Volume 28, 2022, Number 2 (Online First)

Volume 28Number 1 ▷ Number 2 (Online First)


  • Volume opened: 23 March 2022
  • Status: In progress

On wide-output sieves
Original research paper. Pages 147—158
Alessandro Cotronei
Download PDF (282 Kb) | Abstract

We describe and compare several novel sieve-like methods. They assign values of several functions (i.e., the prime omega functions ω and Ω and the divisor function d) to each natural number in the considered range of integer numbers. We prove that in some cases the algorithms presented have a relatively small computational complexity. A more detailed output is indeed obtained with respect to the original Sieve of Eratosthenes.


Remark on the transcendence of real numbers generated by Thue–Morse along squares
Original research paper. Pages 159—166
Eiji Miyanohara
Download PDF (205 Kb) | Abstract

In 1929, Mahler proved that the real number generated by Thue–Morse sequence is transcendental. Later, Adamczewski and Bugeaud gave a different proof of the transcendence of this number using a combinatorial transcendence criterion. Moreover, Kumar and Meher gave the generalization of the combinatorial transcendence criterion under the subspace Lang conjecture. In this paper, we prove under the subspace Lang conjecture that the real number generated by Thue–Morse along squares is transcendental by using the combinatorial transcendence criterion of Kumar and Meher.


A generalization of multiple zeta values. Part 1: Recurrent sums
Original research paper. Pages 167—199
Roudy El Haddad
Download PDF (340Kb) | Abstract

Multiple zeta star values have become a central concept in number theory with a wide variety of applications. In this article, we propose a generalization, which we will refer to as recurrent sums, where the reciprocals are replaced by arbitrary sequences. We introduce a toolbox of formulas for the manipulation of such sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order m to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we use these theorems to derive new results for multiple zeta star values and recurrent sums of powers.


A generalization of multiple zeta values. Part 2: Multiple sums
Original research paper. Pages 200—233
Roudy El Haddad
Download PDF (348 Kb) | Abstract

Multiple zeta values have become of great interest due to their numerous applications in mathematics and physics. In this article, we present a generalization, which we will refer to as multiple sums, where the reciprocals are replaced with arbitrary sequences. We develop formulae to help with manipulating such sums. We develop variation formulae that express the variation of multiple sums in terms of lower order multiple sums. Additionally, we derive a set of partition identities that we use to prove a reduction theorem that expresses multiple sums as a combination of simple sums. We present a variety of applications including applications concerning polynomials and MZVs such as generating functions and expressions for \zeta(\{2p\}_m) and \zeta^\star(\{2p\}_m). Finally, we establish the connection between multiple sums and a type of sums called recurrent sums. By exploiting this connection, we provide additional partition identities for odd and even partitions.


Note on some sequences having periods that divide (p p − 1) / (p − 1)
Original research paper. Pages 234—239
Abdelkader Benyattou and Miloud Mihoubi
Download PDF (185 Kb) | Abstract

In this paper, we use the properties of the classical umbral calculus to determine sequences related to the Bell numbers and having periods divide \left(p^{ \, p}-1\right) / \left(p-1\right).


On the equation f(n2Dnm + m2) = f 2(n) − Df(n)f(m) + f 2(m)
Original research paper. Pages 240—251
B. M. Phong and R. B. Szeidl
Download PDF (192 Kb) | Abstract

We give all solutions f:\Bbb N \to \Bbb C of the functional equation

    \[f(n^2-Dnm+m^2)=f^2(n)-Df(n)f(m)+f^2(m),\]

where D\in\{1,2\}.


Some equalities and binomial sums about the generalized Fibonacci number un
Original research paper. Pages 252—260
Yücel Türker Ulutaş and Derya Toy
Download PDF (146 Kb) | Abstract

In this study, we take the generalized Fibonacci sequence \{u_{n}\} as u_{0}=0,u_{1}=1 and \ u_{n}=ru_{n-1}+u_{n-2} for n>1, where r is a non-zero integer. Based on Halton’s paper in [4], we derive three interrelated functions involving the terms of generalized Fibonacci sequence \{u_{n}\}. Using these three functions we introduce a simple approach to obtain a lot of identities, binomial sums and alternate binomial sums involving the terms of generalized Fibonacci sequence \{u_{n}\}.


q-Fibonacci bicomplex and q-Lucas bicomplex numbers
Original research paper. Pages 261—275
Fügen Torunbalcı Aydın
Download PDF (252 Kb) | Abstract

In the paper, we define the q-Fibonacci bicomplex numbers and the q-Lucas bicomplex numbers, respectively. Then, we give some algebraic properties of the q-Fibonacci bicomplex numbers and the q-Lucas bicomplex numbers.


A note on the Aiello–Subbarao conjecture on addition chains
Original research paper. Pages 276—280
Hatem M. Bahig
Download PDF (231 Kb) | Abstract

Given a positive integer x, an addition chain for x is an increasing sequence of positive integers 1=c_0,c_1, \ldots , c_n=x such that for each 1\leq k\leq n, c_k=c_i+c_j for some 0\leq i\leq j\leq k-1. In 1937, Scholz conjectured that for each positive integer x, \ell(2^x-1) \leq \ell(x)+ x-1, where \ell(x) denotes the minimal length of an addition chain for x. In 1993, Aiello and Subbarao stated the apparently stronger conjecture that there is an addition chain for 2^x-1 with length equals to \ell(x)+x-1. We note that the Aiello–Subbarao conjecture is not stronger than the Scholz (also called the Scholz–Brauer) conjecture.


On certain rational perfect numbers
Original research paper. Pages 281—285
József Sándor
Download PDF (146 Kb) | Abstract

We study equations of type \sigma(n) = \dfrac{k+1}{k} \cdot n+a, where a\in \{0, 1, 2, 3\}, where k and n are positive integers, while \sigma(n) denotes the sum of divisors of n.


Perfect squares in the sum and difference of balancing-like numbers
Original research paper. Pages 286—301
M. K. Sahukar, Zafer Şiar, Refik Keskin and G. K. Panda
Download PDF (231 Kb) | Abstract

In this study, we deal with the existence of perfect powers which are sum and difference of two balancing numbers. Moreover, as a generalization we explore the perfect squares which are sum and difference of two balancing-like numbers, where balancing-like sequence is defined recursively as G_{n+1}=AG_n-G_{n-1} with initial terms G_0=0,G_1=1 for A \geq 3.


This volume of the International Journal “Notes on Number Theory and Discrete Mathematics”
is published with the financial support of the Bulgarian National Science Fund, Grant Ref. No.
KP-06-NP3/43/2021.


Volume 28Number 1 ▷ Number 2 (Online First)

Comments are closed.