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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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

AMS Classification

  • 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.
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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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