On the Congruence ax −by ≡ c (mod p) and the Finite Field Z_p

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
Full paper (PDF, 152 Kb)

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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

1. Ayyad, A., Cochrane, T. & Zheng, Z. (1996) The congruence x₁x₂ ≡x₃x₄ (mod p), the equation and mean values of character sums. J. Number Theory, 59, 398–413.
2. Bourgain, J., Katz N. & Tao, T. (2004) A sum-product estimate in finite fields and their applications, Geom. Funct. Anal. 14, 27–57.
3. 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.