scholarly journals Repdigits as difference of two Fibonacci or Lucas numbers

2021 ◽  
Vol 56 (2) ◽  
pp. 124-132
Author(s):  
P. Ray ◽  
K. Bhoi
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Lucas Numbers ◽  
Lucas Number

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).

scholarly journals A curious property of series involving terms of generalized sequences

2000 ◽  
Vol 23 (1) ◽  
pp. 55-63
Author(s):  
Odoardo Brugia ◽  
Piero Filipponi
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Lucas Numbers ◽  
Curious Property

Here we are concerned with series involving generalized Fibonacci numbersUn  (p,q)and generalized Lucas numbersVn  (p,q). The aim of this paper is to find triples(p,q,r)for which the seriesUn  (p,q)/rnandVn  (p,q)/rn(forrrunning from 0 to infinity) are unconcerned at the introduction of the factorn. The results established in this paper generalize the known fact that the seriesFn/2n(Fnthenth Fibonacci number) and the seriesnFn/2ngive the same result, namely−2/5.

scholarly journals Finite groups of deficiency zero involving the Lucas numbers

1990 ◽  
Vol 33 (1) ◽  
pp. 1-10 ◽  
Author(s):  
C. M. Campbell ◽  
E. F. Robertson ◽  
R. M. Thomas
Keyword(s):  
Finite Groups ◽  
Soluble Group ◽  
Fibonacci Number ◽  
Single Parameter ◽  
The Other ◽  
Finite Soluble Group ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Derived Length ◽  
New Class

In this paper, we investigate a class of 2-generator 2-relator groups G(n) related to the Fibonacci groups F(2,n), each of the groups in this new class also being defined by a single parameter n, though here n can take negative, as well as positive, values. If n is odd, we show that G(n) is a finite soluble group of derived length 2 (if n is coprime to 3) or 3 (otherwise), and order |2n(n + 2)gnf(n, 3)|, where fn is the Fibonacci number defined by f0=0,f1=1,fn+2=fn+fn+1 and gn is the Lucas number defined by g0 = 2, g1 = 1, gn+2 = gn + gn+1 for n≧0. On the other hand, if n is even then, with three exceptions, namely the cases n = 2,4 or –4, G(n) is infinite; the groups G(2), G(4) and G(–4) have orders 16, 240 and 80 respectively.

scholarly journals Interesting Explicit Expressions of Determinants and Inverse Matrices for Foeplitz and Loeplitz Matrices

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 939
Author(s):  
Zhaolin Jiang ◽  
Weiping Wang ◽  
Yanpeng Zheng ◽  
Baishuai Zuo ◽  
Bei Niu
Keyword(s):  
Toeplitz Matrices ◽  
Fibonacci Number ◽  
Inverse Matrix ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Numerical Examples ◽  
Fibonacci And Lucas Numbers ◽  
Theoretical Results ◽  
Explicit Expressions

Foeplitz and Loeplitz matrices are Toeplitz matrices with entries being Fibonacci and Lucas numbers, respectively. In this paper, explicit expressions of determinants and inverse matrices of Foeplitz and Loeplitz matrices are studied. Specifically, the determinant of the n × n Foeplitz matrix is the ( n + 1 ) th Fibonacci number, while the inverse matrix of the n × n Foeplitz matrix is sparse and can be expressed by the nth and the ( n + 1 ) th Fibonacci number. Similarly, the determinant of the n × n Loeplitz matrix can be expressed by use of the ( n + 1 ) th Lucas number, and the inverse matrix of the n × n ( n > 3 ) Loeplitz matrix can be expressed by only seven elements with each element being the explicit expressions of Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our new theoretical results.

The order of appearance of the sum and difference between two Fibonacci numbers

2019 ◽  
Vol 12 (03) ◽  
pp. 1950046 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Natural Number ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Greatest Common Divisor ◽  
Lucas Number ◽  
Number Formula

Let [Formula: see text] be the [Formula: see text]th Fibonacci number and [Formula: see text] be the [Formula: see text]th Lucas number. The order of appearance [Formula: see text] of a natural number [Formula: see text] is defined as the smallest natural number [Formula: see text] such that [Formula: see text] divides [Formula: see text]. For instance, [Formula: see text], for all [Formula: see text]. In this paper, among other things, we prove that [Formula: see text] depends on [Formula: see text], where [Formula: see text] is the greatest common divisor of numbers [Formula: see text] and [Formula: see text], which fulfill the condition [Formula: see text].

scholarly journals Congruences for $q$-Lucas Numbers

10.37236/2557 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Hao Pan
Keyword(s):  
Fibonacci Numbers ◽  
Cyclotomic Polynomial ◽  
Gamma Delta ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Alpha Beta

For $\alpha,\beta,\gamma,\delta\in{\mathbb Z}$ and ${\rm\nu}=(\alpha,\beta,\gamma,\delta)$, the $q$-Fibonacci numbers are given by$$F_0^{{\rm\nu}}(q)=0,\ F_1^{{\rm\nu}}(q)=1\text{ and }F_{n+1}^{{\rm\nu}}(q)=q^{\alpha n-\beta}F_{n}^{{\rm\nu}}(q)+q^{\gamma n-\delta}F_{n-1}^{{\rm\nu}}(q)\text{ for }n\geq 1.$$And define the $q$-Lucas number $L_{n}^{{\rm\nu}}(q)=F_{n+1}^{{\rm\nu}}(q)+q^{\gamma-\delta}F_{n-1}^{{\rm\nu}_*}(q)$, where ${\rm\nu}_*=(\alpha,\beta-\alpha,\gamma,\delta-\gamma)$. Suppose that $\alpha=0$ and $\gamma$ is prime to $n$, or $\alpha=\gamma$ is prime to $n$. We prove that$$L_{n}^{{\rm\nu}}(q)\equiv(-1)^{\alpha(n+1)}\pmod{\Phi_n(q)}$$for $n\geq 3$, where $\Phi_n(q)$ is the $n$-th cyclotomic polynomial. A similar congruence for $q$-Pell-Lucas numbers is also established.

scholarly journals The Order of Appearance of the Product of Five Consecutive Lucas Numbers

2014 ◽  
Vol 59 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Diego Marques ◽  
Pavel Trojovský
Keyword(s):  
Natural Number ◽  
Fibonacci Number ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Positive Integers

Abstract Let Fn be the nth Fibonacci number and let Ln be the nth Lucas number. The order of appearance z(n) of a natural number n is defined as the smallest natural number k such that n divides Fk. For instance, z(Fn) = n = z(Ln)/2 for all n > 2. In this paper, among other things, we prove that for all positive integers n ≡ 0,8 (mod 12).

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

scholarly journals On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 962
Author(s):  
Eva Trojovská  ◽  
Pavel Trojovský
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Initial Values

Let (tn(r))n≥0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence tn(r)=tn−1(r)+⋯+tn−r(r) for n≥r, with initial values t0(r)=0 and ti(r)=1, for all 1≤i≤r. In 2002, Grossman and Luca searched for terms of the sequence (tn(2))n, which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ≥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 tm(r)=n!+(n+1)!+⋯+(n+ℓ)!, with r even, then max{m,n,r}<C. As an application, we solve the previous equation for all 1≤ℓ≤5.

scholarly journals On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

2016 ◽  
Vol 67 (1) ◽  
pp. 41-46
Author(s):  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Initial Values ◽  
Lucas Sequence ◽  
Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

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