On the Congruence axbyc (mod p) and the Finite Field Zp

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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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 axbyc (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 Zp we also obtain a necessary and a sufficient conditions on the cardinalities of S, T so that S + T = Zp.

Keywords

  • Congruence
  • Lattices
  • Solutions

AMS Classification

  • 11D79
  • 11H06

References

  1. Ayyad, A., Cochrane, T. & Zheng, Z. (1996) The congruence x1x2x3x4 (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.

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Cite this paper

APA

Ayyad, A. (2016). On the congruence axbyc (mod p) and the finite field Zp. Notes on Number Theory and Discrete Mathematics, 22(1), 29-32.

Chicago

Ayyad, Anwar. “On the Congruence axbyc (mod p) and the Finite Field Zp.” Notes on Number Theory and Discrete Mathematics 22, no. 1 (2016): 29-32.

MLA

Ayyad, Ayyad. “On the Congruence axbyc (mod p) and the Finite Field Zp.” Notes on Number Theory and Discrete Mathematics 22.1 (2016): 29-32. Print.

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