Volume 31, 2025, Number 4 (Online First)

Volume 31 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 (Online First)

Volume opened: 1 October 2025
Status: In progress

Theta function identities involving fourth power
Original research paper. Pages 683–688
Praveenkumar, Siddaraju, R. Rangarajan
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On page 241 of his Second Notebook, Ramanujan recorded one of his theta function identity, which involves the ratio of the fourth power of theta functions with respect to $\psi(q)$ . In this article, we give a new proof for this theta function identity. Also, we give a new proof of another identity with respect to $\varphi(q)$ established by B. C. Berndt and we establish two new theta function identities analogous to Ramanujan’s theta function identities.

Infinite series containing quotients of central binomial coefficients
Original research paper. Pages 689–695
Zhiling Fan
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By making use of the Wallis’ integral formulae and integration by parts, two classes of infinite series are evaluated, in closed form, in terms of $\pi$ and Riemann zeta function.

On split Narayana and Narayana–Lucas hybrid quaternions
Original research paper. Pages 696–717
Pankaj Kumar and Shilpa Kapoor
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In this paper, we introduce the novel concepts of split Narayana quaternions and split Narayana–Lucas quaternions within the innovative framework of hybrid numbers. We explore their deep connections with Narayana and Narayana–Lucas quaternions, uncovering new perspectives in this mathematical domain. Furthermore, we establish several fundamental properties, including recurrence relations, Binet formulas, generating functions, exponential generating functions, and other significant identities associated with these newly defined quaternions. Finally, to better illustrate these theoretical findings, we also provide a numerical simulation of split Narayana quaternions and split Narayana–Lucas hybrid quaternions.

On the equation $F(n^k - 1) = D$
Original research paper. Pages 718–727
I. Kátai, B. M. M. Khanh, B. M. Phong
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We prove that if $F$ is a completely multiplicative function and $k \in \{2, 3\}$ such that the equation $F(n^k - 1) = 1$ holds for every $n \in \mathbb{N}$ , $n > 1,$ then $F$ is the identity function. A similar result is proved for the equation $F(n^4 - 1) = 1$ assuming a suitable conjecture concerning prime numbers. The equation $F(n^3 + 1) = 1$ is also studied.

On some classes of binary matrices
Original research paper. Pages 728–735
Krasimir Yordzhev
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The work considers the set $\mathcal{L}_n^k$ of all $n\times n$ binary matrices having the same number of $k$ units in each row and each column. The article specifically focuses on the matrices whose rows and columns are sorted lexicographically. We examine some particular cases and special properties of this matrices. Finally, we demonstrate the relationship between the Fibonacci numbers and the cardinality of two classes of $\mathcal{L}_n^k$ -matrices with lexicographically sorted rows and columns.

Three Diophantine equations concerning the polygonal numbers
Original research paper. Pages 736–746
Yong Zhang, Mei Jiang and Qiongzhi Tang
Download paper (PDF, 201 Kb) | Abstract

Many authors investigated the problem about the linear combination of two polygonal numbers is a perfect square, i.e., the Diophantine equation

$mP_k(x)+nP_k(y)=z^2,$

where $P_k(x)$ denotes the $x$ -th $k$ -polygonal number and $m,n$ are positive integers. In this note, we continue the study of this problem in another direction and consider three Diophantine equations

$mP_k(x)-1=z^2,\quad mP_k(x)-nP_k(y)=z^2,\quad mP_k(x)-nP_k(y)=1.$

By the theory of Pell equation and congruence, we obtain some conditions such that the above three Diophantine equations have infinitely many positive integer solutions.

(G,F)-points on ℚ-algebraic varieties
Original research paper. Pages 747–760
Yangcheng Li and Hongjian Li
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Let $\mathbb{Q}$ be the field of rational numbers, and let $C$ be an algebraically closed field containing $\mathbb{Q}$ . Let $G\in \mathbb{Q}[x,y,z]$ be a polynomial, then the zero set of $G$ is $Z(G)=\{P\in C^n \mid G(P)=0\}$ . A set $V\subset C^n$ is called a $\mathbb{Q}$ -algebraic variety if $V = Z(G)$ for some polynomial $G$ in $\mathbb{Q}[x,y,z]$ . The set $V(G)=\{P\in\mathbb{Q}^3~|~G(P)=0\}$ is called the set of $\mathbb{Q}$ -rational points of $V$ . Let

$\begin{split} F:\quad &\mathbb{Q}^3\rightarrow \mathbb{Q}^3,\\ &(x,y,z)\mapsto (f(x),f(y),f(z)) \end{split}$

be a vector function, where $f\in \mathbb{Q}[x]$ . It is easy to show that the function obtained by the composition of $G$ and $F$ , denoted as $G\circ F$ , is still in $\mathbb{Q}[x,y,z]$ . Moreover, let $V(G\circ F)$ be the set of $\mathbb{Q}$ -rational points of the $\mathbb{Q}$ -algebraic variety corresponding to $G\circ F$ , i.e., $V(G\circ F)=\{P\in\mathbb{Q}^3~|~G\circ F(P)=0\}$ . A rational point $P$ is called a $(G,F)$ -point on $V(G)$ if $P$ belongs to the intersection of $V(G)$ and $V(G\circ F)$ , that is $P\in V(G)\cap V(G\circ F)$ . Denote $\langle G,F\rangle$ as the set consisting of all $(G,F)$ -points on $V(G).$ Obviously, $\langle G,F\rangle$ is the set of $\mathbb{Q}$ -rational points of a $\mathbb{Q}$ -algebraic variety, that is, $\langle G,F\rangle=\{P\in\mathbb{Q}^3~|~G(P)=0~\text{and}~G\circ F(P)=0\}$ . In this paper, we consider the algebraic variety $\langle G,F\rangle$ for some specific functions $G$ and $F$ . For these specific functions $G$ and $F$ , we prove that $\langle G,F\rangle$ will be isomorphic to a certain elliptic curve. We also analyze some properties of these elliptic curves.

Relationship between alternating sums of powers of integers and sums of powers of integers
Original research paper. Pages 761–767
Minoru Yamamoto
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In this note, we consider the alternating sums of powers of integers. We write alternating sum of powers of integers as the linear combination of sums of powers of integers. As the coefficients, the special value of the Euler polynomial appears.

On ψ-amicable numbers and their generalizations
Original research paper. Pages 768–775
Stoyan Dimitrov
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In this article, we study the properties of ψ-amicable numbers. We prove that their asymptotic density relative to the positive integers is zero. We also propose generalizations of ψ-amicable numbers.

Proofs of some geometric conjectures on the power sum congruence modulo a prime
Original research paper. Pages 776–784
Pentti Haukkanen
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The main purpose of this paper is to verify the geometric conjectures of Mustonen (2022) concerning the solutions and the number of solutions of the congruence

$x^n+y^n \equiv 0 \pmod{p},$

where $p$ is a prime. For $p>2$ , the nontrivial solutions lie on the “lines” $y \equiv cx \pmod{p},$ where $c$ ranges over the $n$ -th roots of $-1$ modulo $p$ . The total number of solutions is $1+(p-1)d$ if $d$ divides $(p-1)/2$ , and $0$ otherwise, where $d=\gcd(n, p-1)$ . For each $c$ , the lines are equally spaced.

A note on normal ordering of degenerate integral powers of number operator
Original research paper. Pages 785–791
Taekyun Kim, Dae San Kim and Kyo-Shin Hwang
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This study derives the normal ordering expansion of degenerate integral powers of the number operator, $(a^{\dagger}a)_{n,\lambda}$ , using recurrence relations for the coefficients and an operator action on number states. Here $a^{\dagger}$ and $a$ are respectively the boson creation and annihilation operators. We also determine the inverse of this normal ordering expansion. By analyzing diagonal coherent state elements of the degenerate integral powers of the number operator, we establish a combinatorial identity which yields a Dobinski-like formula for the degenerate Bell numbers at a specific value, connecting degenerate quantum operator calculus with combinatorics.

Cryptography using Fibonacci–Mersenne and Fibonacci-balancing p-sequences with a self-invertible matrix and the Affine–Hill cipher
Original research paper. Pages 792–818
Elahe Mehraban, T. Aaron Gulliver, Ömür Deveci and Evren Hincal
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In this paper, we define two new sequences using the Fibonacci $p$ -numbers, the generalized Mersenne numbers, and $m$ -balancing numbers. These sequences are obtained from the corresponding characteristic polynomials. The determinants and combinatorial and exponential representations of these new sequences are given. As an application, we provide two algorithms using these new sequences with self-invertible matrices and the Affine–Hill cipher.

On b-repdigit polygonal numbers
Original research paper. Pages 819–828
Adriana Mora and Eric Bravo
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We prove a finiteness theorem concerning repdigits in base $b\ge 2$ represented by a fixed quadratic polynomial. We also show that there is a finite number of polygonal numbers that are also $b$ -repdigits for all $b\ge 2$ provided that $(b,s)\notin \left\{\left(\frac{8(s-2)}{(s-4)^{2}}d+1,s\right):s\in [3,13]-\{4\}\right\}$ , where $s\ge 3$ denotes the number of sides of the polygon and $d\in \{1,2,\ldots,b-1\}$ . We illustrate this result by finding all triangular, pentagonal and heptagonal numbers that are also $b$ -repdigits for $b\in [2,9]$ . This paper is motivated by a previous work of Kafle, Luca, and Togbé who considered the same finiteness problem for $b=10$ to find all pentagonal and heptagonal numbers that are also repdigits.

Alternating generalized Fibonacci sequences
Original research paper. Pages 829–838
Carlos M. da Fonseca and Paulo Saraiva
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In a recent paper, K. T. Atanassov and A. G. Shannon introduced a Fibonacci-like sequence derived from the generalized Fibonacci sequence by incorporating alternating signs into the recurrence relation. They also proposed explicit formulas for this sequence. In this work, we present the generating function for the sequence using a matrix-based approach. Furthermore, we explore additional variations of the original definition.

Inequalities between some arithmetic functions, II
Original research paper. Pages 839–845
Krassimir Atanassov, József Sándor and Mladen Vassilev-Missana
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As a continuation of Part I (see [1]), we offer new inequalities for classical arithmetic functions such as the Euler’s totient function, the Dedekind’s psi function, the sum of the positive divisors function, the number of divisors function, extended Jordan’s totient function, generalized Dedekind’s psi function.

On an infinite family of unipotent Sylvester–Kac-like matrices
Original research paper. Pages 846–850
Zhibin Du and Carlos M. da Fonseca
Download paper (PDF, 156 Kb) | Abstract

Classical Sylvester–Kac matrices are tridiagonal integral matrices with positive off-diagonal entries and fully integral spectra. Here, by relaxing the positivity requirement and using a lower Pascal triangle framework, we define, for each positive integer $n$ , a unipotent Sylvester–Kac-like matrix in which $n$ is the only eigenvalue. This construction highlights the connection to the original Sylvester–Kac matrices while introducing a new family of unipotent matrices with distinctive properties.

Power series expansions of arbitrary order functional difference equations
Original research paper. Pages 851–858
Anthony G. Shannon, Ömür Deveci and Özgür Erdağ
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This paper looks at some real and complex generalizations of power series associated with some arbitrary order functional difference equations considered as generalizations and extensions of Fibonacci and Lucas numbers. It does this by drawing on, and interconnecting, some classic number theoretic results of Carlitz, Fasenmyer and Horadam.

Research on splitting quaternions with generalized Tribonacci hybrid number components
Original research paper. Pages 859–874
Yanni Yang and Yong Deng
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This paper introduces the Generalized Tribonacci Hybrid Split Quaternion (GTHSQ), a novel split quaternion with coefficients derived from generalized Tribonacci hybrid numbers. This form unifies various existing number types, such as generalized hybrid Tribonacci numbers and Tribonacci numbers, offering a fresh perspective on quaternion theory. To systematically characterize GTHSQ, we develop a comprehensive mathematical framework. This includes defining GTHSQ through related number expressions, specifying its operational rules, and proving that it retains the third-order linear recurrence relation of generalized Tribonacci sequences. We derive its Binet-type formula and generating functions while exploring its core properties. Additionally, we extend classic combinatorial identities (Vajda, Catalan, Cassini) to GTHSQ, define the GTHSQ matrix, and analyze its product with the S-matrix—a generalized third-order linear recurrence sequence representation matrix—to obtain matrix and determinant expressions for GTHSQ. These findings verify the closed-form solution of GTHSQ in terms of combinatorial identities and matrix representation. Furthermore, we discuss potential applications of GTHSQ, including advancements in quaternion algebra, support for encryption algorithms in cryptography, and simplification of spatial transformations in physics, thereby providing new tools and insights for future research in quaternion theory and interdisciplinary studies.

Coefficients of symmetric power L-functions on integers under digital constraints
Original research paper. Pages 875–883
Khadija Mbarki
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Let $\lambda_{{\rm sym}^{r}f}(n)$ be the $n$ -th coefficient in the Dirichlet series representing the symmetric power $L$ -function attached to a primitive form $f$ of weight $k$ and level $N$ . In this paper, we give asymptotic formulas for the arithmetic mean of $\lambda_{{\rm sym}^{r}f}(n)$ on integers under digital constraints related to the sum of digits function. Our results throw the light on the behavior of the classical function $\lambda_{{\rm sym}^{r}f}(n)$ on integers in arithmetic progression related to the sum of digits function.

On q-generalized hyperharmonic numbers with two parameters
Original research paper. Pages 884–898
Neşe Ömür, Sibel Koparal and Ömer Duran
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In this paper, we introduce $q$ -generalized harmonic numbers with two parameters $\omega$ and $\xi ,$ $H_{\mu,\lambda,q}(\omega ,\xi )$ for integers $\mu,\lambda$ such that $\lambda\geq \mu$ . With the help of these numbers, we define a new family of numbers which is called $q$ -generalized hyperharmonic numbers with two parameters $\omega$ and $\xi$ of order $r$ , $H_{\mu,\lambda,q}^{r}(\omega,\xi )$ for integer $r$ . Then, we consider special matrices whose entries are given by these numbers and give some matrix multiplications. Additionally, we derive some combinatorial identities for $H_{\mu,\lambda,q}(\omega ,\xi )$ and $H_{\mu,\lambda,q}^{r}(\omega ,\xi )$ by matrix methods.

Divisibility and sequence properties of σ⁺ and φ⁺
Original research paper. Pages 899–907
Sagar Mandal
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Inspired by Lehmer’s and Deaconescu’s conjectures, as well as various analogue problems concerning Euler’s totient function $\varphi(n)$ , Schemmel’s totient function $S_{2}(n)$ , Jordan totient function $J_k$ , and the unitary totient function $\varphi^{*}(n)$ , we investigate analogous divisibility problems involving the functions $\sigma(n)$ , $\sigma^{+}(n)$ , and $\varphi^{+}(n)$ . Further, we establish some interesting properties of the sequences $\left\{\sigma^+(n)\right\}_{n=1}^\infty$ and $\left\{\varphi^+(n)\right\}_{n=1}^\infty$ , in particular, we prove that each of these sequences contains infinitely many arithmetic progressions of length $3$ .

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-NP6/12/02.12.2024. Bulgarian National Science Fund bears no liability for the content of the published materials.

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