J. V. Leyendekkers and A. G. Shannon

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

Volume 11, 2005, Number 2, Pages 13–22

**Full paper (PDF, 118 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

Modular rings are used to analyse integers of the form *N* = 2^{m} +1. When *m* is odd, the integer structure prevents the formation of primes. When *m* is even, *N* ‘commonly’ has a right-end-digit of 5 and so is not a prime then. However, a sequence defined by *m* = 4 + 4*q*, *q* = 0, 1, 2, 3, can generate some primes as the right-end-digit is 7. Elements of this sequence satisfy the non-linear recurrence relation *G _{m}* =

*G*

^{2}

_{m–1}– 2

*G*

_{m–1}+ 2. Fermat numbers, where

*m*= 2

*n*satisfy this recurrence relation. However, in this case, the integer structure reveals that primes are limited to

*n*< 5.

### AMS Classification

- 11A41
- 11A07

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## Related papers

## Cite this paper

Leyendekkers, J. V., and Shannon, A. G. (2005). Fermat’s theorem on binary powers. *Notes on Number Theory and Discrete Mathematics*, 11(2), 13-22.