Volume 26, 2020, Number 2

A divisor of a positive integer is called a unitary divisor if and is called a bi-unitary divisor of if the greatest common unitary divisor of and is unity. The concept of a bi-unitary divisor is due to D. Surynarayana (1972). Let denote the sum of the bi-unitary divisors of . A positive integer is called a bi-unitary multiperfect number if for some . For we obtain the bi-unitary triperfect numbers.

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is Part II in a series of papers on even bi-unitary multiperfect numbers. In the first part we found all bi-unitary triperfect numbers of the form , where and is odd; the only one being . In this second part we find all bi-unitary triperfect numbers in the cases and . For the only one is , and for they are , , and

Let be an odd perfect number with special prime . In this article, we provide an alternative proof for the biconditional that holds if and only if . We then give an application of this result to the case when is a square.

Restrictive factor and extension factor are two arithmetic functions, introduced by the authors. In the paper, some of their extensions are introduced and some of the basic properties of the newly defined functions are studied.

We obtain explicit lower bound on multiplicative orders of all elements in finite field extensions generated by a root of unity. The bound does not depend on any unknown constant. The result of Ahmadi, Shparlinski and Voloch [1] is a consequence of our main result.

In this paper, for a given natural number we count the number of -tuples with certain conditions such that lcm In the process, we derived different arithmetic functions.

The purpose of this study is to obtain a new generalized quaternions sequences by using hyperbolic functions with second order recurrence sequences. First of all, we define the symmetrical second order hyperbolic sine and the symmetrical second order hyperbolic cosine quaternions. Then, we investigate norms and some relations between these type of quaternions. We also obtain generating functions, Binet formulas, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity of second order hyperbolic quaternions sequences.

Let be the -th balancing number. In this paper, we give some explicit expressions of and . We also consider the convolution identities with binomial coefficients:
.
This type can be generalized, so that is a special case of the number , where () with and .

We use Abel’s summation formula and the method of partial fraction decomposition to study infinite series involving generalized harmonic numbers of any positive integral order, with any positive integral power.

By the theory of Pell equation and congruence, we study the problem about the linear combination of two polygonal numbers is a perfect square. Let denote the -th -gonal number. We show that if , is not a perfect square, and there is a positive integer solution of satisfying

   

then the Diophantine equation has infinitely many positive integer solutions . Moreover, we give conditions about such that the Diophantine equation has infinitely many positive integer solutions .

Let be a positive integer and be a non-negative integer. We define

   

and to be the number of ways can be expressed as a difference of two elements from the sequence . Nyblom found closed expressions for and in terms of some restricted number-of-divisors functions. Here we re-establish these two results of Nyblom in a relatively simple way. Along with the other interpretations for , an expression for is presented in terms of restricted form of and . Also we consider another function due to Nyblom, denoted , which counts the number of partitions of with parts in arithmetic progression having common difference . Nyblom and Evan found a simple expression for and put in terms of a divisor-counting functions when . Here we re-establish Nyblom’s expression for , and find equinumerous expressions for when . Finally, we present the following generalised version of : given a set of positive integers say, , we denote by , the number of ways can be written as a difference of two elements from the set . And we express in terms of partition enumerations when some restrictions are imposed upon the elements of . We close with the hint that, boundedness of together with the divergence of disproves Erdős arithmetic progression conjecture.

In this paper, a new class of q-Hermite-based Frobenius-type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

This paper extends some work of Leonard Carlitz on rising binomial coefficients and hypergeometric series in the context of a result of Louis Saalschütz which has animated further work in a number of branches of mathematics as well as physics.

Let denote the Bernoulli numbers, and denote the Stirling numbers of the second kind. We prove the following explicit formula

   

To the best of our knowledge, the formula is new.

We obtain some closed form expressions for certain infinite sums related to Lucas and associated Lucas sequences. Some of our results generalize existing results concerning infinite sums involving Lucas sequences and others are variants of some existing results.

Two new closed forms for infinite series involving Fibonacci numbers and the Riemann zeta function are derived using standard methods from complex analysis. Also, expressions for the companion series with Lucas numbers are presented.

In this paper, circular-hyperbolic Fibonacci numbers and quaternions are defined. Also, some algebraic properties of circular-hyperbolic Fibonacci numbers and quaternions which are connected with circular-hyperbolic numbers and Fibonacci numbers are investigated. Furthermore, Honsberger’s identity, the generating function, Binet’s formula, d’Ocagne’s identity, Cassini’s identity, and Catalan’s identity for these quaternions are given.

We investigate an r-circulant matrix whose entries are Horadam numbers having arithmetic indices. We then solve for the eigenvalues, determinant, Euclidean norm and spectral norm of the matrix. Lastly, we present some special cases and some results on identities and divisibility.

In this paper, we consider generalized
Fibonacci quadratics and give solutions of them under certain conditions.
For example, for odd number under condition , the equation

   

has rational roots.

This study shows that the generalized order-k Pell–Lucas and Modified Pell numbers can be expressed in terms of the well-known Fibonacci numbers. Certain n-square Hessenberg matrices with permanents equal to the Fibonacci numbers are defined. These Hessenberg matrices are then extended to super-diagonal (0,1,2)-matrices. In particular, the permanents of the super-diagonal matrices are shown to equal the components of the generalized order-k Pell–Lucas and Modified Pell numbers, and also their sums. In addition, two computer algorithms regarding our results are composed.

The classic way to write down Pascal’s triangle leads to entries in alternating rows being vertically aligned. In this paper, we prove a linear relation on vertically aligned entries in Pascal’s triangle. Furthermore, we give an application of this relation to morphisms between hyperelliptic curves.