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Lucas sequences whose 12th or 9th term is a square [PDF]

open access: bronze, 2004
Let P and Q be non-zero relatively prime integers. The Lucas sequence {U_n(P,Q) is defined by U_0=0, U_1=1, U_n = P U_{n-1}-Q U_{n-2} for n>1. The sequence {U_n(1,-1)} is the familiar Fibonacci sequence, and it was proved by Cohn that the only perfect ...
Bremner, Andrew, Tzanakis, Nikos
core   +3 more sources

On squares in Lucas sequences [PDF]

open access: yesJournal of Number Theory, 2006
Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q) is defined by U_0=0, U_1=1, U_n= P*U_{n-1}-Q*U_{n-2} for n >1. The question of when U_n(P,Q) can be a perfect square has generated interest in the literature.
Bremner, A., Tzanakis, N.
core   +4 more sources

A method for computing Lucas sequences [PDF]

open access: yesComputers & Mathematics with Applications, 1999
Most of public-key cryptosystems rely on one-way functions, which can be used to encrypt and sign messages. Their encryption and signature operations are based on the computation of exponentiation. Recently, some public-key cryptosystems are proposed and
Chang, Chin-Chen   +2 more
core   +3 more sources

Lucas sequences and repdigits [PDF]

open access: yesMathematica Bohemica, 2022
Let $(G_n)_{n \geq1}$ be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are $\{U_n\}$ and $\{V_n\}$, respectively.
Hayder Raheem Hashim, Szabolcs Tengely
doaj   +3 more sources

On a divisibility relation for Lucas sequences

open access: greenJournal of Number Theory, 2016
In this note, we study the divisibility relation $U_m\mid U_{n+k}^s-U_n^s$, where ${\bf U}:=\{U_n\}_{n\ge 0}$ is the Lucas sequence of characteristic polynomial $x^2-ax\pm 1$ and $k,m,n,s$ are positive integers.
Yuri Bilu   +4 more
openalex   +6 more sources

The GCD Sequences of the Altered Lucas Sequences [PDF]

open access: yesAnnales Mathematicae Silesianae, 2020
In this study, we give two sequences {L+n}n≥1 and {L−n}n≥1 derived by altering the Lucas numbers with {±1, ±3}, terms of which are called as altered Lucas numbers.
Koken Fikri
doaj   +2 more sources

Coincidences in generalized Lucas sequences [PDF]

open access: yesThe Fibonacci Quarterly, 2014
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.
Bravo, Eric F.   +2 more
core   +2 more sources

A generalization of Lucas polynomial sequence

open access: bronzeDiscrete Applied Mathematics, 2008
In this paper, we obtain a generalized Lucas polynomial sequence from the lattice paths for the Delannoy numbers by allowing weights on the steps (1,0),(0,1) and (1,1). These weighted lattice paths lead us to a combinatorial interpretation for such a Lucas polynomial sequence. The concept of Riordan arrays is extensively used throughout this paper.
Gi‐Sang Cheon   +2 more
openalex   +3 more sources

Almost powers in the Lucas sequence [PDF]

open access: bronzeJournal de Théorie des Nombres de Bordeaux, 2008
The {\it Lucas sequence} $(L_n)_{n\geq 0}$ is defined by $L_0=2, L_1=1$ and $L_n=L_{n-1}+L_{n-2}$ for $n\geq 2$. The first, third and fourth authors have proved, among other things, that the only perfect powers in the Lucas sequence are $L_1=1$ and $L_3=4$ [{\it Y. Bugeaud, M. Mignotte} and {\it S. Siksek}, Ann. Math. (2) 163, No.
Yann Bugeaud   +3 more
openalex   +4 more sources

The degree sequence of Fibonacci and Lucas cubes [PDF]

open access: yesDiscrete Mathematics, 2011
The Fibonacci cube Γn is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λn is obtained from Γn by removing vertices that start and end with 1.
Klavžar, Sandi   +2 more
core   +4 more sources

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