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 $.
Citation: Jiayuan Hu, Yu Zhan. Pythagorean triples and quadratic residues modulo an odd prime[J]. AIMS Mathematics, 2022, 7(1): 957-966. doi: 10.3934/math.2022057
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 $.
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Jiayuan Hu, Yu Zhan. Pythagorean triples and quadratic residues modulo an odd prime[J]. AIMS Mathematics, 2022, 7(1): 957-966. doi: 10.3934/math.2022057
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