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

**Alwyn Horadam: The man and his mathematics**

*In Memoriam. Pages 1—4 *

A. G. Shannon

Full paper (PDF, 349 Kb)

**Combined 3-Fibonacci sequences from a new type**

*Original research paper. Pages 5—8*

Krassimir T. Atanassov and Anthony G. Shannon

Full paper (PDF, 124 Kb) | Abstract

New combined 3-Fibonacci sequences are introduced and the explicit formulae for their *n*-th members are given.

**A generalization of Euler’s Criterion to composite moduli**

*Original research paper. Pages 9—19 *

József Vass

Full paper (PDF, 226 Kb) | Abstract

A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler’s Criterion through that of Euler’s Theorem, and the concepts of order and primitive roots. Idempotent numbers play a central role in this effort.

**Upper bound of embedding index in grid graphs **

*Original research paper. Pages 20—35 *

M. Kamal Kumar and R. Murali

Full paper (PDF, 359 Kb) | Abstract

A subset *S* of the vertex set of a graph *G* is called a dominating set of *G* if each vertex of *G* is either in *S* or adjacent to at least one vertex in *S*. A partition *D* = {*D*_{1}, *D*_{2}, …, *D*_{k}} of the vertex set of *G* is said to be a domatic partition or simply a d-partition of *G* if each class *D*_{i} of *D* is a dominating set in *G*. The maximum cardinality taken over all d-partitions of G is called the domatic number of *G *denoted by *d*(*G*). A graph *G* is said to be domatically critical or *d*-critical if for every edge *x* in *G*, *d*(*G–x*) < *d*(*G*), otherwise *G* is said to be domatically non *d*-critical. The embedding index of a non d-critical graph *G* is defined to be the smallest order of a *d*-critical graph *H* containing *G* as an induced subgraph denoted by *q*(*G*). In this paper, we find the upper bound of *q*(*G*) for grid graphs.

**On integers that are uniquely representable by modified arithmetic progressions**

*Original research paper. Pages 36—44 *

Sarthak Chimni, Soumya Sankar and Amitabha Tripathi

Full paper (PDF, 204 Kb) | Abstract

For positive integers *a, d, h, k*, gcd(*a, d*) = 1, let *A* = {*a, ha+d, ha+2d, …, ha+kd*}. We characterize the set of nonnegative integers that are uniquely representable by nonnegative integer linear combinations of elements of *A*.

**On sum and ratio formulas for balancing-like sequences**

*Original research paper. Pages 45—53 *

Ravi Kumar Davala and G. K. Panda

Full paper (PDF, 172 Kb) | Abstract

Certain sum formulas with terms from balancing-like and Lucas-balancing-like sequences are discussed. The resemblance of some of these formulas with corresponding sum formulas involving natural numbers are exhibited

**Small primitive zeros of quadratic forms mod ***P*^{3}

*Original research paper. Pages 54—67 *

Ali H. Hakami

Full paper (PDF, 245 Kb) | Abstract

Let *Q*(*x*) = *Q*(*x*_{1}, *x*_{2}, …, *x*_{n}) be a quadratic form with integer coefficients, *p* be an odd prime and ||*x*|| = max_{i}|*x*_{i}|. A solution of the congruence *Q*(*x*) ≡ 0 (mod *p*^{3}) is said to be a primitive solution if *p* ∤ *x*_{i} for some *i*. We prove that if *p *>* A*; where *A* = 5·2^{41}; then this congruence has a primitive solution, with ||*x*|| < 34*p*^{3/2}; provided that *n* ≥ 6 is even and *Q* is nonsinqular (mod *p*). Moreover, similar result is proven for cube boxes centered at the origin with edges of arbitrary lengths. These two results are extension of the quadratic forms problems

**Embedding the unitary divisor meet semilattice in a lattice**

*Original research paper. Pages 68—78 *

Pentti Haukkanen

Full paper (PDF, 245 Kb) | Abstract

A positive divisor *d* of a positive integer *n* is said to be a unitary divisor of *n* if (*d, n/d*) = 1. The set of positive integers is a meet semilattice under the unitary divisibility relation but not a lattice since the least common unitary multiple (lcum) does not always exist. This meet semilattice can be embedded to a lattice; two such constructions have hitherto been presented in the literature. Neither of them is distributive nor locally finite. In this paper we embed this meet semilattice to a locally finite distributive lattice. As applications we consider semimultiplicative type functions, meet and join type matrices and the Möbius function of this lattice.

**Right circulant matrices with ratio of the elements of Fibonacci and geometric sequence**

*Original research paper. Pages 79—83*

Aldous Cesar F. Bueno

Full paper (PDF, 164 Kb) | Abstract

We introduce the right circulant matrices with ratio of the elements of Fibonacci and geometric sequence. Furthermore, we investigate their eigenvalues, determinant, Euclidean norm, and inverse.

**Some characteristics of the Golden Ratio family**

*Original research paper. Pages 84—89 *

J. V. Leyendekkers and A. G. Shannon

Full paper (PDF, 157 Kb) | Abstract

Research into properties of generalizations of the Golden Ratio has considered various forms of extreme and mean ratios. This is considered here too within the framework of a family of surds, ½ √(1+*a*), and generalized Fibonacci numbers, *F*_{n}(*a*), with the ordinary Fibonacci numbers being the particular case when a = 5.

**On the irrationality of √N**

*Original research paper. Pages 90—91 *

József Sándor and Edith Egri

Full paper (PDF, 107 Kb) | Abstract

We offer a new proof of the classical fact that √*N* is irrational, when *N* is not a perfect square.

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