Volume 23, 2017, Number 3

In this paper, we prove that if , where , is a -quadruple in the ring of Gaussian integers, then .

Associated Stirling numbers of first and second kind are usually found in the literature in various forms of stairs depending on their order . Yet, it is shown in this note that all of these numbers can be arranged, through a linear transformation, in the same arithmetical triangle structure as the “Pascal’s triangle”.

We give an alternative proof of a formula that generalizes Hermite’s identity. Instead involving modular arithmetic, our short proof relies on the Fourier-type expansion for the floor function and on a trigonometric formula.

In this work, we consider some algebraic properties of -balancing numbers. We deduce some formulas for the greatest common divisor of -balancing numbers, divisibility properties of -balancing numbers, sums of -balancing numbers and simple continued fraction expansion of -balancing numbers.

Dris gave numerical bounds for the sum of the abundancy indices of and , where is an odd perfect number, in his master’s thesis. In this note, we show that improving the limits for this sum is equivalent to obtaining nontrivial bounds for the Euler prime .

In this article we show that the following Pillai’s conjecture is the -th prime number)

can be established in terms of gaps between consecutive primes. We also study general sequences that have this property. We call these sequences Pillai sequences. We prove that the sequence of perfect powers is a Pillai-sequence.

A natural number is said to be quasiperfect if where is the sum of the positive divisors of . No quasiperfect number is known. If a quasiperfect number exists and if is the number of distinct prime factors of then G. L. Cohen has proved while H. L. Abbott et. al have shown if . In this paper we first prove that every quasiperfect numbers has an odd number of special factors (see definition 2.3 below) and use it to show that if which refines the result of Abbott et.al. Also we provide an alternate proof of Cohen’s result when .

In this paper, we make a contribution to the enumeration of permutations avoiding a quadruples of 4-letter patterns by establishing a Wilf class composed of 19 symmetry classes.

This paper develops properties of recurrence sequences defined from circulant matrices obtained from the characteristic polynomial of the Pell-Padovan sequence. The study of these sequences modulo m yields cyclic groups and semigroups from the generating matrices. Finally, we obtain the lengths of the periods of the extended sequences in the extended triangle groups and   for as applications of the results obtained.

Two new two-dimensional extensions of the Fibonacci sequence are introduced. Explicit formulas for their -th members are given.

A set of vertices S is said to dominate the graph G if for each v ∉ S, there is a vertex u ∈ S with v adjacent to u. The minimum cardinality of any dominating set is called the domination number of G and is denoted by γ(G). A dominating set D of a graph G = (V, E) is a non-split dominating set if the induced graph ⟨V – D⟩ is connected. The non-split domination number γ_ns(G) is the minimum cardinality of a non-split domination set. The purpose of this paper is to initiate the investigation of those graphs which are critical in the following sense: A graph G is called vertex domination critical if γ(G − v) < γ(G) for every vertex v in G. A graph G is called vertex non-split critical if γ_ns(G − v) < γ_ns(G) for every vertex v in G. Thus, G is k–γ_ns-critical if γ_ns(G) = k, for each vertex v ∈ V(G), γ_ns(G − v) < k. A graph G is called edge domination critical if (G + e) < (G) for every edge e in G. A graph G is called edge non-split critical if γ_ns(G + e) < γ_ns(G) for every edge e ∈ G. Thus, G is k–γ_ns-critical if γ_ns(G) = k, for each edge e ∈ G, γ_ns(G + e) < k. First we have constructed a bound for a non-split domination number of a subdivision graph S(G) of some particular classes of graph in terms of vertices and edges of a graph G. Then we discuss whether these particular classes of subdivision graph S(G) are γ_ns-critical or not with respect to vertex removal and edge addition.