**J. V. Leyendekkers and A. G. Shannon**

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

Volume 11, 2005, Number 4, Pages 17–24

**Full paper (PDF, 114 Kb)**

## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney, 2006 Australia*

A. G. Shannon

*Warrane College, Kensington, NSW 1465,
& KvB Institute of Technology, North Sydney, NSW 2060, Australia *

### Abstract

Fermat numbers (*F _{n}* = 2

^{2n}+ 1) and Mersenne numbers (

*M*= 2

_{m}*− 1),*

^{m}*m*odd, are compared on the basis of integer structure, using the modular rings Z

_{4}and Z

_{6}. The two numbers fall in different classes and this results in different composite row structures and different potentials for the formation of primes. The constraints on 2

*and the right end digits for*

^{n}*F*result in fewer numbers over a given range than those for

_{n}*M*. This is shown with two functions, which link the two numbers and show that

_{m}*F*= (2

_{n}^{x2 + y2 − 1}+ 1): for primes

*y*=

*n*, but when

*n*> 4,

*y*≠

*n*.

### AMS Classification

- 11A41
- 11A07

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## Related papers

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*Notes on Number Theory and Discrete Mathematics*, 30(1), 116-140. - Ramasamy, A. M. S. (2024). Sequences in finite fields yielding divisors of Mersenne, Fermat and Lehmer numbers, II.
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## Cite this paper

Leyendekkers, J. V., & Shannon, A. G. (2005). Fermat and Mersenne numbers. *Notes on Number Theory and Discrete Mathematics*, 11(4), 17-24.