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 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).

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 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.

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.

Sufficient Conditions for the Oscillation of Delay and Neutral Delay Equations

1988 ◽  
Vol 31 (4) ◽  
pp. 459-466 ◽  
Author(s):  
E. A. Grove ◽  
G. Ladas ◽  
J. Schinas
Keyword(s):  
Differential Equation ◽  
Natural Number ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Delay Equations ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
All Solutions ◽  
Delay Differential

AbstractWe established sufficient conditions for the oscillation of all solutions of the delay differential equationand of the neutral delay differential equationwhere p, q, r and a are nonnegative constants and n is an odd natural number.

On the oscillatory behavior of solutions of higher order nonlinear fractional differential equations

2018 ◽  
Vol 25 (3) ◽  
pp. 363-369
Author(s):  
Said R. Grace ◽  
Ercan Tunç
Keyword(s):  
Differential Equations ◽  
Natural Number ◽  
Fractional Differential Equations ◽  
Caputo Derivative ◽  
Oscillation Theory ◽  
Higher Order ◽  
Oscillatory Behavior ◽  
All Solutions ◽  
Fractional Differential ◽  
New Criteria

AbstractThe study of oscillation theory for fractional differential equations has been initiated by Grace et al. [5]. In this paper we establish some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form {{}^{C}D_{a}^{r}x(t)=e(t)+f(t,x(t)),t>0,a>1}, where {r=\alpha+n-1,\alpha\in(0,1)}, and {n\geq 1} is a natural number. We also present the conditions under which all solutions of this equation are asymptotic to {t^{n-1}} as {t\to\infty}.

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 On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1073 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Natural Number ◽  
Diophantine Equations ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Prime Numbers ◽  
All Solutions ◽  
Adic Valuation

Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z ( n ) = n ± 1 are prime numbers. In this paper, we solve equation z ( n ) = n + ℓ for | ℓ | ∈ { 1 , … , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers.

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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