**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

Let p

_{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

We consider the Fermat’s equation *X*^{n} + *Y*^{n} = *Z*^{n} in the set of 2 × 2 rational matrices. We give some necessary condition of solvabillity of this equation.

**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

In the paper

A. Schinzel and H. Zassenhaus posed the following conjecture: If

*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*_{n} are the conjugates of

*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

The paper is a continuation of

. The considerations are over the class of multiplicative functions with strictly positive values and more precisely, over the pairs (

*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

^{ω(d)}σ(n/d), where

*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

The paper is a continuation of

and

. The considerations are over the class of multiplicative functions with strictly positive values and more precisely, over the pairs (

*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

^{ω(m)}, where

*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

Structural analysis (via the modular rings Z_{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

The characteristics within modular rings of the integer three are discussed. This integer has unique row structures in modular rings which appear to underlie restraints on various aspects of triples, particularly the factors and powers structure of the components. The function *N* = *x*^{m} + 2^{n}, with *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*_{T}} are such that 3|*N*_{T}. The modular ring Z_{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

According to the number of sub-SL(8)s (sub-STS(7)s), there are five classes of sloops SL(16)s (STS(15)s)

.) In

the author has classified SL(20)s into 11 classes. Using computer technique in

the authors gave a large number for each class of SL(20)s. There are only simple SL(22)s and simple SL(26)s. So the next admissible cardinality is 28. Also, all SL(32)s are classified into 14 classes in

. We try to generalize the classification of SL(20)s given in

for SL(2

*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.

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