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 *φ*_{5} 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 *φ*_{5}. Only restricted modular class structures allow prime subscripted members of the sequence to be a sum of squares. Furthermore, other properties of *φ*_{5} 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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## Cite this paper

Leyendekkers, J. V., & Shannon, A. G. (2015). Primes within generalized Fibonacci sequences. *Notes on Number Theory and Discrete Mathematics*, 21(4), 56-63.