡ 0 ( mod p 2 ) is said to be a primitive solution if p x i for some i . In this paper, we seek to obtain primitive solutions of this congruence in small rectangular boxes of the typeIn particular, we show that if n ≥ 4, n even, l ≤ n 2 − 2, and Q is nonsingular (mod p), then there exists a primitive solution with x i = 0, 1 ≤ i ≤ l , and |x i | ≤ 2