The characteristics of primes and other integers within the modular ring Z₄ and in class ̅3

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
Full paper (PDF, 911 Kb)

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 Z₄ have been analysed in detail. All n equal the difference of squares, x² – y², with 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 Z₄ 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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