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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## 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

Campion College, PO Box 3052, Toongabbie East, NSW 2146, Australia

### Abstract

Integers in class ̅3_{4} of the modular ring Z_{4} equal *x*^{2} – *y*^{2} but not *x*^{2} + *y*^{2} whereas integers in class ̅1_{4} can equal both *x*^{2} + *y*^{2} and *x*^{2} – *y*^{2}. 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

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

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## Cite this paper

APALeyendekkers, J. V., & Shannon, A. G. (2016). Sequences obtained from *x*^{2} ± *y*^{2}. Notes on Number Theory and Discrete Mathematics, 22(2), 58-63.

Leyendekkers, J. V., and A. G. Shannon. “Sequences Obtained from *x*^{2} ± *y*^{2}.” Notes on Number Theory and Discrete Mathematics 22, no. 2 (2016): 58-63.

Leyendekkers, J. V., and A. G. Shannon. “Sequences Obtained from *x*^{2} ± *y*^{2}.” Notes on Number Theory and Discrete Mathematics 22.2 (2016): 58-63. Print.