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, Geometry and Topology".

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 31733

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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
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Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania
Interests: inequalities; generalized entropies; Euclidean geometry; operator theory
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Department of Economic Studies, Universita' degli Studi "G. D'Annunzio", Viale Pindaro 42, 65127 Pescara, Italy
Interests: algorithms; computational number theory; algebraic number theory; computational algebra
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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

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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

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Published Papers (22 papers)

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Editorial

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7 pages, 242 KiB  
Editorial
Algebraic, Analytic, and Computational Number Theory and Its Applications
by Diana Savin, Nicusor Minculete and Vincenzo Acciaro
Mathematics 2024, 12(1), 10; https://doi.org/10.3390/math12010010 - 20 Dec 2023
Viewed by 1194
Abstract
Analytic number theory is a branch of number theory which inherits methods from mathematical analysis in order to solve difficult problems about the integers [...] Full article

Research

Jump to: Editorial

14 pages, 300 KiB  
Article
A New Class of Leonardo Hybrid Numbers and Some Remarks on Leonardo Quaternions over Finite Fields
by Elif Tan, Diana Savin and Semih Yılmaz
Mathematics 2023, 11(22), 4701; https://doi.org/10.3390/math11224701 - 20 Nov 2023
Cited by 6 | Viewed by 855
Abstract
In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study [...] Read more.
In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Zp for special values of prime integer p. Full article
17 pages, 312 KiB  
Article
Generalized Fibonacci Sequences for Elliptic Curve Cryptography
by Zakariae Cheddour, Abdelhakim Chillali and Ali Mouhib
Mathematics 2023, 11(22), 4656; https://doi.org/10.3390/math11224656 - 15 Nov 2023
Viewed by 1104
Abstract
The Fibonacci sequence is a well-known sequence of numbers with numerous applications in mathematics, computer science, and other fields. In recent years, there has been growing interest in studying Fibonacci-like sequences on elliptic curves. These sequences have a number of exciting properties and [...] Read more.
The Fibonacci sequence is a well-known sequence of numbers with numerous applications in mathematics, computer science, and other fields. In recent years, there has been growing interest in studying Fibonacci-like sequences on elliptic curves. These sequences have a number of exciting properties and can be used to build new encryption systems. This paper presents a further generalization of the Fibonacci sequence defined on elliptic curves. We also describe an encryption system using this sequence which is based on the discrete logarithm problem on elliptic curves. Full article
12 pages, 303 KiB  
Article
On Indices of Septic Number Fields Defined by Trinomials x7 + ax + b
by Lhoussain El Fadil
Mathematics 2023, 11(21), 4441; https://doi.org/10.3390/math11214441 - 26 Oct 2023
Viewed by 524
Abstract
Let K be a septic number field generated by a root, α, of an irreducible trinomial, x7+ax+bZ[x]. In this paper, for every prime integer, p, we calculate [...] Read more.
Let K be a septic number field generated by a root, α, of an irreducible trinomial, x7+ax+bZ[x]. In this paper, for every prime integer, p, we calculate νp(i(K)); the highest power of p dividing the index, i(K), of the number field, K. In particular, we calculate the index, i(K). In application, when the index of K is not trivial, then K is not monogenic. Full article
16 pages, 319 KiB  
Article
Remarks on the Coefficients of Inverse Cyclotomic Polynomials
by Dorin Andrica and Ovidiu Bagdasar
Mathematics 2023, 11(17), 3622; https://doi.org/10.3390/math11173622 - 22 Aug 2023
Viewed by 661
Abstract
Cyclotomic polynomials play an imporant role in discrete mathematics. Recently, inverse cyclotomic polynomials have been defined and investigated. In this paper, we present some recent advances related to the coefficients of inverse cyclotomic polynomials, including a practical recursive formula for their calculation and [...] Read more.
Cyclotomic polynomials play an imporant role in discrete mathematics. Recently, inverse cyclotomic polynomials have been defined and investigated. In this paper, we present some recent advances related to the coefficients of inverse cyclotomic polynomials, including a practical recursive formula for their calculation and numerical simulations. Full article
13 pages, 291 KiB  
Article
A New Semi-Inner Product and pn-Angle in the Space of p-Summable Sequences
by Muh Nur, Mawardi Bahri, Anna Islamiyati and Harmanus Batkunde
Mathematics 2023, 11(14), 3139; https://doi.org/10.3390/math11143139 - 16 Jul 2023
Viewed by 664
Abstract
In this paper, we propose a definition for a semi-inner product in the space of p-summable sequences equipped with an n-norm. Using this definition, we introduce the concepts of pn-orthogonality and the pn-angle between two vectors in [...] Read more.
In this paper, we propose a definition for a semi-inner product in the space of p-summable sequences equipped with an n-norm. Using this definition, we introduce the concepts of pn-orthogonality and the pn-angle between two vectors in the space of p-summable sequences. For the special case n = 1, these concepts are identical to previous studies. We also introduce the notion of the pn-angle between one-dimensional subspaces and arbitrary-dimensional subspaces. The authors believe that the results obtained in this paper are very significant, especially in the theory of n-normed space in functional analysis. Full article
8 pages, 245 KiB  
Article
A Bound for a Sum of Products of Two Characters and Its Application
by Teerapat Srichan
Mathematics 2023, 11(11), 2507; https://doi.org/10.3390/math11112507 - 30 May 2023
Viewed by 778
Abstract
Using the exponent pair method, a bound is derived for the sum manbxχ1a(m)χ2b(n), where a,b are fixed positive integers, [...] Read more.
Using the exponent pair method, a bound is derived for the sum manbxχ1a(m)χ2b(n), where a,b are fixed positive integers, χ1,χ2 are primitive Dirichlet characters modulo q1 and q2, respectively, and χ1a,χ2b are not principal characters. As an application, an estimate for the error term in an asymptotic formula for the number of square-full integers simultaneously belonging to two arithmetic progressions is obtained. Full article
10 pages, 289 KiB  
Article
Density of Some Special Sequences Modulo 1
by Artūras Dubickas
Mathematics 2023, 11(7), 1727; https://doi.org/10.3390/math11071727 - 4 Apr 2023
Cited by 1 | Viewed by 963
Abstract
In this paper, we explicitly describe all the elements of the sequence of fractional parts {af(n)/n}, n=1,2,3,, where [...] Read more.
In this paper, we explicitly describe all the elements of the sequence of fractional parts {af(n)/n}, n=1,2,3,, where f(x)Z[x] is a nonconstant polynomial with positive leading coefficient and a2 is an integer. We also show that each value w={af(n)/n}, where nnf and nf is the least positive integer such that f(n)n/2 for every nnf, is attained by infinitely many terms of this sequence. These results combined with some earlier estimates on the gaps between two elements of a subgroup of the multiplicative group Zm* of the residue ring Zm imply that this sequence is everywhere dense in [0,1]. In the case when f(x)=x this was first established by Cilleruelo et al. by a different method. More generally, we show that the sequence {af(n)/nd}, n=1,2,3,, is everywhere dense in [0,1] if fZ[x] is a nonconstant polynomial with positive leading coefficient and a2, d1 are integers such that d has no prime divisors other than those of a. In particular, this implies that for any integers a2 and b1 the sequence of fractional parts {an/nb}, n=1,2,3,, is everywhere dense in [0,1]. Full article
16 pages, 330 KiB  
Article
Almost Repdigit k-Fibonacci Numbers with an Application of k-Generalized Fibonacci Sequences
by Alaa Altassan and Murat Alan
Mathematics 2023, 11(2), 455; https://doi.org/10.3390/math11020455 - 14 Jan 2023
Viewed by 1783
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
10 pages, 268 KiB  
Article
Pauli Gaussian Fibonacci and Pauli Gaussian Lucas Quaternions
by Ayşe Zeynep Azak
Mathematics 2022, 10(24), 4655; https://doi.org/10.3390/math10244655 - 8 Dec 2022
Cited by 1 | Viewed by 1176
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
9 pages, 754 KiB  
Article
On the Classification of Telescopic Numerical Semigroups of Some Fixed Multiplicity
by Ying Wang, Muhammad Ahsan Binyamin, Iqra Amin, Adnan Aslam and Yongsheng Rao
Mathematics 2022, 10(20), 3871; https://doi.org/10.3390/math10203871 - 18 Oct 2022
Viewed by 1317
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
10 pages, 359 KiB  
Article
Novel Authentication Protocols Based on Quadratic Diophantine Equations
by Avinash Vijayarangan, Veena Narayanan, Vijayarangan Natarajan and Srikanth Raghavendran
Mathematics 2022, 10(17), 3136; https://doi.org/10.3390/math10173136 - 1 Sep 2022
Cited by 1 | Viewed by 1438
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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10 pages, 277 KiB  
Article
Some Remarks on the Divisibility of the Class Numbers of Imaginary Quadratic Fields
by Kwang-Seob Kim
Mathematics 2022, 10(14), 2488; https://doi.org/10.3390/math10142488 - 17 Jul 2022
Cited by 1 | Viewed by 1086
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
11 pages, 280 KiB  
Article
A Generalized Bohr–Jessen Type Theorem for the Epstein Zeta-Function
by Antanas Laurinčikas and Renata Macaitienė
Mathematics 2022, 10(12), 2042; https://doi.org/10.3390/math10122042 - 13 Jun 2022
Cited by 1 | Viewed by 1224
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
13 pages, 298 KiB  
Article
Residuated Lattices with Noetherian Spectrum
by Dana Piciu and Diana Savin
Mathematics 2022, 10(11), 1831; https://doi.org/10.3390/math10111831 - 26 May 2022
Viewed by 1104
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
13 pages, 284 KiB  
Article
New Properties and Identities for Fibonacci Finite Operator Quaternions
by Nazlıhan Terzioğlu, Can Kızılateş and Wei-Shih Du
Mathematics 2022, 10(10), 1719; https://doi.org/10.3390/math10101719 - 17 May 2022
Cited by 5 | Viewed by 1558
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
13 pages, 305 KiB  
Article
New Zero-Density Results for Automorphic L-Functions of GL(n)
by Wenjing Ding, Huafeng Liu and Deyu Zhang
Mathematics 2021, 9(17), 2061; https://doi.org/10.3390/math9172061 - 26 Aug 2021
Viewed by 1112
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
21 pages, 314 KiB  
Article
Regularities in Ordered n-Ary Semihypergroups
by Jukkrit Daengsaen and Sorasak Leeratanavalee
Mathematics 2021, 9(16), 1857; https://doi.org/10.3390/math9161857 - 5 Aug 2021
Cited by 4 | Viewed by 1579
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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10 pages, 249 KiB  
Article
Some Properties of Euler’s Function and of the Function τ and Their Generalizations in Algebraic Number Fields
by Nicuşor Minculete and Diana Savin
Mathematics 2021, 9(15), 1710; https://doi.org/10.3390/math9151710 - 21 Jul 2021
Viewed by 1787
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
27 pages, 669 KiB  
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
by Busakorn Aiewcharoen, Ratinan Boonklurb and Nanthiya Konglawan
Mathematics 2021, 9(12), 1390; https://doi.org/10.3390/math9121390 - 15 Jun 2021
Cited by 2 | Viewed by 1876
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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9 pages, 315 KiB  
Article
On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers
by Eva Trojovská  and Pavel Trojovský
Mathematics 2021, 9(9), 962; https://doi.org/10.3390/math9090962 - 25 Apr 2021
Cited by 1 | Viewed by 1682
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
17 pages, 321 KiB  
Article
On Generalized Lucas Pseudoprimality of Level k
by Dorin Andrica and Ovidiu Bagdasar
Mathematics 2021, 9(8), 838; https://doi.org/10.3390/math9080838 - 12 Apr 2021
Cited by 4 | Viewed by 1838
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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