**Volume 16** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**On the Diophantine equation Σ**

_{i=1}*1/*^{k}*X*= 1 in distinct integers of the form_{i}*x*_{i}p^{α}q^{β}*Original research paper. Pages 1—5*

Nechemia Burshtein

Full paper (PDF, 156 Kb) | Abstract

*p*<

*q*are primes and

*α*,

*β*are positive integers. Among the several examples exhibited, Example 3 provides a new solution containing eighty-five even numbers

*x*all of which are of the required form. Certain questions and modifications of the equation are also discussed.

_{i}** Integer structure analysis of odd powered triples: The significance of triangular versus pentagonal numbers**

*Original research paper. Pages 6—13*

J. V. Leyendekkers, A. Shannon

Full paper (PDF, 168 Kb) | Abstract

*Z*

_{4}. The critical structure factor is that the rows of integers,

*N*

^{2}, with 3 |

*N*

^{2}, follow the triangular numbers, whereas 3 ∤

*N*

^{2}rows follow the pentagonal numbers. This structural characteristic is the reason for the importance of primitive Pythagorean triples (in which either the smallest odd component or the even component always has a factor 3).

** Fermatian analogues of Gould’s generalized Bernoulli polynomials**

*Original research paper. Pages 14—17*

A. G. Shannon

Full paper (PDF, 117 Kb) | Abstract

** Combined 2-Fibonacci sequences. Part 2**

*Original research paper. Pages 18—24*

Krassimir T. Atanassov

Full paper (PDF, 126 Kb) | Abstract

*n*-th members are given.

** Sharp concentration of the rainbow connection of random graphs**

*Original research paper. Pages 25—28*

Yilun Shang

Full paper (PDF, 145 Kb) | Abstract

*G*is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph

*G*, denoted by

*rc*(

*G*), is the smallest number of colors that are needed in order to make G rainbow connected. Similarly, a vertex-colored graph

*G*is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph

*G*, denoted by

*rvc*(

*G*), is the smallest number of colors that are needed in order to make

*G*rainbow vertex-connected. We prove that both

*rc*(

*G*) and

*rvc*(

*G*) have sharp concentration in classical random graph model

*G*(

*n*,

*p*).

** Some results on multiplicative functions**

*Original research paper. Pages 29—40*

Mladen Vassilev-Missana

Full paper (PDF, 237 Kb) | Abstract

*f*,

*g*) has a certain property (called in the paper

**S**), then for every fixed positive integer

*n*; the minimal and the maximal elements of the set {

*f*(

*d*)

*g*(

*n/d*) :

*d*runs over all divisors of

*n*} are obtained at least for some unitary divisors of

*n*. For these divisors if the maximum of

*f(d)g(n/d)*is reached for

*d**; then the minimum is reached for

*n*/

*d** and vice versa (the main results here are Theorems 1-4). The same investigation is made, but when d runs over the set of all divisors of n different than 1 and

*n*(the main result here is Theorem 5). Also corollaries of the mentioned results are obtained and some particular cases are considered.

**Volume 16**▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4