Volume 32, 2026, Number 1 (Online First)

Volume 32 ▶ Number 1 (Online First)

Volume opened: 1 February 2026
Status: In progress

A note on periodic linear recurrence relations
Original research paper. Pages 1–4
József Bukor
Full paper (PDF, 173 Kb) | Abstract

We provide an elementary proof of the fact that a sequence defined by a linear recurrence relation with integer coefficients is periodic if and only if all characteristic roots are distinct roots of unity. Additionally, we discuss the case in which the coefficients of the recurrence relation are restricted to the set {–1,0,1}.

Note on the irrationality of certain infinite series
Original research paper. Pages 5–14
Pavel Rucki
Full paper (PDF, 213 Kb) | Abstract

The aim of this paper is to introduce new criteria for real infinite series that satisfy a specific property and yield an irrational sum. These criteria are based on an extension of previous ideas proposed by Erdős. The paper includes several illustrative examples.

Characterizations of L-additive functions via generalized arithmetic convolutions
Original research paper. Pages 15–22
Champak Talukdar, Debashis Bhattacharjee and Helen K. Saikia
Full paper (PDF, 189 Kb) | Abstract

This paper investigates the properties of L-additive functions within the algebraic frameworks of two generalized arithmetic convolutions: the K-convolution and Narkiewicz’s A-convolution. We establish the foundational algebraic context for these operations by citing the established conditions for their associativity and commutativity. Our main results provide rigorous characterization theorems for completely additive and L-additive functions, which manifest as Leibniz-type rules that these functions satisfy with respect to the convolutions. Furthermore, we provide insightful, non-trivial examples using classical arithmetic functions to illustrate the mechanics of these characterizations, thereby demonstrating the utility of the generalized convolution framework in the study of arithmetic derivatives and their generalizations.

Computing a maximal clique of graphs of cofinite submonoids
Original research paper. Pages 23–42
Anam Shahzadi and Muhammad Ahsan Binyamin
Full paper (PDF, 312 Kb) | Abstract

A graph $G_{\mathfrak{S}}$ is called an $\mathfrak{S}(\tau,\mathfrak{e})$ -graph if there exists a numerical semigroup $\mathfrak{S}$ with multiplicity $\tau$ and embedding dimension $\mathfrak{e}$ such that $V(G_{\mathfrak{S}})=\{v_{\alpha}:\alpha \in \N_{0}\setminus \mathfrak{S} \}$ and $E(G_{\mathfrak{S}})=\{v_{\alpha}v_{\beta}\Leftrightarrow \alpha+\beta \in \mathfrak{S}\}.$ In this article, we give an algorithmic way to compute the clique~number and the minimum degree of $\mathfrak{S}(\tau,3)$ -graphs, where $\mathfrak{S}$ is a class of symmetric numerical semigroups with arbitrary multiplicity and embedding dimension $3$ . On this basis, we give some bounds for the atom bond connectivity index of graphs $G_{\mathfrak{S}}$ in terms of Randić connectivity index, the first and second Zagreb indices, the maximum and minimum degrees, and the clique number.

Construction of generalized bicomplex Leonardo numbers
Original research paper. Pages 43–51
Murat Turan and Sıddıka Özkaldı Karakuş
Full paper (PDF, 220 Kb) | Abstract

In this paper, we introduce a new class of bicomplex numbers whose components are expressed in terms of bicomplex Leonardo numbers. The motivation for this study arises from the growing interest in generalizations of well-known integer sequences within hypercomplex number systems, which reveal deeper algebraic and geometric properties. First, we define the bicomplex Leonardo numbers and establish their fundamental recurrence relation. Then, we derive a Binet-like formula, which serves as a powerful analytical tool for exploring further identities and relationships.

By employing this Binet-like representation, we obtain several new results, including summation formulas, d’Ocagne’s identity, Catalan’s identity, and Cassini’s identity for bicomplex Leonardo numbers. These identities not only extend classical number-theoretic properties into the bicomplex domain but also demonstrate structural consistencies across related algebraic systems. Furthermore, we establish an important connection between the Catalan and Cassini identities, revealing an intrinsic relationship that enhances the understanding of their interdependence within the bicomplex setting.

Explicit evaluation of some families of log-sine integrals via the hypergeometric mechanism and their applications
Original research paper. Pages 52–75
Shakir Hussain Malik and Mohammad Idris Qureshi
Full paper (PDF, 311 Kb) | Abstract

In this paper, we present explicit analytical expressions for certain families of log-sine definite integrals: $\int_0^{2\pi} x^{m}\{ \ln(2\sin\frac{x}{2})\}^{n}dx \ (m\in\mathbb{N}_{0}, n\in\mathbb{N})$ , expressed in terms of multiple hypergeometric functions of the Kampé de Fériet with arguments $\pm 1$ and the Riemann zeta functions. As applications, we establish several mixed summation formulas (79), (81) and (83) involving the generalized hypergeometric functions $_3F_2(1)$ , $_5F_4(1)$ and $_7F_6(1)$ . Moreover, a collection of possibly new summation formulas (42), (52), (54), (56), (58), (62), (64), (66), (70), (72), (74) and (76) for multiple hypergeometric functions of the Kampé de Fériet are derived. In addition, mixed relations (80), (82) and (84) involving the Riemann zeta functions are also established.

On a family of sums of powers of the floor function and their links with generalized Dedekind sums
Original research paper. Pages 76–87
Steven Brown
Full paper (PDF, 265 Kb) | Abstract

In this paper we are concerned with a family of sums involving the floor function. With $r$ a nonnegative integer and $n$ and $m$ positive integers we consider the sums

$\mathbf{S}_{r}(n,m):=\sum_{k=1}^{n-1}{\left\lfloor \frac{km}{n}}\right\rfloor ^r.$

While a formula for $\mathbf{S}_1$ is well known, we provide closed-form formulas for $\mathbf{S}_2$ and $\mathbf{S}_3$ as well as the reciprocity laws they satisfy. Additionally, one can find a closed-form formula for the classical Dedekind sum using the Euclidean algorithm. Finally, we provide a general formula for $\mathbf{S}_r$ showing its dependency on generalized Dedekind sums.

On certain arithmetical functions connected with the prime factorization of an integer
Original research paper. Pages 88–95
József Sándor
Full paper (PDF, 250 Kb) | Abstract

If $1< n = \displaystyle\prod_{i=1}^r p_i^{a_i}$ is the prime factorization of integer $n,$ we study the arithmetical functions

$M(n) = \displaystyle\prod_{i=1}^r a_i^{p_i}, \quad F(n) = \displaystyle\prod_{i=1}^r p_i^{1/a_i},\quad G(n) = \displaystyle\prod_{i=1}^r a_i^{1/p_i}.$

Determinant-preserving blind signatures from Hadamard-type t-Jacobsthal–Leonardo sequences
Original research paper. Pages 96–111
Elahe Mehraban, Reza Ebrahimi Atani, Ömür Deveci and Ghadir Golkarian
Full paper (PDF, 272 Kb) | Abstract

In this paper, we introduce a new class of sequences called the termed Hadamard-type t-Jacobsthal Leonardo sequence which is generated by applying a Hadamard-type product to the characteristic polynomials of the t-Jacobsthal and Leonardo sequences. We derive fundamental algebraic properties of these sequences including determinant formulas, combinatorial identities, and exponential representations, and building on these mathematical results, we construct a novel blind signature scheme in which the public and secret keys are represented as companion matrices derived from the new sequences. The proposed scheme ensures correctness through determinant-preserving transformations and achieves blindness and unforgeability under matrix- based key assumptions. We provide security analysis within the standard cryptographic framework and discuss efficiency aspects compared with existing blind signature constructions. Our results demonstrate that Hadamard-type t-Jacobsthal–Leonardo matrices can serve as a new algebraic foundation for cryptographic protocols, thereby linking structured number-theoretic sequences with provably secure digital signature mechanisms.

Diophantine equations for additive Pell numbers in Pell, Pell–Lucas, and Modified Pell numbers
Original research paper. Pages 112–119
Ahmet Emin and Ahmet Daşdemir
Full paper (PDF, 171 Kb) | Abstract

This paper investigates the Diophantine equations arising from ternary additive problems of Pell, Pell–Lucas, and Modified Pell numbers. Specifically, we characterize all integer solutions to the equation ${P_n} + {P_m} + {P_r} = {X_k}$ , $X \in \left\{ {{P},{Q},{R}} \right\}$ , where ${P_i}$ , ${Q_i}$ , and ${R_i}$ denote the $i$ -th terms of the Pell, Pell–Lucas, and Modified Pell sequences, respectively. By leveraging recurrence relations, Binet’s formulas, and Carmichael’s Primitive Divisor Theorem, we provide the first complete classification of solutions to this ternary additive problem. Our results reveal several parametric and singular solutions. Furthermore, we reduce prior results to binary sums of the form ${P_n} + {P_m} = {X_k}$ as special instances of our framework.

Star complementary characterization of oriented graphs whose skew spectral radius does not exceed 2
Original research paper. Pages 120–132
Zoran Stanić
Full paper (PDF, 300 Kb) | Abstract

We employ the method of star complements to classify all oriented graphs whose skew spectrum lies within the interval [–2, 2]. At the same time, we provide a structural characterisation of these graphs, showing that, with the sole exception of exactly one graph of order 14, every maximal oriented graph possessing this spectral property is determined by a fixed oriented cycle serving as a star complement for either –2 or 2. The exceptional oriented graph is uniquely determined by a fixed 7-vertex oriented path acting as the star complement. This work may be regarded as a counterpart to [13], where the corresponding oriented graphs were determined via associated signed graphs, without the present characterisation.

A note on the self-convolution of the Tribonacci sequence
Original research paper. Pages 133–136
Karol Gryszka
Full paper (PDF, 168 Kb) | Abstract

We present a simple formula for the self-convolution of the Tribonacci numbers. The resulting identity is considerably simpler than that obtained in a recent publication.

Padovan numbers which are concatenations of three Padovan or Perrin numbers
Original research paper. Pages 137–149
Fatih Erduvan
Full paper (PDF, 281 Kb) | Abstract

This paper presents all Padovan numbers that can be written as the concatenation of three Padovan or Perrin numbers under a certain constraint. Namely, we consider the Diophantine equations

$P_{k}=10^{d+l}P_{m}+10^{l}P_{n}+P_{r}$

and

$P_{k}=10^{d+l}R_{m}+10^{l}R_{n}+R_{r}$

where $k,m,n,r,d$ and $l$ are positive integers satisfying $n\leq m$ . The parameters $d$ and $l$ denote the numbers of digits in the integers $P_{n}$ (or $R_{n}$ ) and $P_{r}$ (or $R_{r}$ ), respectively. The solutions to these equations can be written in the form $P_{18}=\overline{P_{2}P_{2}P_{6}}=114$ for all $m,n,r\geq2$ and, similarly, $P_{30}=\overline{R_{3}R_{3}R_{12}}=3329$ , $P_{33}=\overline{R_{7}R_{7}R_{13}}=7739$ for all $m\geq3$ and $n,r\geq1$ .

A note on generalized Zumkeller numbers
Original research paper. Pages 150–161
Jayanta Kalita and Helen K. Saikia
Full paper (PDF, 254 Kb) | Abstract

A positive integer n is called an e-Zumkeller number if the exponential divisors of n can be partitioned into two disjoint subsets of equal sum. Generalizing the concept of e-Zumkeller numbers, we define multiplicatively e-Zumkeller numbers. In addition, generalizing the concepts of s-Zumkeller numbers and m-Zumkeller numbers, we define two new variants of Zumkeller numbers called (+s)-Zumkeller numbers and (+m)-Zumkeller numbers, considering even positive divisors. We present some examples in support of these two types of positive integers and study their characteristics.

On ψ-quadratic k-tuples and their generalizations
Original research paper. Pages 162–172
Stoyan Dimitrov
Full paper (PDF, 196 Kb) | Abstract

In this paper, we introduce the notion of ψ-quadratic k-tuples. We establish the non-existence of ψ-quadratic pairs and provide a partial analysis of ψ-quadratic triples. We also give examples and propose generalizations of these new concepts.

The generalized Horadam model and the fundamental sequence of a nonhomogeneous linear recursive relation
Original research paper. Pages 173–186
Mustapha Rachidi, Irene Magalhães Craveiro and Elen Viviani Pereira Spreafico
Full paper (PDF, 301 Kb) | Abstract

This study concerns some properties of the generalized nonhomogeneous Horadam model and its application to four types of known sequences of numbers. Regarding the four sequences of this model, we provide results on combinatorial and linear expressions, and also on the analytical representation of Binet. Our approach is based on results we have developed regarding the general setting of nonhomogeneous linear recursive sequences, especially the fundamental sequence related to their homogeneous part.

On a generalization of the Hosoya triangle
Original research paper. Pages 187–197
Turhan Çifçi, Hamza Menken and Orhan Dişkaya
Full paper (PDF, 342 Kb) | Abstract

This paper introduces the Fibonacci polynomial triangle, inspired by the structure of the Hosoya triangle and constructed using Fibonacci polynomials. We then present and rigorously prove a series of novel identities and fundamental properties specifically associated with this Fibonacci polynomial triangle. These findings contribute to a deeper understanding of the algebraic structures and combinatorial patterns that emerge when Fibonacci polynomials are organized in such a triangular fashion, revealing new relationships and characteristics within this framework. This exploration aims to further elucidate the rich interplay between polynomial sequences and triangular constructions.

On the existence of Hamiltonian cycles in hypercubes
Original research paper. Pages 198–206
Gabriele Di Pietro and Marco Ripà
Full paper (PDF, 222 Kb) | Abstract

Building on the results of our previous work on Euclidean leaper tours, considering all integers $k>1$ and $h>0$ , we study the existence of Hamiltonian cycles in the vertex set $C(2,k):=\{0,1\}^k$ of the $k$ -dimensional hypercube when the Euclidean distance between consecutive vertices is fixed. Since the distance between two vertices of $C(2,k)$ is $\sqrt{h}$ for some integer $h$ , the problem amounts to determining for which integers $k$ and $h$ there exists a Hamiltonian cycle whose associated Euclidean distance is $\sqrt{h}$ . In this paper, we prove that such cycles exist if and only if $h$ is odd and $1 \leq h \leq k-1$ . As a result, for all integers $a \geq 0$ , $b \geq a$ with $b>0$ , we provide a necessary and sufficient condition for the existence of closed Euclidean $(a,b)$ -leaper tours on $2 \times 2 \times \cdots \times 2$ chessboards, where the associated distance equals $\sqrt{a^2+b^2}$ .

Two new arithmetic operations
Original research paper. Pages 207–213
Krassimir Atanassov and József Sándor
Full paper (PDF, 167 Kb) | Abstract

Two arithmetic operations are introduced and some of their properties are studied. It is proved that they can be operations of semi-groups, but not of monoids. It is shown that their reverse operations are not one-valued. Some connections between the new operations and the well-known arithmetic functions $\varphi, \psi, \sigma$ are shown.

Matrix representations of an extended family of Horadam polynomials
Original research paper. Pages 214–228
Hayrullah Özimamoğlu
Full paper (PDF, 291 Kb) | Abstract

In this article, we describe the concept of $d$ -Horadam polynomials, which is a generalization of the classical Horadam polynomials $W_{n}(0,W_{1}(x);p(x),q(x))$ . We derive essential characteristics of these newly defined polynomials, such as their generating function, a Binet-type expression, several combinatorial relations, and summation identities. Subsequently, we construct the infinite matrix of d-Horadam polynomials, which can be represented as a Riordan array. Moreover, by employing the Riordan approach, we establish two distinct decompositions of the infinite Pascal matrix involving the d-Horadam polynomials.

Some Fibonacci congruences with square moduli
Original research paper. Pages 229–241
Anthony G. Shannon, Peter J.-S. Shiue and Christopher Saito
Full paper (PDF, 715 Kb) | Abstract

Fibonacci congruence with prime moduli have been extensively studied. Square moduli are obviously not prime numbers, so why study such congruences? The answer is to investigate patterns for a special class of moduli, since there are many Fibonacci identities containing squares of Fibonacci numbers. Previous studies do not seem to have revealed anything of note that is notably distinct from other composite moduli.

Extremal graphs of multiplicative sum Zagreb index
Original research paper. Pages 242–253
Chunlei Xu, Guanru Li, Yiming Lei and Junming Hu
Full paper (PDF, 269 Kb) | Abstract

The multiplicative sum Zagreb index, regarded as the multiplicative version of the first Zagreb index, is defined as the product of the degree sum over all pairs of adjacent vertices. In this paper, we determine the extremal values of the multiplicative sum Zagreb index for three class of trees. Some bounds on the multiplicative sum Zagreb index are obtained, and the corresponding extremal graphs with these bounds are characterized.

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-NP7/2/03.12.2025. Bulgarian National Science Fund bears no liability for the content of the published materials.

Volume 32 ▶ Number 1 (Online First)

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