On Pythagorean Triples and the Primitive Roots Modulo a Prime

Abstract

In this paper, we use the elementary methods and the estimates for character sums to study a problem related to primitive roots and the Pythagorean triples and prove the following result: let $p$ be an odd prime large enough. Then, there must exist three primitive roots $x,y$ , and $z$ modulo $p$ such that $x^2+y^2=z^2$ .