Mathematics

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11 pages, 278 KiB

Open AccessArticle

Reciprocal Formulae among Pell and Lucas Polynomials

by Mei Bai, Wenchang Chu and Dongwei Guo

Mathematics 2022, 10(15), 2691; https://doi.org/10.3390/math10152691 - 29 Jul 2022

Cited by 3 | Viewed by 1374

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

Open AccessArticle

Rings of Multisets and Integer Multinumbers

by Yuriy Chopyuk, Taras Vasylyshyn and Andriy Zagorodnyuk

Mathematics 2022, 10(5), 778; https://doi.org/10.3390/math10050778 - 28 Feb 2022

Cited by 11 | Viewed by 2429

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

Open AccessArticle

On Two Problems Related to Divisibility Properties of z(n)

by Pavel Trojovský

Mathematics 2021, 9(24), 3273; https://doi.org/10.3390/math9243273 - 16 Dec 2021

Viewed by 1806

Abstract

The order of appearance (in the Fibonacci sequence) function

z : Z_{\geq 1} \to Z_{\geq 1}

is an arithmetic function defined for a positive integer n as [...] Read more.

The order of appearance (in the Fibonacci sequence) function

z : Z_{\geq 1} \to Z_{\geq 1}

is an arithmetic function defined for a positive integer n as

z (n) = min {k \geq 1 : F_{k} \equiv 0 (mod n)}

. 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 (n^{2})

, 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

lim sup_{n \to \infty} (z (n + 1) - z (n)) / {(log n)}^{2 - ϵ} = \infty

, for all

ϵ \in (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

Open AccessArticle

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://doi.org/10.3390/math9222912 - 16 Nov 2021

Viewed by 1304

Abstract

Let

{(F_{n})}_{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

{(F_{n})}_{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 {k \geq 1 : n ∣ F_{k}}

. Very recently, Trojovská and Venkatachalam proved that, for any

k \geq 1

, the number

z (n)

is divisible by

2^{k}

, for almost all integers

n \geq 1

(in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing

2^{k}

by any integer

m \geq 1

. 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

Open AccessArticle

On Andrews’ Partitions with Parts Separated by Parity

by Abdulaziz M. Alanazi and Darlison Nyirenda

Mathematics 2021, 9(21), 2693; https://doi.org/10.3390/math9212693 - 23 Oct 2021

Cited by 1 | Viewed by 1546

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

Open AccessArticle

The Proof of a Conjecture Related to Divisibility Properties of z(n)

by Eva Trojovská and Kandasamy Venkatachalam

Mathematics 2021, 9(20), 2638; https://doi.org/10.3390/math9202638 - 19 Oct 2021

Cited by 1 | Viewed by 1353

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

F_{k}

. 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

F_{k}

. 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

n \geq 1

for which the integer 4 divides

z (n)

. In this paper, among other things, we prove that for any

k \geq 1

, the number

z (n)

is divisible by

2^{k}

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

Open AccessArticle

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://doi.org/10.3390/math9161971 - 18 Aug 2021

Cited by 2 | Viewed by 1323

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

Open AccessArticle

On Some Properties of the Limit Points of (z(n)/n)_n

by Eva Trojovská and Kandasamy Venkatachalam

Mathematics 2021, 9(16), 1931; https://doi.org/10.3390/math9161931 - 13 Aug 2021

Viewed by 1313

Abstract

Let

{(F_{n})}_{n \geq 0}

be the sequence of Fibonacci numbers. The order of appearance of an integer

n \geq 1

is defined as [...] Read more.

Let

{(F_{n})}_{n \geq 0}

be the sequence of Fibonacci numbers. The order of appearance of an integer

n \geq 1

is defined as

z (n) : = min {k \geq 1 : n ∣ F_{k}}

. Let

Z^{'}

be the set of all limit points of

{z (n) / n : n \geq 1}

. By some theoretical results on the growth of the sequence

{(z (n) / n)}_{n \geq 1}

, we gain a better understanding of the topological structure of the derived set

Z^{'}

. For instance,

{0, 1, \frac{3}{2}, 2} \subseteq Z^{'} \subseteq [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^{'} \cap (t, 2)

is the empty set. In this paper, we improve this result by proving that

(\frac{12}{7}, 2)

is the largest subinterval of

[0, 2]

which does not intersect

Z^{'}

. In addition, we show a connection between the sequence

{(x_{n})}_{n}

, for which

z (x_{n}) / x_{n}

tends to

r > 0

(as

n \to \infty

), and the number of preimages of r under the map

m \mapsto z (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

Open AccessArticle

On the Parity of the Order of Appearance in the Fibonacci Sequence

by Pavel Trojovský

Mathematics 2021, 9(16), 1928; https://doi.org/10.3390/math9161928 - 13 Aug 2021

Cited by 2 | Viewed by 1670

Abstract

Let

{(F_{n})}_{n \geq 0}

be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence)

z : Z_{\geq 1} \to Z_{\geq 1}

is defined as [...] Read more.

Let

{(F_{n})}_{n \geq 0}

be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence)

z : Z_{\geq 1} \to Z_{\geq 1}

is defined as

z (n) : = min {k \geq 1 : F_{k} \equiv 0 (mod n)}

. 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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