**Volume 13** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Arithmetical function characterizations and identities induced through equivalence relations**

*Original research paper. Pages 1—19*

Temba Shonhiwa

Full paper (6018 Kb) | Abstract

Let *A* denote the set of arithmetic functions and ∗ Dirichlet convolution. The paper presents and alternative approach to the study of arithmetic functions by introducing a homomorphism between the subgroup <*U*, ∗> of the group of units in <*A*, ∗> and the quotient ring induces through an equivalence relation. The same notion is extended to the case of unitary convolution.

**On some Pascal’s like triangles. Part 3**

*Original research paper. Pages 20—26*

Krassimir T. Atanassov

Full paper (87 Kb) | Abstract

In a series of papers, starting with

, we discuss new types of Pascal’s like triangles. Triangles of the present form, but not with the present sense, are described in different publications, e.g.

, but at least the author had not found a research with similar idea.

In the first part of our research we studied properties of some standard sequences and in the second part — of some “special” sequences. Now, we shall construct (0, 1)-analogous of the Pascal’s like triangles (or “(mod 2)-triangles”) from the both previous papers, i.e., we will construct (mod 2)-values of their elements and will discuss the obtained configurations. We will call the new triangles “(0, 1)-triangles”.

**Modular-ring class structures of ***x*^{n} ± *y*^{n}

*Original research paper. Pages 27—35*

J. V. Leyendekkers and A. G. Shannon

Full paper (355 Kb) | Abstract

Integer structure theory is used to analyse the factors of sums and differences of two identical powers of two integers, *x* and *y*. For instance, the sum of two identical powers, *m*, cannot form primes when *m* is odd or when *m* is even if the powers are odd and of the form *m*/2. The expanded forms of the factors indicate why the structure acts against the sum ever equalling an identical power. The difference of odd powers can yield primes when *x* − *y* = 1. The difference of even powers cannot yield primes whereas the sum can when *m*/2^{n} is even. However, *x*^{2} − *y*^{2} can equal a prime when *x* − *y* = 1.

**Remark on Jacobsthal numbers**

*Original research paper. Page 36*

Krassimir T. Atanassov

Full paper (83 Kb)

**Volume 13** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4