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
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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


Integers in class ̅34 of the modular ring Z4 equal x2 – y2 but not x2y2 whereas integers in class ̅14 can equal both x2y2 and x2 – y2. This structure generates an infinity of sequences with neat curious patterns.


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

AMS Classification

  • 11B39
  • 11B50


  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 primesNotes 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.

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Cite this paper

Leyendekkers, J. V., & Shannon, A. G. (2016). Sequences obtained from x2 ± y2. Notes on Number Theory and Discrete Mathematics, 22(2), 58-63.

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