J. V. Leyendekkers, J. M. Rybak and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 5, 1999, Number 1, Pages 1–26

**Full paper (PDF, 1.3 Mb**

## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney, 2006, Australia
*

J. M. Rybak

*The University of Sydney, 2006, Australia*

A. G. Shannon

*University of Technology, Sydney, 2007, Australia*

### Abstract

Mersenne-Fibonacci primes, (2^{p} − 1), (the classical Mersenne primes), and Mersenne-Lucas primes ((2^{p} + l)/3), *p* a prime, are analysed within the framework of the modular ring ℤ_{4}. These primes are restricted to Class 3 of ℤ_{4} and are composed of single-class “nests” of integers. The reason some p values do not give (2^{p} − 1) as a prime is explained in terms of the two-parameter equation for composite integers given in a previous paper by the authors, the difference of squares, and Fermat’s “Little Theorem”. The Fermat characteristics are explored up to exponents of 257. The reasons for the terminology are given in the context that the two types of Mersenne primes are related to one another analogously to the way the Fibonacci and Lucas numbers are inter-related.

### Keywords

- Fermat primes
- Fermat’s Little Theorem
- Fibonacci numbers
- Lucas numbers
- Mersenne primes

### AMS Classification

- 11A41
- 11B37
- 11B39

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## Related papers

- Leyendekkers, J. V., & Shannon, A. G. (2003). Some characteristics of primes within modular rings. Notes on Number Theory and Discrete Mathematics, 9(3), 49-58.

## Cite this paper

Leyendekkers, J. V., Rybak, J. M. & Shannon, A. G. (1999). An analysis of Mersenne–Fibonacci and Mersenne–Lucas primes. *Notes on Number Theory and Discrete Mathematics*, 5(1), 1-26.