J. V. Leyendekkers and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 21, 2015, Number 4, Pages 17–21

**Full paper (PDF, 87 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

Only prime integers that are in Class ̅1_{4} of the Modular Ring Z_{4} equate to a sum of squares of integers *x* and *y*. A simple equation to predict these integers is developed which distinguishes prime and composite numbers in that one (*x*, *y*) couple exists for primes, but composites have either one couple with a common factor or the same number of couples as there are factors. In particular, composite Fibonacci numbers always have multiple (*x*, *y*) couples because the factors are all elements of ̅1_{4}.

### Keywords

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

### AMS Classification

- 11B39
- 11B50

### References

- Borwein, J, & D. Bailey. (2004)
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*Z*5.*Notes on Number**Theory and Discrete Mathematics.*18(2), 28–33. - Leyendekkers, J. V. & A. G. Shannon (2013) Fibonacci and Lucas Primes.
*Notes on**Number Theory and Discrete Mathematics.*19(2), 49–59. - Leyendekkers, J. V. & A. G. Shannon (2013) The Pascal-Fibonacci Numbers.
*Notes on**Number Theory and Discrete Mathematics.*19(3), 5–11. - Leyendekkers, J. V. & A. G. Shannon (2014) Fibonacci Primes.
*Notes on Number**Theory and Discrete Mathematics.*20(2), 6–9. - Leyendekkers, J. V., A. G. Shannon, & J. M. Rybak (2007)
*Pattern Recognition:**Modular Rings and Integer Structure.*North Sydney: Raffles KvB Monograph No. 2. - Livio, M. (2002)
*The Golden Ratio.*New York: Golden Books. - Omey, E., S. Van Gulck. (2015) What are the last digits of … ?
*International Journal of**Mathematical Education in Science and Technology*. 46(1), 147–155. - Phillips, G. M. (2005)
*Mathematics is not a Spectator Sport.*New York: Springer.

## Related papers

- Leyendekkers, J. V., & Shannon, A. G. (2016). Some Golden Ratio generalized Fibonacci and Lucas sequences.
*Notes on Number Theory and Discrete Mathematics*, 22(1), 33-41.

## Cite this paper

Leyendekkers, J. V., & Shannon, A. G. (2015). The sum of squares for primes. *Notes on Number Theory and Discrete Mathematics*, 21(4), 17-21.