Volume 31, 2025, Number 3 (Online First)

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

Volume opened: 17 June 2025
Status: In progress

Editorial: A note on Papal Mathematics
Editorial. Pages 429–432
Anthony G. Shannon
Editorial (PDF, 393 Kb) | Abstract

Many recent media claims that Pope Leo XIV, Robert F. Prevost, born in the USA and elected in 2025 in succession to Pope Francis, is the first mathematician to become the Pope of the Catholic Church are questionable. That honour probably belongs to Pope Sylvester II (999–1003), Gerbert d’Aurillac, a Frenchman, educated in Moorish Spain which was then relatively advanced in mathematics and philosophy.

Fermatian row and column sums as a family of generalized integers
Original research paper. Pages 433–442
Anthony G. Shannon, Mine Uysal and Engin Özkan
Download paper (PDF, 1036 Kb) | Abstract

In this paper, we introduce some feature of the Fermatian numbers. The finite sum formulas of these numbers is calculate. The exponential generating function of Fermatian numbers is found and some of its identities is calculated. Another number sequence is obtained from the partial row sums of these numbers and these numbers were examined. At the same time, another polynomial has been defined as a generalization of these numbers, depending on powers of z.

A new symmetric weighing matrix SW(22,16)
Original research paper. Pages 443–447
Sheet Nihal Topno and Shyam Saurabh
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The existence of symmetric weighing matrix $SW(22,16)$ is settled in this note through a theorem and exhaustive search.

On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers
Original research paper. Pages 448–459
Hunar Sherzad Taher and Saroj Kumar Dash
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Let $(F_{r}^{(k)})_{r\geq2-k}$ and $(L_{r}^{(k)})_{r\geq2-k}$ be generalizations of the Fibonacci and Lucas sequences, where $k\geq2$ . For these sequences the initial $k$ terms are $0, 0, \ldots , 0, 1$ and $0, 0, \ldots , 2, 1$ , and each subsequent term is the sum of the preceding $k$ terms. In this paper, we determined all first and second kinds of Thabit numbers that can be expressed as the sums of $k$ -Fibonacci and $k$ -Lucas numbers. We employed the theory of linear forms in logarithms of algebraic numbers and a reduction method based on the continued fraction.

Solution of an odds inversion problem
Original research paper. Pages 460–470
Robert K. Moniot
Download paper (PDF, 228 Kb) | Abstract

Consider the problem of determining the possible numbers of balls of two different colors in an urn such that if two are drawn out at random, the odds that they are different colors are a given value. We present a general solution of this problem for all odds from nil to certainty. The solution methods use relatively simple concepts from number theory such as modular inverses and the Pell equation. We find upper bounds on the number of solutions and the magnitude of solutions for those cases that have at most a finite number of solutions. We also define solution classes for cases that have an infinite number of solutions, and identify cases having a determinate number of solution classes.

Recursive sufficiency for the Collatz conjecture and computational verification
Original research paper. Pages 471–480
Mohammad Ansari
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We define the notion of recursive sufficiency for the Collatz conjecture and we use it to present some results concerning the computational verification of the conjecture. For any integer $N\ge 1$ and any recursively sufficient set $F$ , it is proved that all integers in the interval $[1, N]$ satisfy the conjecture if and only if $F\cap [1, N]$ satisfies the conjecture. We offer a sequence of sieves for which the corresponding sequence of elimination percentages tends to $100\%$ , and as a result, for any integer $P$ arbitrarily close to $100$ , we give a sieve whose elimination percentage is at least $P\%$ . Also, we prove that if $N=2(3^n)+1$ is the largest known integer for which all integers $1, 2, \ldots , N$ satisfy the conjecture, then all integers $N+1, N+2, \ldots, 2N$ will satisfy the conjecture as well, and hence, they can be eliminated from the verification process.

The Bombieri–Vinogradov theorem for exponential sums over primes
Original research paper. Pages 481–493
Stoyan Dimitrov
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In this paper, we revisit Lemma 18 from [2], which concerns a Bombieri–Vinogradov type theorem for exponential sums over primes. We provide a corrected version of the lemma, clarify the original arguments, and address certain inaccuracies present in the initial proof.

Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often
Original research paper. Pages 494–503
Joel E. Cohen and Dexter Senft
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The infinite sequence of gaps (first differences) between successive odd composite numbers contains only the numbers 2, 4, and 6. We prove that, for any natural number k, the sequence of gaps contains infinitely many k-tuplets of consecutive gaps all equal to 2. Infinitely many gaps equal 4. The sequence of gaps includes infinitely many gap pairs (4, 4) if the sequence of positive primes has infinitely many pairs of successive primes that differ by 4 (cousin primes), which is unproved but holds under a conjecture of Hardy and Littlewood. Gap triplets (4, 4, 4) never occur. Infinitely many gaps equal 6 if and only if there are infinitely many twin primes. Moreover, gap pairs (6, 6) occur infinitely often if other conjectures of Hardy and Littlewood are true. Six of the 27 potential triplets of values of gaps between successive odd composite numbers never occur: (4, 4, 4), (6, 6, 6), (6, 4, 4), (4, 4, 6), (6, 2, 6), and (6, 4, 6).

On the special cases of Carmichael’s totient conjecture
Original research paper. Pages 504–534
Anthony G. Shannon, Peter J.-S. Shiue, Tian-Xiao He, and Christopher Saito
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Euler’s totient function, $\varphi(n)$ , is the arithmetic function defined as the number of positive integers less than or equal to $n$ that are relatively prime to $n$ . In his 1922 paper [3], Professor R. D. Carmichael conjectured that for each positive integer $n$ , there exists at least one positive integer $m \neq n$ such that $\varphi(m) = \varphi(n).$

In this paper, we consider some relevant literature and explore Carmichael’s totient conjecture for particular values of $\varphi(n)=k.$ Our main consideration will be the set $X_k=\left\{n\in\mathbb{N}:\varphi(n)=k\right\}$ . In identifying $X_k$ for $k=2^t, 2p^s, 2^2p,$ and $2pq$ , where $p$ and $q$ are distinct prime numbers, we find that Carmichael’s conjecture holds for those select cases, provide an algorithm, and some related results. The conjecture remains an open problem in number theory [9].

On an analytical study of the generalized Fibonacci polynomials
Original research paper. Pages 535–546
Leandro Rocha, Gabriel F. Pinheiro and Elen V. P. Spreafico
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In this work, we presented an analytical study of the generalized Fibonacci polynomial of order $r\geq 2,$ by using properties of the fundamental system associated with the generalized Fibonacci polynomial. We established the generating function and provided the asymptotic behavior for each system sequence. Moreover, the properties are extended to any generalized Fibonacci type, given the general case’s generating function and asymptotic behavior.

Binomial transform of the bivariate Fibonacci quaternion polynomials and its properties
Original research paper. Pages 547–562
Faruk Kaplan and Arzu Özkoç Öztürk
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The primary aim of this work is to deal with binomial transforms of bivariate Fibonacci quaternion polynomial sequence. The binomial sequence of the bivariate Fibonacci quaternion polynomial is found, and then results are obtained for the recurrence relation, generating function and Binet formula. Furthermore, different types of sums are being sought for the polynomials. While working with bivariate Fibonacci quaternion polynomial sequence, we found the general formula for all sequences with quadratic recurrence relation, which includes the formulas in general terms for binomial transform. In the last part, matrix representations are derived for the corresponding binomial sequence.

Another six Fibonacci-like sequences
Original research paper. Pages 563–569
Karol Gryszka
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We briefly describe six Fibonacci-like sequences or arbitrary order that give rise to a periodic or eventually periodic sequences. We provide some examples and demonstrate the explicit periods of these sequences.

New summation identities of hyperbolic k-Fibonacci and k-Lucas quaternions
Original research paper. Pages 570–582
A. D. Godase
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In this paper, we introduce a set of identities involving hyperbolic $k$ -Fibonacci quaternions and $k$ -Lucas quaternions. Moreover, we derive summation identities for hyperbolic $k$ -Fibonacci and $k$ -Lucas quaternions by utilizing established properties of $k$ -Fibonacci and $k$ -Lucas numbers. These findings add valuable insight into the relationships between these quaternion sequences and offer valuable insights into their properties.

On a logarithmic inequality by Shenton and Kemp
Original research paper. Pages 583–587
József Sándor
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We offer new proof and refinement of a double inequality for $\ln^2 (1+x),$ by L. R. Shenton and A. W. Kemp [11].

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.

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