Research

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://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

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

Open AccessArticle

Pauli Gaussian Fibonacci and Pauli Gaussian Lucas Quaternions

by

Ayşe Zeynep Azak

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

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

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://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

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

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://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

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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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://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 (\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)

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://doi.org/10.3390/math10122042 - 13 Jun 2022

Cited by 1 | Viewed by 644

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))

as

T \to \infty

. Full article

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

Open AccessFeature PaperArticle

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 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

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

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://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

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

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://doi.org/10.3390/math9172061 - 26 Aug 2021

Viewed by 680

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)

Open AccessArticle

Regularities in Ordered n-Ary Semihypergroups

by

Jukkrit Daengsaen

and

Sorasak Leeratanavalee

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

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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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://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

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

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://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

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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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://doi.org/10.3390/math9090962 - 25 Apr 2021

Cited by 1 | Viewed by 1165

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)

Open AccessArticle

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 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

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

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