# The sum of squares for primes

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
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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

### Abstract

Only prime integers that are in Class  ̅14 of the Modular Ring Z4 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 (xy) 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 (xy) couples because the factors are all elements of  ̅14.

### Keywords

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

• 11B39
• 11B50

### References

1. Borwein, J, & D. Bailey. (2004) Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, MA: A K Peters.
2. Leyendekkers, J. V. & A. G. Shannon (2012) The Modular Ring Z5. Notes on Number Theory and Discrete Mathematics. 18(2), 28–33.
3. Leyendekkers, J. V. & A. G. Shannon (2013) Fibonacci and Lucas Primes. Notes on Number Theory and Discrete Mathematics. 19(2), 49–59.
4. Leyendekkers, J. V. & A. G. Shannon (2013) The Pascal-Fibonacci Numbers. Notes on Number Theory and Discrete Mathematics. 19(3), 5–11.
5. Leyendekkers, J. V. & A. G. Shannon (2014) Fibonacci Primes. Notes on Number Theory and Discrete Mathematics. 20(2), 6–9.
6. Leyendekkers, J. V., A. G. Shannon, & J. M. Rybak (2007) Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No. 2.
7. Livio, M. (2002) The Golden Ratio. New York: Golden Books.
8. 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.
9. Phillips, G. M. (2005) Mathematics is not a Spectator Sport. New York: Springer.

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## Cite this paper

APA

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

Chicago

Leyendekkers, J. V., and A. G. Shannon. “The Sum of Squares for Primes.” Notes on Number Theory and Discrete Mathematics 21, no. 4 (2015): 17-21.

MLA

Leyendekkers, J. V., and A. G. Shannon. “The Sum of Squares for Primes.” Notes on Number Theory and Discrete Mathematics 21.4 (2015): 17-21. Print.

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