Primitive Pythagorean triples and generalized Fibonacci sequences

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
Full paper (PDF, 206 Kb)

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 (4r₁ + 1) where r₁ belongs to the class in the modular ring Z₄.

Keywords

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

AMS Classification

• 11B39
• 11B50

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