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Algebraic, Analytic, and Computational Number Theory and Its Applications

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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 31041

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Special Issue Editors

Dr. Diana Savin

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Guest Editor

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
Special Issues, Collections and Topics in MDPI journals

Dr. Nicusor Minculete

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Guest Editor

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
Special Issues, Collections and Topics in MDPI journals

Dr. Vincenzo Acciaro

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Guest Editor

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
Special Issues, Collections and Topics in MDPI journals

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

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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

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

Related Special Issue

Algebraic, Analytic, and Computational Number Theory and Its Applications, 2nd Edition in Mathematics (2 articles)

Published Papers (22 papers)

Download All Papers

Editorial

Jump to: Research

7 pages, 242 KiB

Open AccessEditorial

Algebraic, Analytic, and Computational Number Theory and Its Applications

by Diana Savin, Nicusor Minculete and Vincenzo Acciaro

Mathematics 2024, 12(1), 10; https://0-doi-org.brum.beds.ac.uk/10.3390/math12010010 - 20 Dec 2023

Viewed by 1131

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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

Research

Jump to: Editorial

14 pages, 300 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math11224701 - 20 Nov 2023

Cited by 6 | Viewed by 822

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 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

Z_{p}

for special values of prime integer p. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

17 pages, 312 KiB

Open AccessArticle

Generalized Fibonacci Sequences for Elliptic Curve Cryptography

by Zakariae Cheddour, Abdelhakim Chillali and Ali Mouhib

Mathematics 2023, 11(22), 4656; https://0-doi-org.brum.beds.ac.uk/10.3390/math11224656 - 15 Nov 2023

Viewed by 1011

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 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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

12 pages, 303 KiB

Open AccessArticle

On Indices of Septic Number Fields Defined by Trinomials x⁷ + ax + b

by Lhoussain El Fadil

Mathematics 2023, 11(21), 4441; https://0-doi-org.brum.beds.ac.uk/10.3390/math11214441 - 26 Oct 2023

Viewed by 510

Abstract

Let

K

be a septic number field generated by a root,

α

, of an irreducible trinomial,

x^{7} + a x + b \in Z [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,

x^{7} + a x + b \in Z [x]

. In this paper, for every prime integer,

p

, we calculate

ν_{p} (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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

16 pages, 319 KiB

Open AccessArticle

Remarks on the Coefficients of Inverse Cyclotomic Polynomials

by Dorin Andrica and Ovidiu Bagdasar

Mathematics 2023, 11(17), 3622; https://0-doi-org.brum.beds.ac.uk/10.3390/math11173622 - 22 Aug 2023

Viewed by 635

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.

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

13 pages, 291 KiB

Open AccessArticle

A New Semi-Inner Product and p_n-Angle in the Space of p-Summable Sequences

by Muh Nur, Mawardi Bahri, Anna Islamiyati and Harmanus Batkunde

Mathematics 2023, 11(14), 3139; https://0-doi-org.brum.beds.ac.uk/10.3390/math11143139 - 16 Jul 2023

Viewed by 645

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

p_{n}

-orthogonality and the

p_{n}

-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

p_{n}

-orthogonality and the

p_{n}

-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

p_{n}

-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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

8 pages, 245 KiB

Open AccessArticle

A Bound for a Sum of Products of Two Characters and Its Application

by Teerapat Srichan

Mathematics 2023, 11(11), 2507; https://0-doi-org.brum.beds.ac.uk/10.3390/math11112507 - 30 May 2023

Viewed by 758

Abstract

Using the exponent pair method, a bound is derived for the sum

\sum_{m^{a} n^{b} \leq x} χ_{1}^{a} (m) χ_{2}^{b} (n)

, where

a, b

are fixed positive integers, [...] Read more.

Using the exponent pair method, a bound is derived for the sum

\sum_{m^{a} n^{b} \leq x} χ_{1}^{a} (m) χ_{2}^{b} (n)

, where

a, b

are fixed positive integers,

χ_{1}, χ_{2}

are primitive Dirichlet characters modulo

q_{1}

and

q_{2}

, respectively, and

χ_{1}^{a}, χ_{2}^{b}

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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

10 pages, 289 KiB

Open AccessFeature PaperArticle

Density of Some Special Sequences Modulo 1

by Artūras Dubickas

Mathematics 2023, 11(7), 1727; https://0-doi-org.brum.beds.ac.uk/10.3390/math11071727 - 4 Apr 2023

Cited by 1 | Viewed by 946

Abstract

In this paper, we explicitly describe all the elements of the sequence of fractional parts

{a^{f (n)} / n}

n = 1, 2, 3, \dots

, where [...] Read more.

In this paper, we explicitly describe all the elements of the sequence of fractional parts

{a^{f (n)} / n}

n = 1, 2, 3, \dots

, where

f (x) \in Z [x]

is a nonconstant polynomial with positive leading coefficient and

a \geq 2

is an integer. We also show that each value

w = {a^{f (n)} / n}

, where

n \geq n_{f}

and

n_{f}

is the least positive integer such that

f (n) \geq n / 2

for every

n \geq n_{f}

, 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

Z_{m}^{*}

of the residue ring

Z_{m}

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

{a^{f (n)} / n^{d}}

n = 1, 2, 3, \dots

, is everywhere dense in

[0, 1]

f \in Z [x]

is a nonconstant polynomial with positive leading coefficient and

a \geq 2

d \geq 1

are integers such that d has no prime divisors other than those of a. In particular, this implies that for any integers

a \geq 2

and

b \geq 1

the sequence of fractional parts

{a^{n} / \sqrt[b]{n}}

n = 1, 2, 3, \dots

, is everywhere dense in

[0, 1]

. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

16 pages, 330 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math11020455 - 14 Jan 2023

Viewed by 1758

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 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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

10 pages, 268 KiB

Open AccessArticle

Pauli Gaussian Fibonacci and Pauli Gaussian Lucas Quaternions

by Ayşe Zeynep Azak

Mathematics 2022, 10(24), 4655; https://0-doi-org.brum.beds.ac.uk/10.3390/math10244655 - 8 Dec 2022

Cited by 1 | Viewed by 1154

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 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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

9 pages, 754 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math10203871 - 18 Oct 2022

Viewed by 1288

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 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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

10 pages, 359 KiB

Open AccessArticle

Novel Authentication Protocols Based on Quadratic Diophantine Equations

by Avinash Vijayarangan, Veena Narayanan, Vijayarangan Natarajan and Srikanth Raghavendran

Mathematics 2022, 10(17), 3136; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173136 - 1 Sep 2022

Cited by 1 | Viewed by 1415

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.

x_{1}^{2} + x_{2}^{2} = y_{1}^{2} + y_{2}^{2} (x_{1}, x_{2}, y_{1}, y_{2} > 0)

, as well as their use in communication protocols. Given one pair

(x_{1}, y_{1})

, finding another pair

(x_{2}, y_{2})

satisfying

x_{1}^{2} + x_{2}^{2} = y_{1}^{2} + y_{2}^{2}

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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

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10 pages, 277 KiB

Open AccessArticle

Some Remarks on the Divisibility of the Class Numbers of Imaginary Quadratic Fields

by Kwang-Seob Kim

Mathematics 2022, 10(14), 2488; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142488 - 17 Jul 2022

Cited by 1 | Viewed by 1063

Abstract

For a given integer n, we provide some families of imaginary quadratic number fields of the form

Q (\sqrt{4 q^{2} - p^{n}})

, whose ideal class group has a subgroup isomorphic to

Z / n Z

. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

11 pages, 280 KiB

Open AccessFeature PaperArticle

A Generalized Bohr–Jessen Type Theorem for the Epstein Zeta-Function

by Antanas Laurinčikas and Renata Macaitienė

Mathematics 2022, 10(12), 2042; https://0-doi-org.brum.beds.ac.uk/10.3390/math10122042 - 13 Jun 2022

Cited by 1 | Viewed by 1193

Abstract

Let Q be a positive defined

n \times n

matrix and

Q [\underset{̲}{x}] = {\underset{̲}{x}}^{T} Q \underset{̲}{x}

. The Epstein zeta-function

ζ (s; Q)

s = σ + i t

, is defined, [...] Read more.

Let Q be a positive defined

n \times n

matrix and

Q [\underset{̲}{x}] = {\underset{̲}{x}}^{T} Q \underset{̲}{x}

. The Epstein zeta-function

ζ (s; Q)

s = σ + i t

, is defined, for

σ > \frac{n}{2}

, by the series

ζ (s; Q) = \sum_{\underset{̲}{x} \in Z^{n} \ {\underset{̲}{0}}}

{(Q [\underset{̲}{x}])}^{- s},

and is meromorphically continued on the whole complex plane. Suppose that

n ⩾ 4

is even and

φ (t)

is a differentiable function with a monotonic derivative. In the paper, it is proved that

\frac{1}{T} meas \{t \in [0, T] : ζ (σ + i φ (t); Q) \in A\}

A \in B (C),

converges weakly to an explicitly given probability measure on

(C, B (C))

T \to \infty

. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

13 pages, 298 KiB

Open AccessFeature PaperArticle

Residuated Lattices with Noetherian Spectrum

by Dana Piciu and Diana Savin

Mathematics 2022, 10(11), 1831; https://0-doi-org.brum.beds.ac.uk/10.3390/math10111831 - 26 May 2022

Viewed by 1086

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.

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

13 pages, 284 KiB

Open AccessArticle

New Properties and Identities for Fibonacci Finite Operator Quaternions

by Nazlıhan Terzioğlu, Can Kızılateş and Wei-Shih Du

Mathematics 2022, 10(10), 1719; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101719 - 17 May 2022

Cited by 5 | Viewed by 1536

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.

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

13 pages, 305 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math9172061 - 26 Aug 2021

Viewed by 1101

Abstract

Let

L (s, π)

be an automorphic L-function of

G L (n)

, where

π

is an automorphic representation of group

G L (n)

over rational number field

Q

. In this paper, we study [...] Read more.

Let

L (s, π)

be an automorphic L-function of

G L (n)

, where

π

is an automorphic representation of group

G L (n)

over rational number field

Q

. In this paper, we study the zero-density estimates for

L (s, π)

. Define

N_{π} (σ, T_{1}, T_{2})

= ♯ {

ρ

β

i γ

L (ρ, π)

= 0,

σ < β < 1

T_{1} \leq γ \leq T_{2}

}, where

0 \leq σ < 1

and

T_{1} < T_{2}

. We first establish an upper bound for

N_{π} (σ, T, 2 T)

when

σ

is close to 1. Then we restrict the imaginary part

γ

into a narrow strip

[T, T + T^{α}]

with

0 < α \leq 1

and prove some new zero-density results on

N_{π} (σ, T, T + T^{α})

under specific conditions, which improves previous results when

σ

near

\frac{3}{4}

and 1, respectively. The proofs rely on the zero detecting method and the Halász-Montgomery method. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

21 pages, 314 KiB

Open AccessArticle

Regularities in Ordered n-Ary Semihypergroups

by Jukkrit Daengsaen and Sorasak Leeratanavalee

Mathematics 2021, 9(16), 1857; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161857 - 5 Aug 2021

Cited by 4 | Viewed by 1559

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

1 \leq j \leq n

and

n \geq 3

. 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

1 \leq j \leq n

and

n \geq 3

. 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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

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10 pages, 249 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math9151710 - 21 Jul 2021

Viewed by 1763

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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

27 pages, 669 KiB

Open AccessArticle

Global and Local Behavior of the System of Piecewise Linear Difference Equations x_n+1 = |x_n| − y_n − b and y_n+1 = x_n − |y_n| + 1 Where b ≥ 4

by Busakorn Aiewcharoen, Ratinan Boonklurb and Nanthiya Konglawan

Mathematics 2021, 9(12), 1390; https://0-doi-org.brum.beds.ac.uk/10.3390/math9121390 - 15 Jun 2021

Cited by 2 | Viewed by 1854

Abstract

The aim of this article is to study the system of piecewise linear difference equations

x_{n + 1} = | x_{n} | - y_{n} - b

and [...] Read more.

The aim of this article is to study the system of piecewise linear difference equations

x_{n + 1} = | x_{n} | - y_{n} - b

and

y_{n + 1} = x_{n} - | y_{n} | + 1

where

n \geq 0

. A global behavior for

b = 4

shows that all solutions become the equilibrium point. For a large value of

| x_{0} |

and

| y_{0} |

, we can prove that (i) if

b = 5

, then the solution becomes the equilibrium point and (ii) if

b \geq 6

, then the solution becomes the periodic solution of prime period 5. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

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9 pages, 315 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/math9090962 - 25 Apr 2021

Cited by 1 | Viewed by 1656

Abstract

Let

{(t_{n}^{(r)})}_{n \geq 0}

be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence [...] Read more.

Let

{(t_{n}^{(r)})}_{n \geq 0}

be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence

t_{n}^{(r)} = t_{n - 1}^{(r)} + \dots + t_{n - r}^{(r)}

for

n \geq r

, with initial values

t_{0}^{(r)} = 0

and

t_{i}^{(r)} = 1

, for all

1 \leq i \leq r

. In 2002, Grossman and Luca searched for terms of the sequence

{(t_{n}^{(2)})}_{n}

, which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any

ℓ \geq 1

, there exists an effectively computable constant

C = C (ℓ) > 0

(only depending on ℓ), such that, if

(m, n, r)

is a solution of

t_{m}^{(r)} = n! + (n + 1)! + \dots + (n + ℓ)!

, with r even, then

max {m, n, r} < C

. As an application, we solve the previous equation for all

1 \leq ℓ \leq 5

. Full article

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

17 pages, 321 KiB

Open AccessArticle

On Generalized Lucas Pseudoprimality of Level k

by Dorin Andrica and Ovidiu Bagdasar

Mathematics 2021, 9(8), 838; https://0-doi-org.brum.beds.ac.uk/10.3390/math9080838 - 12 Apr 2021

Cited by 4 | Viewed by 1810

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 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

(This article belongs to the Special Issue Algebraic, Analytic, and Computational Number Theory and Its Applications)

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