The Ramanujan Journal

, Volume 38, Issue 1, pp 189–198

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

Article

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 $$n$$.

Mathematics Subject Classification

11D79 11E08 11H50 11H55

Notes

Acknowledgments

The author would like to thank the anonymous referee for his helpful and constructive comments and suggestions. He would also like to thank the Editors for their generous comments and support during the review process.

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