scholarly journals Lucas numbers of the form (2t/k)

2019 ◽  
Vol 23 (1) ◽  
pp. 65-70
Author(s):  
Nurettin Irmak ◽  
László Szalay
Keyword(s):  
Diophantine Equation ◽  
Lucas Numbers ◽  
Lucas Number

Let Lm denote the mth Lucas number. We show that the solutions to the diophantine equation (2t/k) = Lm, in non-negative integers t, k ≤ 2t−1, and m, are (t, k, m) = (1, 1, 0), (2, 1, 3), and (a, 0, 1) with non-negative integers a.

Pell Numbers, Pell–Lucas Numbers and Modular Group

2007 ◽  
Vol 14 (01) ◽  
pp. 97-102 ◽  
Author(s):  
Q. Mushtaq ◽  
U. Hayat
Keyword(s):  
Modular Group ◽  
Symmetric Matrix ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Pell Numbers ◽  
The Matrix

We show that the matrix A(g), representing the element g = ((xy)2(xy2)2)m (m ≥ 1) of the modular group PSL(2,Z) = 〈x,y : x2 = y3 = 1〉, where [Formula: see text] and [Formula: see text], is a 2 × 2 symmetric matrix whose entries are Pell numbers and whose trace is a Pell–Lucas number. If g fixes elements of [Formula: see text], where d is a square-free positive number, on the circuit of the coset diagram, then d = 2 and there are only four pairs of ambiguous numbers on the circuit.

scholarly journals Lucas numbers of the formPX2, wherePis prime

1991 ◽  
Vol 14 (4) ◽  
pp. 697-703 ◽  
Author(s):  
Neville Robbins
Keyword(s):  
Natural Number ◽  
Lucas Numbers ◽  
Lucas Number ◽  
All Solutions

LetLndenote thenthLucas number, wherenis a natural number. Using elementary techniques, we find all solutions of the equation:Ln=px2wherepis prime andp<1000.

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.

On generalised Fibonaccian and Lucasian numbers

2006 ◽  
Vol 90 (518) ◽  
pp. 215-222 ◽  
Author(s):  
Peter Hilton ◽  
Jean Pedersen
Keyword(s):  
Residue Class ◽  
The Other ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Residue Classes ◽  
Other Hand ◽  
Positive Integers ◽  
Striking Fact

In [1, Chapter 3, Section 2], we collected together results we had previously obtained relating to the question of which positive integers m were Lucasian, that is, factors of some Lucas number L n. We pointed out that the behaviors of odd and even numbers m were quite different. Thus, for example, 2 and 4 are both Lucasian but 8 is not; for the sequence of residue classes mod 8 of the Lucas numbers, n ⩾ 0, reads and thus does not contain the residue class 0*. On the other hand, it is a striking fact that, if the odd number s is Lucasian, then so are all of its positive powers.

scholarly journals Solutions of the Diophantine Equation 7X2 + Y7 = Z2 from Recurrence Sequences

2020 ◽  
Vol 28 (1) ◽  
pp. 55-66
Author(s):  
Hayder R. Hashim
Keyword(s):  
Diophantine Equation ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Quadratic Reciprocity ◽  
Integer Solutions ◽  
Integer Polynomial ◽  
Recurrence Sequences

AbstractConsider the system x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation 7X2 + Y7 = Z2 if (X, Y) = (Ln, Fn) (or (X, Y) = (Fn, Ln)) where {Fn} and {Ln} represent the sequences of Fibonacci numbers and Lucas numbers respectively.

scholarly journals Generalized Lucas numbers of the form 3 × 2^m

2021 ◽  
Vol 27 (2) ◽  
pp. 129-136
Author(s):  
Salah Eddine Rihane ◽  
◽  
Chefiath Awero Adegbindin ◽  
Alain Togbé ◽  
◽  
...  
Keyword(s):  
Diophantine Equation ◽  
Lucas Numbers ◽  
Lucas Sequence ◽  
Generalized Lucas Numbers ◽  
Positive Integers

For an integer $k\geq 2$, let $(L_n^{(k)})_n$ be the k-generalized Lucas sequence which starts with $0,\ldots,0,2,1$ (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we look the k-generalized Lucas numbers of the form $3\times 2^m$ i.e. we study the Diophantine equation $L^{(k)}_n = 3\times 2^m$ in positive integers n, k, m with $k \geq 2$.

Determining Lucas identities by using Hosoya index

2017 ◽  
Vol 26 (2) ◽  
pp. 145-151
Author(s):  
HACENE BELBACHIR ◽  
HAKIM HARIK ◽  
S. PIRZADA
Keyword(s):  
Hosoya Index ◽  
Lucas Numbers ◽  
Lucas Number

We introduce a new identity of Lucas number by using the Hosoya index. As a consequence we give some properties of Lucas numbers and the extension of the work of Hillard and Windfeldt.

scholarly journals Repdigits as Euler functions of Lucas numbers

2016 ◽  
Vol 24 (2) ◽  
pp. 105-126
Author(s):  
Jhon J. Bravo ◽  
Bernadette Faye ◽  
Florian Luca ◽  
Amadou Tall
Keyword(s):  
Lucas Numbers ◽  
Euler Function ◽  
Lucas Number

Abstract We prove some results about the structure of all Lucas numbers whose Euler function is a repdigit in base 10. For example, we show that if Ln is such a Lucas number, then n < 10111 is of the form p or p2, where p3 | 10p-1 -1.

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 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.

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