Volume 32, 2026, Number 3 (Online First)

Volume 32 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ (Online First)

Volume opened: 1 September 2026
Status: In progress

Editorial: An introduction to the Fields Medal and the work of Yu Deng and Hong Wang
Book review. Pages 455–476
Chungwu Ho
Editorial (PDF, 917 Kb)

A compact notation for peculiar properties characterizing integer tetration
Original research paper. Pages 477–496
Marco Ripà and Gabriele Di Pietro
Full paper (PDF, 281 Kb) | Abstract

By initially working in the decimal numeral system, we introduce a compact notation to express the congruence speed of an integer tetration base $a$ , along with the cycle of the rightmost non-stable digits of $^{b}a$ for unit increments of $b$ . The resulting discrete function provides a useful tool for efficiently computing the exact number of frozen digits that characterize the right tail of each nontrivial integer tetration. We also establish an improved upper bound for the minimum hyperexponent $\bar{b}(a)$ that guarantees the constancy of the congruence speed of $a$ for all heights $b \geq \bar{b}(a)$ . Moreover, we prove that the minimum between the constant congruence speeds of any two integers greater than $1$ , whose product is not divisible by $10$ , is always less than or equal to the constant congruence speed of their product. Additionally, within the decimal numeral system, we give examples of infinitely many perfect powers whose degree matches their constant congruence speed at every height above $2$ , emphasizing the peculiar recurrence relations of hyper- $4$ . Finally, Appendix B generalizes the decimal framework described above to all squarefree numeral systems, showing that only in such systems the congruence speed stabilizes to a fixed (positive) value for all $a>1$ not divisible by the radical of the radix. Furthermore, we derive compact formulas for all prime-radix numeral systems and for the composite squarefree senary case.

Combinatorial and arithmetic properties of n^r-partitions: A unified analytic generalization
Original research paper. Pages 497–513
Mateus Alegri, A. David Christopher and Elen Viviani Pereira Spreafico
Full paper (PDF, 310 Kb) | Abstract

This paper presents a comprehensive study of $n^r$ -partitions, defined by the generating function $\prod_{n=1}^{\infty} (1 - q^n)^{-n^r}$ , which interpolates analytically between ordinary partitions ( $r=0$ ), plane partitions ( $r=1$ ), and higher-dimensional combinatorial structures ( $r\geq2$ ).

Our main contributions include: (1) explicit formulas relating $p_{(r)}(n)$ to divisor sums $\sigma_{r+1}(n)$ and integer compositions; (2) novel recurrence relations and congruence properties modulo primes and composite numbers; (3) complete Bell polynomial representations; and (4) combinatorial interpretations through generating function techniques. For weighted restricted $n^r$ -partitions into three parts, we establish exact quasi-polynomial formulas.

The results extend classical partition theory while providing new connections between combinatorics and number theory. The methods developed here offer a unified framework for investigating partition-like structures across dimensions, with applications in enumerative combinatorics.

The Hosoya-type Pell r-lengths of certain 2-generator groups
Original research paper. Pages 514–525
Ömür Deveci and Dürdane Banu Arıkanoğlu
Full paper (PDF, 238 Kb) | Abstract

In this paper, we give the lengths of the periods of the Hosoya-type Pell $r$ -orbits of the dihedral group $D_{2m}$ , the semidihedral group $SD_{2^{m}}$ and the generalized quaternion group $Q_{2^{m+1}}$ by using some special recurrence sequences.

Rising factorials and hypergeometric functions
Original research paper. Pages 526–535
Carlos M. da Fonseca and Anthony G. Shannon
Full paper (PDF, 207 Kb) | Abstract

This note considers factorial binomial coefficients, focusing on both rising and falling factorial powers and their generalisations. We recall a factorial-binomial formulation of classical hypergeometric identities, including Saalschütz’s ${}_3F_2$ summation, and highlight the combinatorial structure underlying the coefficients. For Type 2 rising factorial binomial coefficients we establish a Pascal-type recurrence relation, identities for negative indices and some generating functions. We also discuss potential extensions to rising factorial Jordan coefficients and congruence properties, motivated by recent results of Kim and Kim. This unified approach offers a combinatorial and algebraic framework for analysing hypergeometric series and related number-theoretic properties.

A note on some identities of degenerate Bernoulli numbers
Original research paper. Pages 536–543
Taekyun Kim, Dae San Kim, Hyunseok Lee, and Misbah Khan
Full paper (PDF, 173 Kb) | Abstract

In this note, we investigate the algebraic and combinatorial properties of degenerate special numbers and polynomials, focusing on their deep connections to classical sequences. By leveraging the framework of degenerate exponential and logarithm functions, we establish several explicit formulas and identities. Specifically, we provide an explicit expression for the Bessel numbers of the first kind and recall a previously established explicit formula for the degenerate Bernoulli numbers that is needed for the subsequent development. Finally, we establish an identity linking the Bessel numbers of the first kind, the degenerate Bernoulli numbers, and the degenerate Stirling numbers of the first kind.

On a conjecture involving twin practical numbers
Original research paper. Pages 544–555
Runbo Li
Full paper (PDF, 199 Kb) | Abstract

In this paper, we state a conjecture that for any positive integer $p,$ there are infinitely many practical numbers $q$ such that both $q^p$ and $q^p + 2$ are practical numbers. We prove the conjecture when $p$ is a prime number between $7$ and $100,$ refining a result involving twin practical numbers of Wang and Sun. We also prove several results related to this conjecture.

Enumeration of cyclic components of the power digraph Γ(n, 13)
Original research paper. Pages 556–575
Prabin Das, Pinkimani Goswami and Riazul Hoque
Full paper (PDF, 296 Kb) | Abstract

For each positive integer $n$ , we assign a digraph $\Gamma(n,13)$ whose set of vertices is $\mathbb{Z}_n=\lbrace 0,1,2, \ldots ,n-1\rbrace$ and attach exactly one directed edge from a vertex $a$ to the vertex $b$ if and only if $a^{13}\equiv b \pmod n$ . In this work, the number of cycles and the number of components in the digraph $\Gamma(n,13)$ are presented for $n=p^k$ with the help of Carmichael’s lambda function. It is proved that for $k\geq 1$ and for the prime $p=3, 5, 7$ and $11$ , the respective numbers of components in the digraph $\Gamma(p^k,13)$ are $(4k-1)$ , $(4k+1)$ , $(18k-11)$ and $(6k-1)$ . The structure of tree associated with power digraph $\Gamma(p^k, 13)$ for $p=3, 5, 7$ and $11$ are also studied. We have also studied the structure of power digraph $\Gamma(p, 13)$ , where $p=2^{2^m}+1, m\geq 0$ is a Fermat prime.

A note on quadratic cyclotomic extensions
Original research paper. Pages 576–589
Sophie Marques and Elizabeth Mrema
Full paper (PDF, 288 Kb) | Abstract

Schinzel’s classification of radical extensions relies on the structure of quadratic cyclotomic extensions, which also play an important role in the study of cyclotomic extensions over general fields. In this paper, we identify properties and invariants associated with quadratic cyclotomic extensions that are relevant to these classification problems. We determine the possible minimal polynomials of primitive roots of unity generating quadratic extensions, describe the prime-power levels at which such extensions occur, and assemble the prime-power information into a global classification of quadratic cyclotomic fields. The treatment applies in arbitrary characteristic, with the usual restriction that, in positive characteristic, the orders of the roots of unity considered are prime to the characteristic.

Recurrence function of the generalized Thue–Morse word
Original research paper. Pages 590–609
Mahamadi Nana, Boucaré Kientéga and Idrissa Kaboré
Full paper (PDF, 272 Kb) | Abstract

This paper is devoted to the study of the recurrence function of q-ary Thue–Morse word t_q. After giving the structure of singular factors, we describe the maximal return time of factors of t_q. We conclude by determining the recurrence function of generalized Thue–Morse word.

An extension of Dedekind’s arithmetic function – II
Original research paper. Pages 610–621
Krassimir Atanassov and József Sándor
Full paper (PDF, 262 Kb) | Abstract

We obtain refinements and new results related to an extension of Dedekind’s arithmetic function, introduced in [1].

On generalized continued fractions converging to an irrational number
Original research paper. Pages 622–628
Hayato Yoshida
Full paper (PDF, 207 Kb) | Abstract

In [5], we give a sufficient condition for generalized continued fractions to converge to an irrational number. Specifically, we determine how many non-integers must exist in $c_{1},c_{2},\ldots, c_{i}$ in order for the continued fraction $[0; c_{1},c_{2},\ldots]$ to be irrational, where $c_{i}$ are rational numbers greater than $M>1$ (denoted by $c_{i}\in \mathbb{Q}_{\ge M}$ ) and the denominator of $c_{i}$ less than or equal to positive integer $N$ . In [6], we extended this result from the case $M>1$ to $M>0$ . In this paper, we give a further extension by removing the restrictions on the denominators of the partial quotients. Let $M$ be a real number greater than $0$ . We consider continued fractions $[0;c_{1},c_{2},\ldots]$ , where $c_{i}$ are rational numbers greater than or equal to $M$ for any positive integer $i$ . We give a sufficient condition for it to converge to an irrational number and some interesting examples.

A short remark on a connection between Goldbach’s conjecture and quadratic forms
Original research paper. Pages 629–631
Mladen V. Vassilev-Missana
Full paper (PDF, 128 Kb) | Abstract

In the paper, a new version of the binary/strong Goldbach’s conjecture is considered and discussed. It is based on quadratic forms representation and on a new conjecture, that establishes equivalence.

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 ▷ Number 2 ▷ Number 3 (Online First)

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