Transformations of Pythagorean triples generated by generalized Fibonacci numbers

Jathan Austin
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 924–928
DOI: 10.7546/nntdm.2025.31.4.924-928
Full paper (PDF, 189 Kb)

Details

Authors and affiliations

Jathan Austin
Department of Mathematical Sciences, Salisbury University
Salisbury, Maryland, USA

Abstract

We present matrices that transform Pythagorean triples arising from generalized Fibonacci sequences into other such triples. We also show that entries in the powers of such matrices can be expressed in terms of generalized Fibonacci sequences.

Keywords

• Pythagorean triples
• Fibonacci numbers
• Lucas numbers

2020 Mathematics Subject Classification

• 11B39
• 11C20

References

1. Austin, J. (2023). A note on generating primitive Pythagorean triples using matrices. Notes on Number Theory and Discrete Mathematics, 29(2), 402–406.
2. Austin, J., & Schneider, L. (2020). Generalized Fibonacci numbers in Pythagorean triple preserving matrices. The Fibonacci Quarterly, 58(4), 340–350.
3. Burton, D. M. (2010). Elementary Number Theory. McGraw Hill, New York.
4. Kalman, D., & Mena, R. (2003). The Fibonacci numbers–exposed. Mathematics Magazine, 76(3), 167–181.
5. Leyendekkers, J. V., & Shannon, A. G. (2016). Primitive Pythagorean triples and generalized Fibonacci sequences. Notes on Number Theory and Discrete Mathematics, 23(1), 54–62.
6. Palmer, L., Ahuja, M., & Tikoo, M. (1998). Finding Pythagorean triple preserving matrices. Missouri Journal of Mathematical Sciences, 10(2), 99–105.

Manuscript history

• Received: 7 August 2025
• Accepted: 9 December 2025
• Online First: 12 December 2025

Copyright information

Ⓒ 2025 by the Author.
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).