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Discrete Mathematics as the Basis and Application of Number Theory

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A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: closed (31 August 2022) | Viewed by 10279

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Special Issue Editors

Prof. Dr. Takao Komatsu

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Guest Editor

Faculty of Education, Nagasaki University, Nagasaki 852-8521, Japan
Interests: combinatorics; discrete mathematics; number theory; pure mathematics; special functions
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Claudio Pita-Ruiz

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Guest Editor

Faculty of Engineering, Universidad Panamericana, Mexico City Campus, Mexico
Interests: math; number theory; integer sequences and their properties; fibonacci sequences; fibonomials

Prof. Dr. Ram Krishna Pandey

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Guest Editor

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
Interests: number theory (additive, combinatorial, and elementary)

Special Issue Information

Dear Colleagues,

The subject of this Special Issue is discrete mathematics. At present, the use of AI is becoming indispensable in all situations and fields, and discrete mathematics is the basis of correct understanding for its use. Discrete mathematics covers a very wide range of fields, but this Special Issue focuses on its fundamental and applications in relevance to combinatorial number theory and elementary number theory. Any related or further idea is greatly welcomed if it does not contain only key words but also some interesting contents, such as sequences, sets, permutations, combinations, inductions, recurrence formulas, graphs, algorithms, and so on.

Prof. Dr. Takao Komatsu
Prof. Dr. Claudio Pita-Ruiz
Prof. Dr. Ram Krishna Pandey
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

discrete mathematics
generating functions
arithmetic functions
fibonacci numbers
bernoulli numbers
stirling numbers
harmonic numbers
matrices and determinants
continued fractions
recurrences
linear diophantine equations
additive bases
arithmetic progressions
numeration systems
beta-expansions
combinatorics on words

Published Papers (6 papers)

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Research

16 pages, 320 KiB

Open AccessArticle

Series of Convergence Rate −1/4 Containing Harmonic Numbers

by Chunli Li and Wenchang Chu

Axioms 2023, 12(6), 513; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12060513 - 24 May 2023

Cited by 1 | Viewed by 761

Abstract

Two general transformations for hypergeometric series are examined by means of the coefficient extraction method. Several interesting closed formulae are shown for infinite series containing harmonic numbers and binomial/multinomial coefficients. Among them, three conjectured identities due to Z.-W. Sun are also confirmed. Full article

(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)

18 pages, 372 KiB

Open AccessArticle

The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions

by Takao Komatsu and Claudio Pita-Ruiz

Axioms 2023, 12(2), 98; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12020098 - 17 Jan 2023

Cited by 5 | Viewed by 1092

Abstract

In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For

p = 0

, the problem is reduced [...] Read more.

p = 0

, the problem is reduced to the famous linear Diophantine problem of Frobenius, the largest integer of which is called the Frobenius number. We also give a closed formula for the number of non-negative integers (p-genus), such that linear equations have at most p representations. Extensions to the Jacobsthal polynomial and the Jacobsthal–Lucas polynomial give more general formulas that include the familiar Fibonacci and Lucas numbers. A basic problem with the Fibonacci triplet was dealt by Marin, Ramírez Alfonsín and M. P. Revuelta for

p = 0

and by Komatsu and Ying for the general non-negative integer p. Full article

(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)

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Chart 1

7 pages, 225 KiB

Open AccessArticle

Iterated Partial Sums of the k-Fibonacci Sequences

by Sergio Falcón and Ángel Plaza

Axioms 2022, 11(10), 542; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11100542 - 11 Oct 2022

Cited by 1 | Viewed by 1778

Abstract

In this paper, we find the sequence of partial sums of the k-Fibonacci sequence, say,

S_{k, n} = \sum_{j = 1}^{n} F_{k, j}

, and then we find the sequence of partial sums of this new [...] Read more.

In this paper, we find the sequence of partial sums of the k-Fibonacci sequence, say,

S_{k, n} = \sum_{j = 1}^{n} F_{k, j}

, and then we find the sequence of partial sums of this new sequence,

S_{k, n}^{2)} = \sum_{j = 1}^{n} S_{k, j}

, and so on. The iterated partial sums of k-Fibonacci numbers are given as a function of k-Fibonacci numbers, in powers of k, and in a recursive way. We finish the topic by indicating a formula to find the first terms of these sequences from the k-Fibonacci numbers themselves. Full article

(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)

14 pages, 788 KiB

Open AccessArticle

Cayley Graphs Defined by Systems of Equations

by Fuyuan Yang, Qiang Sun, Hongbo Zhou and Chao Zhang

Axioms 2022, 11(3), 100; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11030100 - 25 Feb 2022

Viewed by 1760

Abstract

Let R be a finite ring. In this paper, we mainly explore the conditions to ensure the graph

B Γ_{n}

defined by a system of equations

{f_{i} | i = 2, \dots, n}

to be a Cayley [...] Read more.

Let R be a finite ring. In this paper, we mainly explore the conditions to ensure the graph

B Γ_{n}

defined by a system of equations

{f_{i} | i = 2, \dots, n}

to be a Cayley graph or a Hamiltonian graph. More precisely, we prove that

B Γ_{n}

is a Cayley graph with

G = ⟨ ϕ, A ⟩

a group of dihedral type if and only if the system

F_{n} = {f_{i} | i = 2, \dots, n}

is Cayley graphic of dihedral type in R. As an application, the well-known Lov

\overset{´}{a}

sz Conjecture, which states that any finite connected Cayley graph has a Hamilton cycle, holds for the connected

B Γ_{n}

defined by Cayley graphic system

F_{n}

of dihedral type in the field

G F (p^{k})

. Full article

(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)

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Figure 1

18 pages, 293 KiB

Open AccessArticle

Fourier Series for the Tangent Polynomials, Tangent–Bernoulli and Tangent–Genocchi Polynomials of Higher Order

by Cristina Bordaje Corcino and Roberto Bagsarsa Corcino

Axioms 2022, 11(3), 86; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11030086 - 22 Feb 2022

Cited by 3 | Viewed by 1735

Abstract

In this paper, the Fourier series expansion of Tangent polynomials of higher order is derived using the Cauchy residue theorem. Moreover, some variations of higher-order Tangent polynomials are defined by mixing the concept of Tangent polynomials with that of Bernoulli and Genocchi polynomials, Tangent–Bernoulli and Tangent–Genocchi polynomials. Furthermore, Fourier series expansions of these variations are also derived using the Cauchy residue theorem. Full article

(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)

9 pages, 265 KiB

Open AccessArticle

Balances in the Set of Arithmetic Progressions

by Chan-Liang Chung, Chunmei Zhong and Kanglun Zhou

Axioms 2021, 10(4), 350; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms10040350 - 20 Dec 2021

Viewed by 2069

Abstract

This article focuses on searching and classifying balancing numbers in a set of arithmetic progressions. The sufficient and necessary conditions for the existence of balancing numbers are presented. Moreover, explicit formulae of balancing numbers and various relations are included. Full article

(This article belongs to the Special Issue Discrete Mathematics as the Basis and Application of Number Theory)

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Discrete Mathematics as the Basis and Application of Number Theory

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