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
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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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Related papers

  1. 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 Z4 and in class ̅3. Notes on Number Theory and Discrete Mathematics, 4(1), 18-37.

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