Small primitive zeros of quadratic forms mod $$p^m$$ p m

Abstract

Let \(Q(\mathbf{{x}}) = Q(x_1 ,x_2 ,\dots ,x_n )\) be a nonsingular quadratic form over \(\mathbb {Z}\), and \(p\) be an odd prime. A solution of the congruence \(Q({\mathbf {x}}) \equiv {\mathbf {0}}\,(\mathrm{mod}\, p^m )\) is said to be a primitive solution if \(p\not \mid x_i \) for some \(i\). We prove that if \(p > A,\) where \( A = 2^{2(n + 1)/(n - 2)} 3^{2/(n - 2)}\), then this congruence has a primitive solution, with \( \left\| \mathbf{{x}} \right\| \le 6^{1/n} p^{(m/2) + (m/n)}\) whenever \(n>m\) and \(m\ge 2,\) for every even \(n\).