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16 pages, 17598 KiB

Open AccessArticle

Perrin Numbers That Are Concatenations of a Perrin Number and a Padovan Number in Base b

by Merve Güney Duman

Symmetry 2025, 17(3), 364; https://doi.org/10.3390/sym17030364 - 27 Feb 2025

Viewed by 422

Let

{(P_{k})}_{k \geq 0}

be a Padovan sequence and

{(R_{k})}_{k \geq 0}

be a Perrin sequence. Let n, m, b, and k be non-negative integers, where

2 \leq b \leq 10

. [...] Read more.

Let

{(P_{k})}_{k \geq 0}

be a Padovan sequence and

{(R_{k})}_{k \geq 0}

be a Perrin sequence. Let n, m, b, and k be non-negative integers, where

2 \leq b \leq 10

. In this paper, we are devoted to delving into the equations

R_{n} = b^{d} P_{m} + R_{k}

and

R_{n} = b^{d} R_{m} + P_{k}

, where d is the number of digits of

R_{k}

or

P_{k}

in base b. We show that the sets of solutions are

R_{n} \in {R_{5}, R_{6}, R_{7}, R_{8}, R_{9}, R_{10}, R_{11}, R_{12}, R_{13}, R_{14}, R_{15},

R_{16}, R_{17}, R_{19}, R_{23}, R_{25}, R_{27}}

for the first equation and

R_{n} \in {R_{0}, R_{2}, R_{3}, R_{4}, R_{5}, R_{6}, R_{7}, R_{8}, R_{9}, R_{10}, R_{11}, R_{12}, R_{13}, R_{14}, R_{15}, R_{16}, R_{17}, R_{18}, R_{20}, R_{21}}

for the second equation. Our approach employs advanced techniques in Diophantine analysis, including linear forms in logarithms, continued fractions, and the properties of Padovan and Perrin sequences in base b. We investigate both the deep structural symmetries and the complex structures that connect recurrence relations and logarithmic forms within Diophantine equations involving special number sequences. Full article

(This article belongs to the Section Mathematics)

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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://doi.org/10.3390/math11020455 - 14 Jan 2023

Cited by 2 | Viewed by 2632

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)

10 pages, 279 KiB

Open AccessArticle

On Diophantine Equations Related to Narayana’s Cows Sequence and Double Factorials or Repdigits

by Yanjiao Ji, Peng Yang and Tianxin Cai

Symmetry 2022, 14(10), 2004; https://doi.org/10.3390/sym14102004 - 24 Sep 2022

Viewed by 1570

Abstract

In this paper, we determine all the Narayana’s cows numbers that are factorials or double factorials. We also show that 88 is the only repdigit (i.e., a class of numbers that has reflectional symmetry across a vertical axis) that can be written as [...] Read more.

In this paper, we determine all the Narayana’s cows numbers that are factorials or double factorials. We also show that 88 is the only repdigit (i.e., a class of numbers that has reflectional symmetry across a vertical axis) that can be written as the product of consecutive Narayana’s cows numbers. Full article

(This article belongs to the Section Mathematics)

15 pages, 302 KiB

Open AccessArticle

Repdigits as Sums of Four Tribonacci Numbers

by Yuetong Zhou, Peng Yang, Shaonan Zhang and Kaiqiang Zhang

Symmetry 2022, 14(9), 1931; https://doi.org/10.3390/sym14091931 - 16 Sep 2022

Viewed by 1776

Abstract

In this paper, we show that 66666 is the largest repdigit expressible as the sum of four tribonacci numbers. We used Binet’s formula, Baker’s theory, and a reduction method during the proving procedure. We also used the periodic properties of tribonacci number modulo [...] Read more.

In this paper, we show that 66666 is the largest repdigit expressible as the sum of four tribonacci numbers. We used Binet’s formula, Baker’s theory, and a reduction method during the proving procedure. We also used the periodic properties of tribonacci number modulo 9 to deal with three individual cases. Full article

(This article belongs to the Topic Engineering Mathematics)

12 pages, 322 KiB

Open AccessArticle

Curious Generalized Fibonacci Numbers

by Jose L. Herrera, Jhon J. Bravo and Carlos A. Gómez

Mathematics 2021, 9(20), 2588; https://doi.org/10.3390/math9202588 - 15 Oct 2021

Cited by 3 | Viewed by 3585

Abstract

A generalization of the well-known Fibonacci sequence is the

k -

Fibonacci sequence whose first k terms are

0, \dots, 0, 1

and each term afterwards is the sum of the preceding k terms. In this paper, we find all [...] Read more.

A generalization of the well-known Fibonacci sequence is the

k -

Fibonacci sequence whose first k terms are

0, \dots, 0, 1

and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i.e., numbers whose base ten representation have the form

a \dots a b \dots b a \dots a)

. This work continues and extends the prior result of Trojovský, who found all Fibonacci numbers with a prescribed block of digits, and the result of Alahmadi et al., who searched for k-Fibonacci numbers, which are concatenation of two repdigits. Full article

10 pages, 282 KiB

Open AccessArticle

Repdigits as Product of Terms of k-Bonacci Sequences

by Petr Coufal and Pavel Trojovský

Mathematics 2021, 9(6), 682; https://doi.org/10.3390/math9060682 - 22 Mar 2021

Cited by 7 | Viewed by 2689

Abstract

For any integer

k \geq 2

, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values [...] Read more.

For any integer

k \geq 2

, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values

F_{- (k - 2)}^{(k)} = \dots = F_{0}^{(k)} = 0

and

F_{1}^{(k)} = 1

and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form

a a \dots a

, with

a \in [1, 9]

) in the sequence

{(F_{n}^{(k)} F_{n}^{(k + m)})}_{n}

, for

m \in [1, 9]

. This result generalizes a recent work of Bednařík and Trojovská (the case in which

(k, m) = (2, 1)

). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method). Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

7 pages, 233 KiB

Open AccessArticle

On Repdigits as Sums of Fibonacci and Tribonacci Numbers

by Pavel Trojovský

Symmetry 2020, 12(11), 1774; https://doi.org/10.3390/sym12111774 - 26 Oct 2020

Cited by 3 | Viewed by 2104

Abstract

In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the [...] Read more.

In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a ”symmetrical” type of numbers) that can be written in the form

F_{n} + T_{n}

, for some

n \geq 1

, where

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

and

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

are the sequences of Fibonacci and Tribonacci numbers, respectively. Full article

(This article belongs to the Section Mathematics)

8 pages, 252 KiB

Open AccessArticle

Repdigits as Product of Fibonacci and Tribonacci Numbers

by Dušan Bednařík and Eva Trojovská

Mathematics 2020, 8(10), 1720; https://doi.org/10.3390/math8101720 - 7 Oct 2020

Cited by 7 | Viewed by 2519

Abstract

In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a [...] Read more.

In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a Tribonacci number (both with the same indexes). To work on this problem, our approach is to combine lower bounds from the Baker’s theory with reduction methods (based on the theory of continued fractions) due to Dujella and Pethö. Full article

(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)

8 pages, 255 KiB

Open AccessArticle

Lucas Numbers Which Are Concatenations of Two Repdigits

by Yunyun Qu and Jiwen Zeng

Mathematics 2020, 8(8), 1360; https://doi.org/10.3390/math8081360 - 13 Aug 2020

Cited by 15 | Viewed by 2981

Abstract

In this paper, we find all Lucas numbers written in the form

\bar{c \dots c d \dots d}

, where

\bar{c \dots c d \dots d}

is the concatenation of two repdigits in base 10 with

c, d \in {0, 1, \dots, 9}

,

c \neq d

and

c > 0

. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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