**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

*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

**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

*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*} of the vertex set of

_{k}*G*is said to be a domatic partition or simply a d-partition of

*G*if each class

*D*of

_{i}*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

*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

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

*Original research paper. Pages 54–67*

Ali H. Hakami

Full paper (PDF, 245 Kb) | Abstract

*Q*(

*x*) =

*Q*(

*x*

_{1},

*x*

_{2}, …,

*x*) be a quadratic form with integer coefficients,

_{n}*p*be an odd prime and ||

*x*|| = max

*|*

_{i}*x*|. A solution of the congruence

_{i}*Q*(

*x*) ≡ 0 (mod

*p*

^{3}) is said to be a primitive solution if

*p*∤

*x*for some

_{i}*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

*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

**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

*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

*N*is irrational, when

*N*is not a perfect square.