Pythagorean triples and quadratic residues modulo an odd prime

Abstract

In this article, we use the elementary methods and the estimate for character sums to study a problem related to quadratic residues and the Pythagorean triples, and prove the following result. Let $ p $ be an odd prime large enough. Then for any positive number $ 0 < \\epsilon < 1 $, there must exist three quadratic residues $ x, \\ y $ and $ z $ modulo $ p $ with $ 1\\leq x, \\ y, \\ z\\leq p^{1+\\epsilon} $ such that the equation $ x^2+y^2 = z^2 $.