**J. V. Leyendekkers and A. G. Shannon**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 22, 2016, Number 4, Pages 54—62

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## Details

### Authors and affiliations

J. V. Leyendekkers

*Faculty of Science, The University of Sydney, NSW 2006, Australia*

A. G. Shannon

*Emeritus Professor, University of Technology Sydney, NSW 2007, Australia
Campion College, PO Box 3052, Toongabbie East, NSW 2146, Australia*

### Abstract

It is proved that infinite sequences of generalized Fibonacci sequences obtained from generalizations of the Golden Ratio can generate all primitive Pythagorean triples. This is a consequence of the integer structure since the major component of a primitive Pythagorean triple always has the form (4*r*_{1} + 1) where *r*_{1} belongs to the class in the modular ring *Z*_{4}.

### Keywords

- Unit digits (right-end-digits)
- Modular rings
- Golden ratio
- Fibonacci and Lucas numbers
- Pythagorean triples

### AMS Classification

- 11B39
- 11B50

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## Cite this paper

APALeyendekkers, J. V., & Shannon, A. G. (2017). Primitive Pythagorean triples and generalized Fibonacci sequences, Notes on Number Theory and Discrete Mathematics, 23(1), 54-62.

ChicagoLeyendekkers, J. V. and A. G. Shannon. “Primitive Pythagorean Triples and Generalized Fibonacci Sequences.” Notes on Number Theory and Discrete Mathematics 23, no. 1 (2017): 54-62.

MLALeyendekkers, J. V. and A. G. Shannon. “Primitive Pythagorean Triples and Generalized Fibonacci Sequences.” Notes on Number Theory and Discrete Mathematics 23.1 (2017): 54-62. Print.