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

**The characteristics of primes and other integers within the modular ring Z _{4} 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

*Z*has been analysed in detail. Most integers of this category equal a sum of two squares (_{4}*x*). Those that do not are non-primes. The primes are distinguished by having a unique <^{2}+ y^{2}*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*Z*and the right-most end digit characteristics. The identification of primes is consequently facilitated._{4}**The characteristics of primes and other integers within the modular ring Z _{4} 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*Z*have been analysed in detail. All_{4}*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._{4}**Numerical properties of Morgan—Voyce Numbers**

*Original research paper. Pages 38—42*

A. F. Horadam

Full paper (PDF, 163 Kb)

**One extremal problem. 8**

*Original research paper. Pages 43—44*

Krassimir T. Atanassov

Full paper (PDF, 131 Kb)