Special Issue "Algebraic, Analytic, and Computational Number Theory and Its Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra and Geometry".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 13739

Special Issue Editors

Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu street 50, 500091 Brasov, Romania
Interests: algebraic number theory; elementary number theory; computational number theory; associative algebras; combinatorics
Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu street 50, 500091 Brasov, Romania
Interests: analytical inequalities; generalized entropies; arithmetic functions; Euclidean geometry
Special Issues, Collections and Topics in MDPI journals
Dipartimento di EconomiaUniversit`a di Chieti PescaraViale della Pineta, 4, 65127 Pescara, Italy
Interests: algorithms; computational number theory; algebraic number theory; computational algebra

Special Issue Information

Dear Colleagues,

The aim and scope of this Special Issue is to publish new results in algebraic number theory and analytic number theory, namely in ramification theory in algebraic number fields, class field theory, arithmetic functions, L-functions, modular forms, elliptic curves, and some close research directions, namely associative algebras, logical algebras, elementary number theory, combinatorics, difference equations, group rings, and algebraic hyperstructures. 

Prof. Dr. Diana Savin
Prof. Dr. Nicusor Minculete
Prof. Dr. Vincenzo Acciaro
Guest Editors

Manuscript Submission Information

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Keywords

  • ramification theory in algebraic number fields
  • quadratic fields
  • biquadratic field
  • cyclotomic fields
  • Kummer fields
  • Dedekind rings
  • p-adic fields
  • class field theory
  • elliptic curves
  • L-functions
  • modular forms
  • quaternion algebras
  • arithmetic functions
  • difference equations
  • Fibonacci, Lucas, Pell, Horadam numbers and quaternions
  • logical algebras
  • algebraic hyperstructures
  • cryptography

Published Papers (14 papers)

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Research

Article
Almost Repdigit k-Fibonacci Numbers with an Application of k-Generalized Fibonacci Sequences
Mathematics 2023, 11(2), 455; https://doi.org/10.3390/math11020455 - 14 Jan 2023
Viewed by 613
Abstract
In this paper, we define the notion of almost repdigit as a positive integer whose digits are all equal except for at most one digit, and we search all terms of the k-generalized Fibonacci sequence which are almost repdigits. In particular, we [...] Read more.
In this paper, we define the notion of almost repdigit as a positive integer whose digits are all equal except for at most one digit, and we search all terms of the k-generalized Fibonacci sequence which are almost repdigits. In particular, we find all k-generalized Fibonacci numbers which are powers of 10 as a special case of almost repdigits. In the second part of the paper, by using the roots of the characteristic polynomial of the k-generalized Fibonacci sequence, we introduce k-generalized tiny golden angles and show the feasibility of this new type of angles in application to magnetic resonance imaging. Full article
Article
Pauli Gaussian Fibonacci and Pauli Gaussian Lucas Quaternions
Mathematics 2022, 10(24), 4655; https://doi.org/10.3390/math10244655 - 08 Dec 2022
Viewed by 443
Abstract
We have investigated new Pauli Fibonacci and Pauli Lucas quaternions by taking the components of these quaternions as Gaussian Fibonacci and Gaussian Lucas numbers, respectively. We have calculated some basic identities for these quaternions. Later, the generating functions and Binet formulas are obtained [...] Read more.
We have investigated new Pauli Fibonacci and Pauli Lucas quaternions by taking the components of these quaternions as Gaussian Fibonacci and Gaussian Lucas numbers, respectively. We have calculated some basic identities for these quaternions. Later, the generating functions and Binet formulas are obtained for Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions. Furthermore, Honsberger’s identity, Catalan’s and Cassini’s identities have been given for Pauli Gaussian Fibonacci quaternions. Full article
Article
On the Classification of Telescopic Numerical Semigroups of Some Fixed Multiplicity
Mathematics 2022, 10(20), 3871; https://doi.org/10.3390/math10203871 - 18 Oct 2022
Viewed by 575
Abstract
The telescopic numerical semigroups are a subclass of symmetric numerical semigroups widely used in algebraic geometric codes. Suer and Ilhan gave the classification of triply generated telescopic numerical semigroups up to multiplicity 10 and by using this classification they computed some important invariants [...] Read more.
The telescopic numerical semigroups are a subclass of symmetric numerical semigroups widely used in algebraic geometric codes. Suer and Ilhan gave the classification of triply generated telescopic numerical semigroups up to multiplicity 10 and by using this classification they computed some important invariants in terms of the minimal system of generators. In this article, we extend the results of Suer and Ilhan for telescopic numerical semigroups of multiplicities 8 and 12 with embedding dimension four. Furthermore, we compute two important invariants namely the Frobenius number and genus for these classes in terms of the minimal system of generators. Full article
Article
Novel Authentication Protocols Based on Quadratic Diophantine Equations
Mathematics 2022, 10(17), 3136; https://doi.org/10.3390/math10173136 - 01 Sep 2022
Cited by 1 | Viewed by 701
Abstract
The Diophantine equation is a strong research domain in number theory with extensive cryptography applications. The goal of this paper is to describe certain geometric properties of positive integral solutions of the quadratic Diophantine equation [...] Read more.
The Diophantine equation is a strong research domain in number theory with extensive cryptography applications. The goal of this paper is to describe certain geometric properties of positive integral solutions of the quadratic Diophantine equation x12+x22=y12+y22(x1,x2,y1,y2>0), as well as their use in communication protocols. Given one pair (x1,y1), finding another pair (x2,y2) satisfying x12+x22=y12+y22 is a challenge. A novel secure authentication mechanism based on the positive integral solutions of the quadratic Diophantine which can be employed in the generation of one-time passwords or e-tokens for cryptography applications is presented. Further, the constructive cost models are applied to predict the initial effort and cost of the proposed authentication schemes. Full article
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Article
Some Remarks on the Divisibility of the Class Numbers of Imaginary Quadratic Fields
Mathematics 2022, 10(14), 2488; https://doi.org/10.3390/math10142488 - 17 Jul 2022
Viewed by 585
Abstract
For a given integer n, we provide some families of imaginary quadratic number fields of the form Q(4q2pn), whose ideal class group has a subgroup isomorphic to Z/nZ. Full article
Article
A Generalized Bohr–Jessen Type Theorem for the Epstein Zeta-Function
Mathematics 2022, 10(12), 2042; https://doi.org/10.3390/math10122042 - 13 Jun 2022
Cited by 1 | Viewed by 644
Abstract
Let Q be a positive defined n×n matrix and Q[x̲]=x̲TQx̲. The Epstein zeta-function ζ(s;Q), s=σ+it, is defined, [...] Read more.
Let Q be a positive defined n×n matrix and Q[x̲]=x̲TQx̲. The Epstein zeta-function ζ(s;Q), s=σ+it, is defined, for σ>n2, by the series ζ(s;Q)=x̲Zn\{0̲}(Q[x̲])s, and is meromorphically continued on the whole complex plane. Suppose that n4 is even and φ(t) is a differentiable function with a monotonic derivative. In the paper, it is proved that 1Tmeast[0,T]:ζ(σ+iφ(t);Q)A, AB(C), converges weakly to an explicitly given probability measure on (C,B(C)) as T. Full article
Article
Residuated Lattices with Noetherian Spectrum
Mathematics 2022, 10(11), 1831; https://doi.org/10.3390/math10111831 - 26 May 2022
Viewed by 634
Abstract
In this paper, we characterize residuated lattices for which the topological space of prime ideals is a Noetherian space. The notion of i-Noetherian residuated lattice is introduced and related properties are investigated. We proved that a residuated lattice is i-Noetherian iff every ideal [...] Read more.
In this paper, we characterize residuated lattices for which the topological space of prime ideals is a Noetherian space. The notion of i-Noetherian residuated lattice is introduced and related properties are investigated. We proved that a residuated lattice is i-Noetherian iff every ideal is principal. Moreover, we show that a residuated lattice has the spectrum of a Noetherian space iff it is i-Noetherian. Full article
Article
New Properties and Identities for Fibonacci Finite Operator Quaternions
Mathematics 2022, 10(10), 1719; https://doi.org/10.3390/math10101719 - 17 May 2022
Viewed by 927
Abstract
In this paper, with the help of the finite operators and Fibonacci numbers, we define a new family of quaternions whose components are the Fibonacci finite operator numbers. We also provide some properties of these types of quaternions. Moreover, we derive many identities [...] Read more.
In this paper, with the help of the finite operators and Fibonacci numbers, we define a new family of quaternions whose components are the Fibonacci finite operator numbers. We also provide some properties of these types of quaternions. Moreover, we derive many identities related to Fibonacci finite operator quaternions by using the matrix representations. Full article
Article
New Zero-Density Results for Automorphic L-Functions of GL(n)
Mathematics 2021, 9(17), 2061; https://doi.org/10.3390/math9172061 - 26 Aug 2021
Viewed by 680
Abstract
Let L(s,π) be an automorphic L-function of GL(n), where π is an automorphic representation of group GL(n) over rational number field Q. In this paper, we study [...] Read more.
Let L(s,π) be an automorphic L-function of GL(n), where π is an automorphic representation of group GL(n) over rational number field Q. In this paper, we study the zero-density estimates for L(s,π). Define Nπ(σ,T1,T2) = ♯ {ρ = β + iγ: L(ρ,π) = 0, σ<β<1, T1γT2}, where 0σ<1 and T1<T2. We first establish an upper bound for Nπ(σ,T,2T) when σ is close to 1. Then we restrict the imaginary part γ into a narrow strip [T,T+Tα] with 0<α1 and prove some new zero-density results on Nπ(σ,T,T+Tα) under specific conditions, which improves previous results when σ near 34 and 1, respectively. The proofs rely on the zero detecting method and the Halász-Montgomery method. Full article
Article
Regularities in Ordered n-Ary Semihypergroups
Mathematics 2021, 9(16), 1857; https://doi.org/10.3390/math9161857 - 05 Aug 2021
Cited by 2 | Viewed by 1000
Abstract
This paper deals with a class of hyperstructures called ordered n-ary semihypergroups which are studied by means of j-hyperideals for all positive integers 1jn and n3. We first introduce the notion of (softly) left [...] Read more.
This paper deals with a class of hyperstructures called ordered n-ary semihypergroups which are studied by means of j-hyperideals for all positive integers 1jn and n3. We first introduce the notion of (softly) left regularity, (softly) right regularity, (softly) intra-regularity, complete regularity, generalized regularity of ordered n-ary semihypergroups and investigate their related properties. Several characterizations of them in terms of j-hyperideals are provided. Finally, the relationships between various classes of regularities in ordered n-ary semihypergroups are also established. Full article
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Article
Some Properties of Euler’s Function and of the Function τ and Their Generalizations in Algebraic Number Fields
Mathematics 2021, 9(15), 1710; https://doi.org/10.3390/math9151710 - 21 Jul 2021
Viewed by 1112
Abstract
In this paper, we find some inequalities which involve Euler’s function, extended Euler’s function, the function τ, and the generalized function τ in algebraic number fields. Full article
Article
Global and Local Behavior of the System of Piecewise Linear Difference Equations xn+1 = |xn| − ynb and yn+1 = xn − |yn| + 1 Where b ≥ 4
Mathematics 2021, 9(12), 1390; https://doi.org/10.3390/math9121390 - 15 Jun 2021
Viewed by 1177
Abstract
The aim of this article is to study the system of piecewise linear difference equations xn+1=|xn|ynb and [...] Read more.
The aim of this article is to study the system of piecewise linear difference equations xn+1=|xn|ynb and yn+1=xn|yn|+1 where n0. A global behavior for b=4 shows that all solutions become the equilibrium point. For a large value of |x0| and |y0|, we can prove that (i) if b=5, then the solution becomes the equilibrium point and (ii) if b6, then the solution becomes the periodic solution of prime period 5. Full article
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Article
On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers
Mathematics 2021, 9(9), 962; https://doi.org/10.3390/math9090962 - 25 Apr 2021
Cited by 1 | Viewed by 1165
Abstract
Let (tn(r))n0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence [...] Read more.
Let (tn(r))n0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence tn(r)=tn1(r)++tnr(r) for nr, with initial values t0(r)=0 and ti(r)=1, for all 1ir. In 2002, Grossman and Luca searched for terms of the sequence (tn(2))n, which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any 1, there exists an effectively computable constant C=C()>0 (only depending on ), such that, if (m,n,r) is a solution of tm(r)=n!+(n+1)!++(n+)!, with r even, then max{m,n,r}<C. As an application, we solve the previous equation for all 15. Full article
Article
On Generalized Lucas Pseudoprimality of Level k
Mathematics 2021, 9(8), 838; https://doi.org/10.3390/math9080838 - 12 Apr 2021
Cited by 3 | Viewed by 1242
Abstract
We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some [...] Read more.
We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels k+ and k and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences. Full article
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