Anwar Ayyad

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 22, 2016, Number 1, Pages 29—32

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## Details

### Authors and affiliations

Anwar Ayyad

*Department of Mathematics, AL-Azhar University – Gaza
P. O. Box 1277, Gaza Strip, Palestine
*

### Abstract

For prime *p* and 1 ≤ *a*, *b*,* c* < *p* let *V* be the algebraic set of the congruence *ax* − *by* ≡ *c* (mod *p*) in the plane. For an arbitrary box of size *B* we obtain a necessary and a sufficient conditions on the size *B* in order for the box to meet *V*. For arbitrary subsets *S*, *T* of *Z _{p}* we also obtain a necessary and a sufficient conditions on the cardinalities of

*S*,

*T*so that

*S*+

*T*=

*Z*.

_{p}### Keywords

- Congruence
- Lattices
- Solutions

### AMS Classification

- 11D79
- 11H06

### References

- Ayyad, A., Cochrane, T. & Zheng, Z. (1996) The congruence
*x*≡_{1}x_{2}*x*(mod_{3}x_{4}*p*), the equation and mean values of character sums. J. Number Theory, 59, 398–413. - Bourgain, J., Katz N. & Tao, T. (2004) A sum-product estimate in finite fields and their applications, Geom. Funct. Anal. 14, 27–57.
- Glibichuk, A. A. (2006) Combinatorial properties of sets of residues modulo a prime and the Erdős–Graham problem, Mat. Zametki, 79, 384-395 (in Russian), English transl.: Math. Notes 79, 2006, 56–65.

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## Cite this paper

APA Ayyad, A. (2016). On the congruence *ax* −*by* ≡ *c* (mod *p*) and the finite field *Z _{p}*. Notes on Number Theory and Discrete Mathematics, 22(1), 29-32.

Ayyad, Anwar. “On the Congruence *ax* −*by* ≡ *c* (mod *p*) and the Finite Field *Z _{p}*.” Notes on Number Theory and Discrete Mathematics 22, no. 1 (2016): 29-32.

Ayyad, Ayyad. “On the Congruence *ax* −*by* ≡ *c* (mod *p*) and the Finite Field *Z _{p}*.” Notes on Number Theory and Discrete Mathematics 22.1 (2016): 29-32. Print.