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Pythagorean triples and quadratic residues modulo an odd prime
- Source :
- AIMS Mathematics, Vol 7, Iss 1, Pp 957-966 (2022)
- Publication Year :
- 2021
- Publisher :
- American Institute of Mathematical Sciences (AIMS), 2021.
-
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 < \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 $.
Details
- ISSN :
- 24736988
- Volume :
- 7
- Database :
- OpenAIRE
- Journal :
- AIMS Mathematics
- Accession number :
- edsair.doi.dedup.....da061e2b378ac022d46ba65d96da7bc4