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
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  ̅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 (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  ̅1₄.

Keywords

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

AMS Classification

• 11B39
• 11B50

References

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