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

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

Volume 4, 1998, Number 1, Pages 18–37

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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

Integers, *n*, in Class ̅3 of the modular ring *Z _{4}* have been analysed in detail. All

*n*equal the difference of squares, x

*– y*

^{2}*, with*

^{2}*x*even and

*y*odd. Primes are distinguished by having only one <

*x, y*> pair: <X, Y> with X – Y = 1, and X = (

*n*+ l ) / 2 . In this paper four methods of calculating die <

*x, y*> values are given. These methods are based on prime factorisation, the

*Z*class structure, and Fermat’s Little Theorem. Mersenne primes are uniquely distributed within Class ̅3 and some new features of these primes are also stated.

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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. (1998). The characteristics of primes and other integers within the modular ring Z_{4} and in class ̅3. *Notes on Number Theory and Discrete Mathematics*, 4(1), 18-37.