Sequences obtained from x² ± y²

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 2, Pages 58–63
Full paper (PDF, 137 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
Campion College, PO Box 3052, Toongabbie East, NSW 2146, Australia

Abstract

Integers in class ̅3₄ of the modular ring Z₄ equal x² – y² but not x² + y² whereas integers in class ̅1₄ can equal both x² + y² and x² – y². This structure generates an infinity of sequences with neat curious patterns.

Keywords

• Modular rings
• Golden Ratio
• Infinite series
• Binet formula
• Right-end-digits
• Fibonacci sequence
• Meta-Fibonacci sequences

AMS Classification

• 11B39
• 11B50

References

1. Atanassov, K., Daryl, T., Deford, R., & Shannon, A. G. (2014) Pulsated Fibonacci Recurrences. The Fibonacci Quarterly. 52(5), 22–27.
2. Leyendekkers, J. V., & Shannon, A. G. (2015) The sum of squares for primes. Notes on Number Theory & Discrete Mathematics. 21(4), 17-21.
3. Leyendekkers, J. V., Shannon, A. G., & Rybak, J. M. (2005) Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No.9.
4. Livio, M. (2002) The Golden Ratio. New York, Broadway Books.
5. Vajda, S. (1989) Fibonacci Numbers & The Golden Section: Theory and Applications. Chichester, Ellis Horwood.