Alternative solutions to the Legendre’s equation x^2+ky^2=z^2

Kanwara Mukkhata and Sompong Chuysurichay
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 228–235
DOI: 10.7546/nntdm.2025.31.2.228-235
Full paper (PDF, 243 Kb)

Details

Authors and affiliations

Kanwara Mukkhata
Division of Computational Science, Faculty of Science, Prince of Songkla University
15 Kanchanavanit Rd, Hatyai, Songkhla 90110, Thailand

Sompong Chuysurichay
Division of Computational Science, Faculty of Science, Prince of Songkla University
15 Kanchanavanit Rd, Hatyai, Songkhla 90110, Thailand

Abstract

In this paper, we aim to provide alternative solutions of the Legendre’s equation x^2 +ky^2 = z^2, where k is a square-free positive integer. The results also lead to solutions of the well-known Pythagorean triples and Eisenstein triples.

Keywords

  • Diophantine equations
  • Pythagorean triples
  • Eisenstein triples

2020 Mathematics Subject Classification

  • 11D09
  • 11D72

References

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Manuscript history

  • Received: 29 October 2024
  • Revised: 4 May 2025
  • Accepted: 5 May 2025
  • Online First: 6 May 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Mukkhata, K. & Chuysurichay, S. (2025). Alternative solutions to the Legendre’s equation x^2+ky^2=z^2. Notes on Number Theory and Discrete Mathematics, 31(2), 228-235, DOI: 10.7546/nntdm.2025.31.2.228-235.

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