**Volume 9** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Some characteristics of primes within modular rings**

*Original research paper. Pages 49—58*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 3999 Kb) | Abstract

_{4}fall in even rows,

*R*

_{1}, in Class ̅1

_{4}with

*R*

_{1}= 6

*K*and

_{j}*K*= ½

_{j}*n*(3

*n*± 1) (depending on the parity of

*j*). The

*n*values are found to equal the rows that the primes occupy when Z

_{6}is set as a tabular array. The primes in Z

_{4}equal a unique sum or difference of squares (

*x*

^{2}±

*y*

^{2}) and via

*n*, the (

*x*,

*y*) pairs can be identified within the Z

_{6}structure. The

*n*values for composites follow well-defined linear functions that permit easy sorting. Finally, the parameter

__in the well-known function__

*n**=*

__P__*h*2

*− 1 has been identified as the row in one or more of the Modular Rings Z*

^{n}_{10}, Z

_{6}or Z

_{4}that contains one or more primes.

**Properties of the restrictive factor**

*Original research paper. Pages 59—61*

Laurențiu Panaitopol

Full paper (PDF, 1087 Kb) | Abstract

*n*) =

*q*

_{1}

^{α1−1}

*q*

_{2}

^{α2−1}…

*q*

_{k}^{αk−1}whenever

*n*=

*q*

_{1}

^{α1}

*q*

_{2}

^{α2}…

*q*, where

_{k}^{αk}*q*

_{1}, …,

*q*are pairwise different prime numbers and we have

_{k}*α*

_{1}…

*α*≥ 1. The function RF(

_{k}*n*) is called the restrictive factor. In the present paper we denote it simply by R(

*n*).

**A property of an arithmetic function**

*Original research paper. Pages 62—64*

Krassimir T. Atanassov

Full paper (PDF, 903 Kb) | Abstract

**Iterated Dirichlet series and the inverse of Ramanujan’s sum**

*Original research paper. Pages 65—72*

Kenneth R. Johnson

Full paper (PDF, 2246 Kb) | Abstract