On Pythagorean Triples and the Primitive Roots Modulo a Prime

Abstract

<jats:p>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 <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M1 >\n <mi>p</mi>\n </math>\n </jats:inline-formula> be an odd prime large enough. Then, there must exist three primitive roots <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M2 >\n <mi>x</mi>\n <mo>,</mo>\n <mtext>\u2009</mtext>\n <mi>y</mi>\n </math>\n </jats:inline-formula>, and <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M3 >\n <mi>z</mi>\n </math>\n </jats:inline-formula> modulo <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M4 >\n <mi>p</mi>\n </math>\n </jats:inline-formula> such that <jats:inline-formula>\n <math xmlns= http://www.w3.org/1998/Math/MathML id= M5 >\n <msup>\n <mrow>\n <mi>x</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msup>\n <mo>+</mo>\n <msup>\n <mrow>\n <mi>y</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msup>\n <mo>=</mo>\n <msup>\n <mrow>\n <mi>z</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula>.</jats:p>