**Volume 22** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**The definitive solution of Gauss’s lattice points problem in the circle**

*Original research paper. Pages 1—3*

Aldo Peretti

Full paper (PDF, 464 Kb) | Abstract

By means of Bienayme’s theorem of Statistics is found that the remainder term in Gauss’s problem about the lattice points in the circle is a function normally distributed with mean value zero and standard deviation 1,10368 multiplied by the fourth root of *x*. This result can not be improved.

**On irrationality of some distances between points on a circle**

*Original research paper. Pages 4—9*

Mladen Vassilev–Missana

Full paper (PDF, 155 Kb) | Abstract

Let

*n* > 3 be arbitrary integer. In the present paper it is shown that if

*K* is an arbitrary circle and

*M*_{i},

*i* = 1,…,

*n* are points on

*K*, dividing

*K* into

*n* equal arcs, then for each point

*M* on

*K*, different from the mentioned above, at least

of the distances |

*MM*_{i}| are irrational numbers.

**Catalan triangles and Finucan’s hidden folders**

*Original research paper. Pages 10—16*

A. G. Shannon

Full paper (PDF, 138 Kb) | Abstract

This note is to capture and extend some of Finucan’s ideas for further exploration by students. These ideas connect with several elementary concepts in combinatorial analysis which lend themselves to undergraduate research projects

**On function “Restrictive factor”**

*Original research paper. Pages 17—22*

Krassimir T. Atanassov

Full paper (PDF, 172 Kb) | Abstract

Some new properties of the arithmetic function called “Restrictive factor” are formulated and studied.

**On transitive polynomials modulo integers**

*Original research paper. Pages 23—35*

Mohammad Javaheri and Gili Rusak

Full paper (PDF, 223 Kb) | Abstract

A polynomial *P*(*x*) with integer coefficients is said to be transitive modulo *m*, if for every *x*, *y* ∈ ℤ there exists *k* ≥ 0 such that *P*^{k}(*x*) = *y* (mod *m*). In this paper, we construct new examples of transitive polynomials modulo prime powers and partially describe cubic and quartic transitive polynomials. We also study the orbit structure of affine maps modulo prime powers.

**The ***n* × *n* × *n* Points Problem optimal solution

*Original research paper. Pages 36—43*

Marco Ripà

Full paper (PDF, 557 Kb) | Abstract

We provide an optimal strategy to solve the *n* × *n* × *n* points problem inside the box, considering only 90° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach that significantly reduces the number of straight lines connected at their end-points necessary to join all the *n*^{3} dots. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference *h*_{u}(*n*) − *h*_{l}(*n*) between the upper and the lower bound, proving that it is ≤ 0.5 ∙ *n* ∙ (*n* + 3), for any *n* > 1.

**Maximal trees with log-concave independence polynomials**

*Original research paper. Pages 44—53*

Eugen Mandrescu and Alexander Spivak

Full paper (PDF, 182 Kb) | Abstract

If

*s*_{k} denotes the number of independent sets of cardinality

*k* in graph

*G*, and

*α*(

*G*) is the size of a maximum independent set, then

is the independence polynomial of

*G* (I. Gutman and F. Harary, 1983, ⟦8⟧). The Merrifield–Simmons index

*σ*(

*G*) (known also as the Fibonacci number) of a graph

*G* is defined as the number of all independent sets of

*G*. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdos (1987, ⟦2⟧) conjectured that

*I*(

*T*,

*x*) is unimodal whenever

*T* is a tree, while, in general, they proved that for each permutation

*π* of {1, 2, …,

*α*} there is a graph

*G* with

*α*(

*G*) =

*α* such that

*s*_{π(1)} <

*s*_{π(2)} < … <

*s*_{π(α)}. By maximal tree on

*n* vertices we mean a tree having a maximum number of maximal independent sets among all the trees of order

*n*. B. Sagan proved that there are three types of maximal trees, which he called batons ⟦24⟧.

In this paper we derive closed formulas for the independence polynomials and Merrifield–Simmons indices of all the batons. In addition, we prove that *I*(*T*,*x*) is log-concave for every maximal tree *T* having an odd number of vertices. Our findings give support to the above mentioned conjecture.

**Series expansions related to the logarithmic mean**

*Original research paper. Pages 54—57*

József Sándor

Full paper (PDF, 126 Kb) | Abstract

We show that the Gregory series combined with Newton’s binomial expansion give a natural approach to the logarithmic mean inequalities.

**Sequences obtained from ***x*^{2} ± *y*^{2}

*Original research paper. Pages 58—63*

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 137 Kb) | Abstract

Integers in class ̅3_{4} of the modular ring Z_{4} equal *x*^{2} – *y*^{2} but not *x*^{2} + *y*^{2} whereas integers in class ̅1_{4} can equal both *x*^{2} + *y*^{2} and *x*^{2} – *y*^{2}. This structure generates an infinity of sequences with neat curious patterns.

**A note on a family of alternating Fibonacci sums**

*Original research paper. Pages 64—71*

Robert Frontczak

Full paper (PDF, 167 Kb) | Abstract

In this note we consider a family of finite and infinite alternating sums containing products of Fibonacci numbers. We derive closed-form expressions for this family of sums. As a consequence of this result we establish new algebraic relationships between certain alternating sums of reciprocals of products of Fibonacci numbers with integer power.

**On subgroups of non-commutative general rhotrix group**

*Original research paper. Pages 72—90*

A. Mohammed and U. E. Okon

Full paper (PDF, 208 Kb) | Abstract

This paper considers the pair (*GR*_{n}(*F*),⚬) consisting of the set of all invertible rhotrices of size n over an arbitrary field *F*; and together with the binary operation of row-column method for rhotrix multiplication; in order to introduce it as the concept of “non-commutative general rhotrix group”. We identify a number of subgroups of (*GR*_{n}(*F*),⚬) and then advance to show that its particular subgroup is embedded in a particular subgroup of the well-known general linear group (*GR*_{n}(*F*),•). Furthermore, we shall investigate isomorphic relationship between some subgroups of (*GR*_{n}(*F*),⚬).

**Volume 22** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4