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The Ramanujan Journal

, Volume 38, Issue 1, pp 189–198 | Cite as

Small primitive zeros of quadratic forms mod \(p^m\)

  • A. H. Hakami
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 \(n\).

Keywords

Quadratic forms Congruences Small solutions 

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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceJazan UniversityJazanSaudi Arabia

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