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

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## 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

### References

- Conway John H & Guy Richard K. 1996. The Book of Numbers. New York: Copernicus.
- Crandall Richard E. 1997. The challenge of large numbers. Scientific American. 276.2. 58-62.
- Glaisher JWL (ed.). 1940. Tables of Powers. British Association Mathematical Tables, Volume 9. Cambridge: Cambridge University Press.
- Griffin William Raymond. 1975. Significance of even-oddness of a prime’s penultimate digit. The Fibonacci Quarterly. 13.3: 204-205.
- Hoggatt Verner E Jr. 1969. Fibonacci and Lucas Numbers. Boston: Houghton- Mifflin.
- Horadam A F. 1965. Generating functions for powers of a certain generalized sequence of integers. Duke Mathematical Journal. 32.3: 437-446.
- Kravitz Sidney. 1970. The Lucas-Lehmer Test for Mersenne numbers. The Fibonacci Quarterly. 8.1:1-3.
- Leyendekkers JV & Rybak JM. 1995. The generation and analysis of Pythagorean triples within a two-parameter grid. International Journal of Mathematical Education in Science & Technology. 26.6: 787-793.
- Leyendekkers JV, Rybak JM & Shannon AG. 1998. The characteristics of primes and other integers within the modular ring Z4 and in Class 3. Notes on Number Theory & Discrete Mathematics. 4.1: 21-40.
- Ligh Steve & Jones Pat. 1981. Generalized Fermat and Mersenne numbers. The Fibonacci Quarterly. 20.1: 12-16.
- Mayer E & Morain F. 1997. http://www.lix.polvtechnique.fr/~morain/.
- Riesel Hans. 1994. Prime Numbers and Computer Methods for Factorization. Second Edition. Boston: Birkhauser.
- Shanks D. 1978. Solved and Unsolved Problems in Number Theory. Second Edition. New York: Chelsea.
- Sloane NJA & Plouffe Simon. 1965. The Encyclopedia of Integer Sequences.San Diego: Academic Press.
- Ulam Stanislaw M. 1964. Computers. In Mathematics in the Modern World. San Francisco: W H Freeman, pp.337-346.

## 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.