**Volume 17** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**On a limit involving the product of prime numbers**

*Original research paper. Pages 1–3*

József Sándor and Antoine Verroken

Full paper (PDF, 142 Kb) | Abstract

_{k}denote the

_{k}th prime number. The aim of this note is to prove that the limit of the sequence is

*e*.

**Note on the matrix Fermat’s equation**

*Original research paper. Pages 4–11*

Aleksander Grytczuk and Izabela Kurzydło

Full paper (PDF, 182 Kb) | Abstract

*X*+

^{n}*Y*=

^{n}*Z*in the set of 2 × 2 rational matrices. We give some necessary condition of solvabillity of this equation.

^{n}**On some application of the spectral properties of the matrices**

*Original research paper. Pages 12–17*

Aleksander Grytczuk and Izabela Kurzydło

Full paper (PDF, 170 Kb) | Abstract

*α*≠ 0 is an algebraic integer of degree n which is not a root of unity, then there exists a constant

*c*> 0 such that

where

*α*=

*α*

_{1}; and

*α*

_{2}, …,

*α*are the conjugates of

_{n}*α*.

In this paper we give some information concerning this conjecture. In the proofs of the theorems we use some spectral properties of matrices.

**New results on some multiplicative functions**

*Original research paper. Pages 18–30*

Mladen Vassilev-Missana and Peter Vassilev

Full paper (PDF, 266 Kb) | Abstract

*f*,

*g*) of such functions, which have a special property, called in the paper property

**S**. For every such pair (

*f*,

*g*) and for every composite number

*n*> 1, the problem of finding the maximum and minimum of the numbers

*f*(

*d*)

*g*(

*n/d*), when

*d*runs over all proper divisors of n, is completely solved. Since some classical multiplicative functions like Euler’s totient function

*φ*, Dedekind’s function

*ψ*, the sum of all divisors of m, i.e.

*σ(m)*, the number of all divisors of

*m*, i.e.

*τ(m)*, and 2

^{ω(m)}(where

*ω(m)*is the number of all prime divisors of m) form pairs having property

**S**, we apply our results to these functions and also resolve the questions of finding the maximum and minimum of the numbers

*φ(d)σ(n/d)*,

*φ(d)ψ(n/d)*,

*τ(d)σ(n/d)*, 2

*, where*

^{ω(d)}σ(n/d)*d*runs over all proper divisors of

*n*. In addition some corollaries from the obtained results, concerning unitary proper divisors, are made. Since many other pairs of multiplicative functions (except the considered in the paper) have property

**S**, they may be investigated in similar manner in a future research.

**On multiplicative functions with strictly positive values**

*Original research paper. Pages 31–36*

Mladen Vassilev-Missana and Peter Vassilev

Full paper (PDF, 209 Kb) | Abstract

*f*,

*g*) of such functions, which have a special property, called in the paper property

**S**. For every two such pairs (

*f*

_{1},

*g*) and (

*f*

_{2},

*g*), with different

*f*

_{1}and

*f*

_{2}, a sufficient condition for the coincidence of the maximum (respectively of the minimum) of the numbers

*f*

_{1}(

*d*)

*g*(

*n*/

*d*) and

*f*

_{2}(

*d*)

*g*(

*n*/

*d*), where d runs over all proper divisors of an arbitrary composite number

*n*> 1, is given. Some applications of the results are made for several classical multiplicative functions like Euler’s totient function

*φ*, Dedekind’s function

*ψ*, the sum of all divisors of m, i.e.

*σ(m)*, the number of all divisors of m, i.e.

*τ(m)*, and 2

*, where*

^{ω(m)}*ω(m)*is the number of all prime divisors of

*m*.

**Remark on Jacobsthal numbers. Part 2**

*Original research paper. Pages 37–39*

Krassimir T. Atanassov

Full paper (PDF, 123 Kb)

**Structure analysis of the perimeters of primitive Pythagorean triples**

*Original research paper. Pages 40–46*

J. V. Leyendekkers, A. G. Shannon

Full paper (PDF, 278 Kb) | Abstract

_{4}, Z

_{6}) shows that the Perimeters, Pr, of primitive Pythagorean Triples (pPts) do not belong to simple functions. However, the factors

*x*, (

*x+y*) of the perimeter do, and the number of pPts in a given interval can be estimated from this. When

*x*is prime, the series for (

*x+y*) is complete and the associated pPts are one third of the total. When

*x*is composite, members of the series for (

*x+y*) are invalid when common factors with

*x*occur. These members are not associated with pPts. When 3|(

*x+y*), Pr ∈ ̅3

_{6}, while if 3 ∤(

*x+y*), Pr ∈ { ̅1

_{6}, ̅3

_{6}}. Class ̅3

_{6}dominates in the distribution.

**Modular rings and the integer 3**

*Original research paper. Pages 47–51*

J. V. Leyendekkers, A. G. Shannon

Full paper (PDF, 176 Kb) | Erratum (PDF, 10 Kb) | Abstract

*N*=

*x*+ 2

^{m}*, with*

^{n}*m*even and

*n*odd but

*x*not divisible by 3, always has 3 as a factor, and a majority of elements of the sequence of triangular numbers {

*N*} are such that 3|

_{T}*N*. The modular ring Z

_{T}_{3}and the distribution of primes within its structure are also discussed.

**On the classes of Steiner loops of small orders**

*Original research paper. Pages 52–68*

M. H. Armanious

Full paper (PDF, 565 Kb) | Abstract

*n*)s for each possible n and applying this method for

*n*= 14 to classify all possible classes of SL(28)s. Consequently, we can establish all classes of nonsimple SL(28)s and all classes of semi-planar SL(28)s (STS(27)s). In this article, we show in section 3 that there are nine classes of SL(28)s (STS(27)s having one sub-SL(14) (sub-STS(13)) and r sub-SL(8)s (sub- STS(7)s) for

*r*= 0, 1, 2, 3, 4, 5, 8, 11 or 16. All these sloops are subdirectly irreducible having exactly one proper homomorphic image isomorphic to SL(2). In section 4, we construct all classes of semi-planar SL(28)s. Such SL(28)s (STS(27)s) have

*r*sub-SL(8)s (sub-STS(7)s) for

*r*=1, 2, 3, 4, 5, 8, 11, 16 but no sub-SL(14) (sub-STS(13)).

In is determined a necessary and sufficient condition for a sub-SL(2) = {1,

*x*} of an SL(2

*n*) to be normal. This result supplies us with the following two facts. First, there is another nonsimple subdirectly irreducible SL(2

*n*) having exactly one proper homomorphic image isomorphic to an SL(n). Accordingly, we can construct all classes of nonsimple subdirectly irreducible SL(28)s. Second fact is that if an SL(2

*n*) has a simple sub-SL(

*n*) and (

*n*– 1)(

*n*– 2)/6 sub-SL(8)s passing through a non-unit element, then SL(2

*n*) is isomorphic to the direct product SL(

*n*) × SL(2). According to the result of section 3 and the above two facts, we may say that there are 8 simple classes of SL(28)s and only 11 classes of nonsimple SL(28)s, all these classes have no sub-SL(10)s. In the last section, we construct an example for each class given above of nonmsimple and simple (semi-planar) SL(28)s (STS(27)s). Finally, we review the classes of SL(2

*n*)s (STS(2

*n*– 1)s) in 3 tables for 2

*n*= 16, 20, and 28.