**Volume 6** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Some remarks concerning the Bernoulli numbers**

*Original research paper. Pages 29—33*

S. Tabirca

Full paper (PDF, 207 Kb) | Abstract

**#!**The aim of this article is to propose some remarks on the Bernoulli numbers. Firstly, a simple proof for the the equation B 2n+ i = 0 is presented. This proof also gives an equation for ((2k). Using a simple computation, the values of ((2k), k = 1,12 are 2k i presented. Finally, an equation for the infinite product is proposed based on the Bernoulli numbers.

**Some new equations concerning the Euler function**

*Original research paper. Pages 34—38*

S. Tabirca and T. Tabirca

Full paper (PDF, 210 Kb) | Abstract

**#!**This article presents some new equations concerning the Euler function. An equation for the sum Y^d\n da ’ Vb{d) is found by using the multiplicative property. This is applied to find the sums Y!i=\{hn)a and ]£”=1 i ■ {i,n)a.

**On prime k-tuples conjectures**

*Original research paper. Pages 39—44*

V. Jotsov

Full paper (PDF, 193 Kb) | Abstract

**On the one-sided orderability for semigroups**

*Original research paper. Pages 45—55*

A. Baxhaku and M. Aslanski

Full paper (PDF, 358 Kb) | Abstract

**On the cardinal of the linear stable orders in a class of semigroups**

*Original research paper. Pages 56—60*

A. Baxhaku and M. Aslanski

Full paper (PDF, 209 Kb) | Abstract

**#!**The main results are: Theorem 1: For any semigroup S generated by the nonidempotent elements x\, x 2, . . . , x n and the generating relations (z) X{Xj — x 2\ (ii) x 2 = Xj and (Hi) x 2 ^ x2 for i ^ j there exist n\2n_1 non dual two-sided stable orders and 2 [(2n)! — n!2n] (non dual) one sided stable orders which are not two sided stable ones and Theorem 2: Let S’ be a semigroup generated by the idempotents xi, x2, the non-idempotents x^+i, Xk+2 , ■ • • ,%n and the relations (i),(ii),(iii). Then the semigroup S’ has n\2n~k~1 non-dual two-sided stable orders and \[(2n — k)\ — n\2n~k] only one-sided stable orders which are not two-sided stable ones.

**Connections in mathematics: Fibonacci sequence via arithmetic progression**

*Original research paper. Pages 61—63*

K. Atanassov

Full paper (PDF, 84 Kb)

**On a formula related to the n-th partial sum of the harmonic series**

*Original research paper. Pages 64—68*

M. Vassilev-Missana

Full paper (PDF, 139 Kb)