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Lucas sequences whose 12th or 9th term is a square [PDF]
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
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On squares in Lucas sequences [PDF]
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.
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A method for computing Lucas sequences [PDF]
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
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Lucas sequences and repdigits [PDF]
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
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On a divisibility relation for Lucas sequences
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
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The GCD Sequences of the Altered Lucas Sequences [PDF]
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
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Coincidences in generalized Lucas sequences [PDF]
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
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A generalization of Lucas polynomial sequence
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
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Almost powers in the Lucas sequence [PDF]
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
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The degree sequence of Fibonacci and Lucas cubes [PDF]
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
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