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
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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

References

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Cite this paper

APA

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

Chicago

Leyendekkers, 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.

MLA

Leyendekkers, 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.

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