Lucas Number Research Articles

K–Generalized Lucas numbers, perfect powers and the problem of Pillai

For an integer k≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 2$$\\end{document}, let L(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{(k)}$$\\end{document} be the k–generalized Lucas sequence which starts with 0,…,2,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0, \\ldots , 2,1$$\\end{document} (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper we assume that an integer c can be represented in at least two ways as the difference between a k–generalized Lucas number and a power of b, then using the theory of nonzero linear forms in logarithms of algebraic numbers, we bound all possible solutions on this representation of c in terms of b. Finally, combination our general result and some known reduction procedures based on the continued fraction algorithm, we find all the integers c and their representations for b∈[2,10]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ b\\in [2,10]$$\\end{document}, this argument can be generalized to any b>10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ b> 10 $$\\end{document}.

Almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers

In this work, the general terms of almost balancers, almost cobalancers, almost Lucas-balancers and almost Lucas-cobalancers of first and second type are determined in terms of balancing and Lucas-balancing numbers. Later some relations on all almost balancing numbers and all almost balancers are obtained. Further the general terms of all balancing numbers, Pell numbers and Pell–Lucas number are determined in terms of almost balancers, almost Lucas-balancers, almost cobalancers and almost Lucas-cobalancers of first and second type.

Cholesky algorithm of a Lucas type matrix

Many generalizations have been made for Fibonacci and Lucas number sequences and many properties have been found about these sequences. In the article [13], the authors obtained many features of these sequences with the Cholesky decomposition algorithm, using the 2 x 2 matrix belonging to a generalization of the Fibonacci sequence. In this study, it is shown that many different features can be found by using a 2 x 2 matrix belonging to the Lucas number sequence with the same method.

The Generalized Lucas Primes in the Landau’s and Shanks’ Conjectures

Landau’s conjecture and Shanks’ conjecture state that there are infinitely many prime numbers of the forms x2+1 and x4+1 for some nonzero integer , respectively. In this paper, we present a technique for studying whether or not there are infinitely many prime numbers of the form x2+1 or x4+1 derived from some Lucas sequences of the first kind {Un(P,Q)} (or simply, {Un}) or the second kind {Vn(P,Q)} (or simply, {Vn}) , where P greater or equal to 1 and Q= 1 or -1. Furthermore, as applications we represent the procedure of this technique in case of x is either an integer or a Lucas number of the first or the second kind with x greater or equal to 1 and 1 less or equal to P less or equal to 20.

Neutrosophic Number Sequences: An introductory Study

In this paper, Neutrosophic definitions and properties of some special number sequences which are frequently found in the science literature, called Neutrosophic Number Sequences (NNSq) via Horadam sequence are studied for the first time. Especially for Neutrosophic Fibonacci (NFNq) and Neutrosophic Lucas (NLNq) number sequences, fundamental properties and identities such as Ruggles, Honsberger, Cassini, Catalan, d’Ocagne, and Tagiuri are given. In addition, Neutrosophic definitions of the sequences of Pell (NPNq), Pell-Lucas (NPLNq), Jacobsthal (NJNq), Jacobsthal-Lucas (NJLNq), Mersenne (NMNq), Mersenne-Lucas (NMLNq), Balancing (NBNq), and Lucas-Balancing (NLBNq) numbers are introduced. Besides defining these numbers and their sequences, since fuzzy and intuitionistic fuzzy sets are restrictions of neutrosophic sets, sequences of numbers within these sets are naturally and indirectly revealed.

On certain prime numbers in Lucas sequences

In 1964, Shanks conjectured that there are infinitely many primes of the form 1/2(x2 + 1). Therefore, the aim of this paper is to introduce a technique for studying whether or not there are infinitely many prime numbers of the form 1/2(x2 + 1) derived from some Lucas sequences of the first kind {Un(P, Q)} or the second kind {Vn(P, Q)}, where P ≥ 1 and Q = ±1. In addition, as applications we represent the procedure of this technique in case of x is an either integer or Lucas number of the first or the second kind with x ≥ 1 and 1 ≤ P ≤ 20.

On matrix sequences represented by negative indices Pell and Pell–Lucas number with the decoding of lucas blocking error correcting codes

This paper focuses on the matrix sequences that represent negative indices Pell and Pell–Lucas number. It will specifically provide the important relationships between negative indices Pell and Pell–Lucas number and Pell and Pell–Lucas matrix sequences. The research of new error correcting codes has received a lot of attention in recent years, specifically due to its potential use in quantum-resistant cryptosystems. These matrices are then applied in cryptology and have a great opportunity to report some cases that might be a mistake between the sender and the receiver by using Lucas blocking algorithm.

A TILING INTERPRETATION FOR (p,q)-FIBONACCI AND (p,q)-LUCAS NUMBERS

In this paper, we introduce a tiling approach to (p,q)-Fibonacci and (p,q)-Lucas numbers that generalize of the well-known Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal ve Jacobsthal-Lucas numbers. We show that nth (p,q)-Fibonacci number is interpreted as the number of ways to tile a 1×n board with cells labeled 1,2,...,n using colored 1×1 squares and 1×2 dominoes, where there are p kind colors for squares and q kind colors for dominoes. Then nth (p,q)-Lucas number is interpreted as the number of ways to tile a circular 1×n board with squares and dominoes. We also present some generalized Fibonacci and Lucas identities using this tiling approach.

Gauss Bronz Lucas Sayıları

The present work aims to introduce and study the Gaussian Bronze Lucas number sequence. Firstly, we define Gaussian Bronze Lucas numbers by extending the Bronze Lucas numbers. Then, we find the Binet formula and generating function for this number sequence. We also investigate some sum formulas and matrices related to the Gaussian Bronze Lucas numbers. Finally, we obtain some known equalities like Catalan, Cassini and d’Ocagne identities by considering the Binet formula of this sequence.

On the exponential Diophantine equation Fnx ± Fmx = a with a ∈{Fr,Lr}

In this paper, we will answer the question of when the sum or the difference of [Formula: see text]th powers of any two Fibonacci numbers becomes a Fibonacci number or a Lucas number. We prove that if this is possible with [Formula: see text], then [Formula: see text] may be only 3,4,5, or 10. Also, we give the result that the Diophantine equation in the title with [Formula: see text] has no solutions for [Formula: see text]

Kuzeydoğu Anadolu’da doğal olarak yayılış gösteren bazı taksonların Fibonacci ve Lucas sayıları

In this study, general information about obtaining Fibonacci and Lucas number sequences from the rabbit problem is given. Obtaining the golden ratio and phyllotaxis ratio with Fibonacci numbers is examined.The Fibonacci and Lucas number characteristics of plants belonging to the same species are the same all over the world. The phyllotaxis ratio of 47 plant taxa and the number of cone spirals of two pines belonging to Gymnospermae distributed in North East Anatolia were determined. 34 of 47 taxa which phyllotaxis ratio investigated are herbs and 13 ones are woody/scrubs. Three of the herbaceous plants are endemic (Psephellus straminicephalus(Hub.-Mor.) Wagenitz, Tripleurospermum fissurale(Sosn.) E.Hossainand Chesneya elegansFomin) and two are rare species (Callicephalus nitens(M.Bieb. ex Willd.) C.A.Mey.and Cirsium rigidumDC.). With these studies, a literature contribution was provided to the studies in the field.

Time-series analysis with smoothed Convolutional Neural Network

CNN originates from image processing and is not commonly known as a forecasting technique in time-series analysis which depends on the quality of input data. One of the methods to improve the quality is by smoothing the data. This study introduces a novel hybrid exponential smoothing using CNN called Smoothed-CNN (S-CNN). The method of combining tactics outperforms the majority of individual solutions in forecasting. The S-CNN was compared with the original CNN method and other forecasting methods such as Multilayer Perceptron (MLP) and Long Short-Term Memory (LSTM). The dataset is a year time-series of daily website visitors. Since there are no special rules for using the number of hidden layers, the Lucas number was used. The results show that S-CNN is better than MLP and LSTM, with the best MSE of 0.012147693 using 76 hidden layers at 80%:20% data composition.

Relation between matrices and the suborbital graphs by the special number sequences

under circuit and forest conditions. Special number sequences and special vertex values of minimal length paths in suborbital graphs have been associated in our previous studies. In these associations, matrix connections of the special continued fractions $\mathcal K (-1/-k)$, where $k\in \mathbb{Z}^{+}, \ k\geq 2$ with the values of the special number sequences are used and new identities are obtained. In this study, by producing new matrices, new identities related to Fibonacci, Lucas, Pell, and Pell-Lucas number sequences are found by using both recurrence relations and matrix connections of the continued fractions. In addition, the farthest vertex values of the minimal length path in the suborbital graph $\mathrm{\mathbf{F}}_{u,N}$ and these number sequences are associated.

Fibonacci–Lucas–Pell–Jacobsthal relations

In this paper, we prove several identities involving linear combinations of convolutions of the generalized Fibonacci and Lucas sequences. Our results apply more generally to broader classes of second-order linearly recurrent sequences with constant coefficients. As a consequence, we obtain as special cases many identities relating exactly four sequences amongst the Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal, and Jacobsthal–Lucas number sequences. We make use of algebraic arguments to establish our results, frequently employing the Binet-like formulas and generating functions of the corresponding sequences. Finally, our identities above may be extended so that they include only terms whose subscripts belong to a given arithmetic progression of the non-negative integers.

Some Divisibility Properties of Lucas Numbers

The Lucas number sequence is a popular number sequence that has been described as similar to the Fibonacci number sequence. A lot of research has been done on this number sequence. Some of these studies are on the divisibility properties of this number sequence. Carlitz (1964) examined the requirement that a given Lucas number can be divided by another Lucas number. After that, many studies have been done on this subject. In the present article, we obtain some divisibility properties of the Lucas Numbers. First, we examine the case $L_{(2n-1)m}/L_{m}$ and then we obtain $L_{\left( 2n-1\right) m}$ using different forms of Lucas numbers.

Repdigits as difference of two Fibonacci or Lucas numbers

In the present study we investigate all repdigits which are expressed as a difference of two Fibonacci or Lucas numbers. We show that if $F_{n}-F_{m}$ is a repdigit, where $F_{n}$ denotes the $n$-th Fibonacci number, then $(n,m)\in \{(7,3),(9,1),(9,2),(11,1),(11,2),$ $(11,9),(12,11),(15,10)\}.$ Further, if $L_{n}$ denotes the $n$-th Lucas number, then $L_{n}-L_{m}$ is a repdigit for $(n,m)\in\{(6,4),(7,4),(7,6),(8,2)\},$ where $n>m.$Namely, the only repdigits that can be expressed as difference of two Fibonacci numbers are $11,33,55,88$ and $555$; their representations are $11=F_{7}-F_{3},\33=F_{9}-F_{1}=F_{9}-F_{2},\55=F_{11}-F_{9}=F_{12}-F_{11},\88=F_{11}-F_{1}=F_{11}-F_{2},\555=F_{15}-F_{10}$ (Theorem 2). Similar result for difference of two Lucas numbers: The only repdigits that can be expressed as difference of two Lucas numbers are $11,22$ and $44;$ their representations are $11=L_{6}-L_{4}=L_{7}-L_{6},\ 22=L_{7}-L_{4},\4=L_{8}-L_{2}$ (Theorem 3).

A new approach to Fibonacci Tessarines with Fibonacci and Lucas number components

In this paper, by using identities related to the tessarines, Fibonacci numbers and Lucas numbers we define Fibonacci tessarines and Lucas tessarines. We obtain Binet formulae, D’ocagnes identity and Cassini identity for these tessarines. We also give the identities of Fibonacci negatessarines and Lucas negatessarines and define new vector which are called Fibonacci tessarine vector.

Transmission of Time and Position Variable Cryptology in Fibonacci and Lucas Number Series with Music

Since people existed, they have prioritized confidentiality in information sharing and communication. Although there are independent studies on encryption and music in literature, no study is seen on encryption methods that are created by using the properties of mathematical number strings and can be expressed with musical instruments. The purpose in this research is to develop ideas for an effective encryption method and to create a time and location variable encryption method considering this deficiency in the literature by getting advantage of the additive feature in Fibonacci and Lucas number sequences and moving from here to develop new perspectives on encryption science. In the research letters in alphabet, numbers and 10 of the most used symbols were selected and ASCII codes were determined. The objects to be encrypted are divided into 6 main groups (uppercase vowel, uppercase consonant, lowercase vowel, lowercase consonant letters, numbers, and symbols). ASCII codes are written with the additive property of the Fibonacci and Lucas numbers (Zeckendorf's Theorem) and matched with the corresponding notes. In addition to the first method in the study, the encryption system is encrypted by shifting depending on time. In addition to this method, the encryption system was encrypted by shifting depending on the location. In the last method, the text to be encrypted was encrypted by shifting depending on both location and time. The software of the first stage of the encryption system has been created. The encryption method we have created can be transmitted in both audio and text. Since encryption can be applied with various instruments, it offers variety in terms of data privacy. In the encryption system, people who have a musical ear can audibly decipher the password regardless of the written source. In the research, the same text differs as time and location change. This method allows multiple transformations of a character in a text. With these features, it differs from the encryption methods made until now.

On k-circulant matrices involving the Pell–Lucas (and the modified Pell) numbers

Let k be a nonzero complex number. In this paper, we consider a k-circulant matrix whose first row is $$(Q_{1},Q_{2},\ldots ,Q_{n})$$ , where $$Q_{n}$$ is the nth Pell–Lucas number. The formulas for the eigenvalues of such matrix are obtained. Namely, the result which can be obtained from the result of Theorem 7. (Yazlik and Taskara, J Inequal Appl 2013:394, 2013) is improved. The obtained formulas for the eigenvalues of a k-circulant matrix involving the Pell–Lucas numbers show that the result of Theorem 8. (Jing, Li and Shen, WSEAS Trans Math 12(3):341-351, 2013) (i.e. Theorem 8. (Yazlik and Taskara 2013)) is not always applicable. The Euclidean norm of such matrix is determined. The upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is $$(Q_{1}^{-1},Q_{2}^{-1},\ldots ,Q_{n}^{-1})$$ are also investigated. The obtained results are illustrated by examples. As a consequence of the previous results, the eigenvalues, the determinant, the Euclidean norm of a k-circulant matrix whose first row is $$(q_{1},q_{2},\ldots ,q_{n})$$ , where $$q_{n}$$ is the nth modified Pell number, are presented. Also, the upper and lower bounds for the spectral norm of a k-circulant matrix whose first row is $$(q_{1}^{-1},q_{2}^{-1},\ldots ,q_{n}^{-1})$$ are given

INTERESTING DETERMINANTS AND INVERSES OF SKEW LOEPLITZ AND FOEPLITZ MATRICES

<abstract> In this paper, we show that there is an intimate relationship between Toeplitz matrix, tridiagonal Toeplitz matrix, the Fibonacci number, the Lucas number, and the Golden Ratio. We introduce skew Loeplitz and skew Foeplitz matrices and derive their determinants and inverses by construction. Specifically, the determiant of $ n\times n $ skew Loeplitz matrix can be expressed by the $ (n+1) $st Fibonacci number. The inverse of skew Loeplitz matrix is sparse and can be expressed by the $ n $th and $ (n+1) $st Fibonacci numbers. Similarly, the determinant of $ n\times n $ skew Foeplitz matrix also can be expressed by the $ (n+1) $st Lucas number. The inverse of skew Foeplitz matrix can be expressed by only seven elements with each element being the explicit expression of the Lucas or Fibonacci numbers. We also calculate the determinants and inverses of skew Lankel and skew Fankel matrices. </abstract>