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Research Article | Open Access

Pythagorean triples and quadratic residues modulo an odd prime

Jiayuan Hu1( ), Yu Zhan2
Department of Mathematics and Computer Science, Hetao College, Bayannur 015000, China
Department of Civil Engineering, Hetao College, Bayannur 015000, China
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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 < ϵ < 1, there must exist three quadratic residues x , y and z modulo p with 1 ≤ x , y , z ≤ p 1 + ϵ such that the equation x 2 + y 2 = z 2 .

Keywords

quadratic residuePythagorean triplesthe estimate for character sumsasymptotic formula
CLC number: 11A15, 11D09

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AIMS Mathematics
Volume 7 Issue 1,
January 2022
Pages 957-966
DOI: 10.3934/math.2022057

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Cite this article:
Hu J, Zhan Y. Pythagorean triples and quadratic residues modulo an odd prime. AIMS Mathematics, 2022, 7(1): 957-966. https://doi.org/10.3934/math.2022057

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Received: 06 July 2021
Accepted: 18 September 2021
Published: 15 January 2022
©2022 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)