Primes within generalized Fibonacci sequences

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 4, Pages 56–63
Full paper (PDF, 95 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

The structure of the ‘Golden Ratio Family’ is consistent enough to permit the primality tests developed for φ₅ to be applicable. Moreover, the factors of the composite numbers formed by a prime subscripted member of the sequence adhere to the same pattern as for φ₅. Only restricted modular class structures allow prime subscripted members of the sequence to be a sum of squares. Furthermore, other properties of φ₅ are found to apply to those other members with structural compatibility.

Keywords

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

AMS Classification

• 11B39
• 11B50

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