**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

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## Details

### 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*(*x*_{1}, *x*_{2}, …, *x _{n}*) be a quadratic form with integer coefficients,

*p*be an odd prime and ||

*x*|| = max

*|*

_{i}*x*|. A solution of the congruence

_{i}*Q*(

*x*) ≡ 0 (mod

*p*

^{3}) is said to be a primitive solution if

*p*∤

*x*for some

_{i}*i*. We prove that if

*p*>

*A*; where

*A*= 5·2

^{41}; then this congruence has a primitive solution, with ||

*x*|| < 34

*p*

^{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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## Cite this paper

APAHakami, A. H. (2016). Small primitive zeros of quadratic forms mod *P*^{3}, Notes on Number Theory and Discrete Mathematics, 22(3), 54-67.

Hakami, Ali H. “Small Primitive Zeros of Quadratic Forms mod *P*^{3}.” Notes on Number Theory and Discrete Mathematics 22, no. 3 (2016): 54-67.

Hakami, Ali H. “Small Primitive Zeros of Quadratic Forms mod *P*^{3}.” Notes on Number Theory and Discrete Mathematics 22.3 (2016): 54-67. Print.