Volume 22, 2016, Number 3

Volume 22Number 1Number 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 = {D1, D2, …, Dk} of the vertex set of G is said to be a domatic partition or simply a d-partition of G if each class Di 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 P3
Original research paper. Pages 54—67
Ali H. Hakami
Full paper (PDF, 245 Kb) | Abstract

Let Q(x) = Q(x1, x2, …, xn) be a quadratic form with integer coefficients, p be an odd prime and ||x|| = maxi|xi|. A solution of the congruence Q(x) ≡ 0 (mod p3) is said to be a primitive solution if pxi for some i. We prove that if p > A; where A = 5·241; then this congruence has a primitive solution, with ||x|| < 34p3/2; 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, Fn(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.

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