New Insights in Algebra, Discrete Mathematics and Number Theory II

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 11719

Special Issue Editors

Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
Interests: number theory; linear algebra; difference equations; computer-aided mathematics
Special Issues, Collections and Topics in MDPI journals
The Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland
Interests: discrete mathematics; graph theory; number theory
Special Issues, Collections and Topics in MDPI journals
Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Rokitasnkého 62, Czech Republic
Interests: number theory; applied mathematics; computer science

Special Issue Information

Dear Colleagues,

We all probably realize, as we study the current breakthrough results in our favorite area of mathematics, that the very different areas of mathematics are now converging again, as the basis for deriving these admirable results is that other discipline methods are crucial for proof assertions in our field of mathematics. The purpose of this Special Issue is to gather a collection of articles reflecting new trends in contemporary elementary, abstract, linear, Boolean, commutative, computer, and homological algebra, analytical, algebraic, combinatorial and computational number theory, modular forms, factors, fractions, arithmetic dynamics, sieve methods, quadratic forms, L-functions, combinatorics, and graph theory. For this Special Issue, we welcome original research articles or review articles focused on recent problems concerned mainly with abstract and linear algebra, algebraic, analytic and combinatorial number theory, combinatorics, and graph theory as well as their multidisciplinary applications.

Dr. Pavel Trojovský
Prof. Dr. Iwona Włoch
Dr. Štěpán Hubálovský
Guest Editors

Manuscript Submission Information

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Keywords

  • Special groups, rings, and fields
  • Applications of linear algebra
  • Algebraic number fields
  • Transcendental number theory
  • Arithmetic functions
  • Diophantine equations and Diophantine approximations
  • Recurrence sequences and difference equations
  • Combinatorics
  • Directed, discrete, and planar graphs

Published Papers (9 papers)

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Research

11 pages, 278 KiB  
Article
Reciprocal Formulae among Pell and Lucas Polynomials
by Mei Bai, Wenchang Chu and Dongwei Guo
Mathematics 2022, 10(15), 2691; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152691 - 29 Jul 2022
Cited by 1 | Viewed by 972
Abstract
Motivated by a problem proposed by Seiffert a quarter of century ago, we explicitly evaluate binomial sums with Pell and Lucas polynomials as weight functions. Their special cases result in several interesting identities concerning Fibonacci and Lucas numbers. Full article
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)
15 pages, 301 KiB  
Article
Rings of Multisets and Integer Multinumbers
by Yuriy Chopyuk, Taras Vasylyshyn and Andriy Zagorodnyuk
Mathematics 2022, 10(5), 778; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050778 - 28 Feb 2022
Cited by 9 | Viewed by 1880
Abstract
In the paper, we consider a ring structure on the Cartesian product of two sets of integer multisets. In this way, we introduce a ring of integer multinumbers as a quotient of the Cartesian product with respect to a natural equivalence. We examine [...] Read more.
In the paper, we consider a ring structure on the Cartesian product of two sets of integer multisets. In this way, we introduce a ring of integer multinumbers as a quotient of the Cartesian product with respect to a natural equivalence. We examine the properties of this ring and construct some isomorphisms to subrings of polynomials and Dirichlet series with integer coefficients. In addition, we introduce finite rings of multinumbers “modulo (p,q)” and propose an algorithm for construction of invertible elements in these rings that may be applicable in Public-key Cryptography. An analog of the Little Fermat Theorem for integer multinumbers is proved. Full article
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)
9 pages, 278 KiB  
Article
On Two Problems Related to Divisibility Properties of z(n)
by Pavel Trojovský
Mathematics 2021, 9(24), 3273; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243273 - 16 Dec 2021
Viewed by 1471
Abstract
The order of appearance (in the Fibonacci sequence) function z:Z1Z1 is an arithmetic function defined for a positive integer n as [...] Read more.
The order of appearance (in the Fibonacci sequence) function z:Z1Z1 is an arithmetic function defined for a positive integer n as z(n)=min{k1:Fk0(modn)}. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z(n)=z(n2), where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z(n)=z(n+1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that limsupn(z(n+1)z(n))/(logn)2ϵ=, for all ϵ(0,2). Full article
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)
7 pages, 256 KiB  
Article
The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)
by Eva Trojovská and Venkatachalam Kandasamy
Mathematics 2021, 9(22), 2912; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222912 - 16 Nov 2021
Viewed by 978
Abstract
Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as [...] Read more.
Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k1:nFk}. Very recently, Trojovská and Venkatachalam proved that, for any k1, the number z(n) is divisible by 2k, for almost all integers n1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m1. In this paper, in particular, we prove this conjecture. Full article
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)
9 pages, 237 KiB  
Article
On Andrews’ Partitions with Parts Separated by Parity
by Abdulaziz M. Alanazi and Darlison Nyirenda
Mathematics 2021, 9(21), 2693; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212693 - 23 Oct 2021
Viewed by 1195
Abstract
In this paper, we present a generalization of one of the theorems in Partitions with parts separated by parity introduced by George E. Andrews, and give its bijective proof. Further variations of related partition functions are studied resulting in a number of interesting [...] Read more.
In this paper, we present a generalization of one of the theorems in Partitions with parts separated by parity introduced by George E. Andrews, and give its bijective proof. Further variations of related partition functions are studied resulting in a number of interesting identities. Full article
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)
8 pages, 268 KiB  
Article
The Proof of a Conjecture Related to Divisibility Properties of z(n)
by Eva Trojovská and Kandasamy Venkatachalam
Mathematics 2021, 9(20), 2638; https://0-doi-org.brum.beds.ac.uk/10.3390/math9202638 - 19 Oct 2021
Cited by 1 | Viewed by 1003
Abstract
The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) [...] Read more.
The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture). Full article
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)
9 pages, 264 KiB  
Article
New Asymptotic Properties of Positive Solutions of Delay Differential Equations and Their Application
by Osama Moaaz and Clemente Cesarano
Mathematics 2021, 9(16), 1971; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161971 - 18 Aug 2021
Cited by 2 | Viewed by 1040
Abstract
In this study, new asymptotic properties of positive solutions of the even-order delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Moreover, we use these properties to [...] Read more.
In this study, new asymptotic properties of positive solutions of the even-order delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Moreover, we use these properties to obtain new criteria for the oscillation of the solutions of the studied equation using the principles of comparison. Full article
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)
8 pages, 1511 KiB  
Article
On Some Properties of the Limit Points of (z(n)/n)n
by Eva Trojovská and Kandasamy Venkatachalam
Mathematics 2021, 9(16), 1931; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161931 - 13 Aug 2021
Viewed by 985
Abstract
Let (Fn)n0 be the sequence of Fibonacci numbers. The order of appearance of an integer n1 is defined as [...] Read more.
Let (Fn)n0 be the sequence of Fibonacci numbers. The order of appearance of an integer n1 is defined as z(n):=min{k1:nFk}. Let Z be the set of all limit points of {z(n)/n:n1}. By some theoretical results on the growth of the sequence (z(n)/n)n1, we gain a better understanding of the topological structure of the derived set Z. For instance, {0,1,32,2}Z[0,2] and Z does not have any interior points. A recent result of Trojovská implies the existence of a positive real number t<2 such that Z(t,2) is the empty set. In this paper, we improve this result by proving that (127,2) is the largest subinterval of [0,2] which does not intersect Z. In addition, we show a connection between the sequence (xn)n, for which z(xn)/xn tends to r>0 (as n), and the number of preimages of r under the map mz(m)/m. Full article
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)
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8 pages, 270 KiB  
Article
On the Parity of the Order of Appearance in the Fibonacci Sequence
by Pavel Trojovský
Mathematics 2021, 9(16), 1928; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161928 - 13 Aug 2021
Cited by 2 | Viewed by 1233
Abstract
Let (Fn)n0 be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z:Z1Z1 is defined as [...] Read more.
Let (Fn)n0 be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z:Z1Z1 is defined as z(n):=min{k1:Fk0(modn)}. In this paper, among other things, we prove that z(n) is an even number for almost all positive integers n (i.e., the set of such n has natural density equal to 1). Full article
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics and Number Theory II)
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