# Volume 4, 1998, Number 1

Volume 4 ▶ Number 1 ▷ Number 2Number 3Number 4

The characteristics of primes and other integers within the modular ring Z4 and in class ̅1
Original research paper. Pages 1-17
J. V. Leyendekkers, J. M. Rybak and A. G. Shannon
Full paper (PDF, 663 Kb) | Abstract

The integer structure of Class ̅1 in the modular ring Z4 has been analysed in detail. Most integers of this category equal a sum of two squares (x2 + y2). Those that do not are non-primes. The primes are distinguished by having a unique < x, y > pair that has no common factors. Other integers in Class  ̅1 have multiple values of < x, y > or more rarely a single < x, y > pair with common factors. Methods of estimating < x ,y > pairs are given. These are based on the class structure within Z4 and the right-most end digit characteristics. The identification of primes is consequently facilitated.

The characteristics of primes and other integers within the modular ring Z4 and in class ̅3
Original research paper. Pages 18—37
J. V. Leyendekkers, J. M. Rybak and A. G. Shannon
Full paper (PDF, 911 Kb) | 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.

Numerical properties of Morgan—Voyce Numbers
Original research paper. Pages 38—42