The structure of the Fibonacci numbers in the modular ring Z₅

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 1, Pages 66–72
Full paper (PDF, 149 Kb)

Details

Authors and affiliations

J. V. Leyendekkers
Faculty of Science, The University of Sydney
NSW 2006, Australia

A. G. Shannon
Faculty of Engineering & IT, University of Technology
Sydney, NSW 2007, Australia

Abstract

Various Fibonacci number identities are analyzed in terms of their underlying integer structure in the modular ring Z₅.

Keywords

• Fibonacci sequence
• Golden Ratio
• Modular rings
• Binet formula

AMS Classification

• 11B39
• 11B50

References

1. Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007. Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No 9.
2. Leyendekkers, J.V., A.G. Shannon. The Modular Ring Z5. Notes on Number Theory and Discrete Mathematics. Vol. 18, 2012, No. 2, 28–33.
3. Leyendekkers, J.V., A.G. Shannon. Geometrical and Pellian Sequences. Advanced Studies in Contemporary Mathematics. Vol. 22, 2012, No. 4, 507–508.
4. Leyendekkers, J.V., A.G. Shannon. On the Golden Ratio (Submitted).
5. Leyendekkers, J.V., A.G. Shannon. The Decimal String of the Golden Ratio (Submitted).
6. Livio, Mario. The Golden Ratio. Golden Books, New York, 2002.
7. Shannon, A.G., A.F. Horadam, S.N. Collings. Some Fibonacci Congruences. The Fibonacci Quarterly. Vol. 12, 1974, No. 4, 351–354.
8. Simons, C.S., M. Wright. Fibonacci Imposters. International Journal of Mathematical Education in Science and Technology. Vol. 38, 2007, No. 5, 677–682.
9. Ward, M. The Algebra of Recurring Series. Annals of Mathematics. Vol. 32, 1931, No. 1, 1–9.