@article{Hu2022, 
author = {Jiayuan Hu and Yu Zhan},
title = {Pythagorean triples and quadratic residues modulo an odd prime},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {1},
pages = {957-966},
keywords = {quadratic residue, Pythagorean triples, the estimate for character sums, asymptotic formula},
url = {https://www.sciopen.com/article/10.3934/math.2022057},
doi = {10.3934/math.2022057},
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  .}
}