Small primitive zeros of quadratic forms mod P³

Ali H. Hakami
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 3, Pages 54–67
Full paper (PDF, 245 Kb)

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Authors and affiliations

Ali H. Hakami
Department of Mathematics, Jazan University
P.O.Box 277, Postal Code: 45142, Saudi Arabia

Abstract

Let Q(x) = Q(x₁, x₂, …, 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 p³) is said to be a primitive solution if p ∤ x_i for some i. We prove that if p > A; where A = 5·2⁴¹; then this congruence has a primitive solution, with ||x|| < 34p^3/2; provided that n ≥ 6 is even and Q is nonsinqular (mod p). Moreover, similar result is proven for cube boxes centered at the origin with edges of arbitrary lengths. These two results are extension of the quadratic forms problems.

Keywords

• Quadratic forms
• Congruences
• Small solutions

AMS Classification

• 11D79
• 11E08
• 11H50
• 11H55

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