Related papers: On Squares in Lucas Sequences

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. We show that for…

Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q), n=0,1,2,... is defined by U_0=0, U_1=1, U_n= P U_{n-1}-Q U_{n-2} for n>1. For each positive integer n<8 we describe all Lucas sequences with (P,Q)=1 having the property that…

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…

(Below, \Box means "perfect square") Let $P$ and $Q$ be non-zero integers. The Lucas sequence $\{U_n(P,Q)\}$ is defined by $U_0=0$, $U_1=1$, $U_n=P U_{n-1}-Q U_{n-2}$, $(n \geq 2)$. Historically, there has been much interest in when the…

Let $(U_{n}(P,Q) $ and $(V_{n}(P,Q) $ denote the generalized Fibonacci and Lucas sequence, respectively. In this study, we assume that $Q=1.$ We determine all indices $n$ such that $U_{n}=5\square $ and $U_{n}=5U_{m}\square $ under some…

For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are either integers or conjugate quadratic integers, we describe the set of indices n for…

For the Lucas sequence $\{U_{k}(P,Q)\}$ we discuss the identities such as the well-known Fibonacci identities. We also propose a method for obtaining identities involving recurrence sequences. With the help of which we find an interpolating…

We observe that a sequence satisfies Lucas congruences modulo $p$ if and only if its values modulo $p$ can be described by a linear $p$-scheme, as introduced by Rowland and Zeilberger, with a single state. This simple observation suggests…

We give an overview about well-known basic properties of two classes of q-Fibonacci and q-Lucas polynomials and offer a common generalization.

Let $P,Q\in\Bbb Z$, $U_0=0,\ U_1=1$ and $U_{n+1}=PU_n-QU_{n+1}$. In this paper we obtain a general congruence for $U_{kmn^r}/U_k\pmod {n^{r+1}}$, where $k,m,n,r$ are positive integers. As applications we extend Lucas' law of repetition and…

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 find all the integers…

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.

We construct the sequences of Fibonacci and Lucas at any quadratic field $\mathbb{Q}(\sqrt{d}\ )$ with $d>0$ square free, noting in general that the properties remain valid as those given by the classical sequences of Fibonacci and Lucas…

We prove a Lucas-type congruence for q-Delannoy numbers.

Let $w_n=w_n(P,Q)$ be numerical sequences which satisfy the recursion relation \begin{equation*} w_{n+2}=Pw_{n+1}-Qw_n. \end{equation*} We consider two special cases $(w_0,w_1)=(0,1)$ and $(w_0,w_1)=(2,P)$ and we denote them by $U_n$ and…

In this paper, we introduce the bi-periodic Lucas matrix sequence and present some fundamental properties of this generalized matrix sequence. Moreover, we investigate the important relationships between the bi-periodic Fibonacci and Lucas…

We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.

In these notes we address the study of the log-concave operator acting on Lucas Sequences of first kind. We will find for which initial values a generic Lucas sequence is log-concave, and using this we show when the same sequence is…

We say that an arithmetical function $S:\mathbb{N}\rightarrow\mathbb{Z}$ has Lucas property if for any prime $p$, \begin{equation*} S(n)\equiv S(n_{0})S(n_{1})\ldots S(n_{r})\pmod p, \end{equation*} where $n=\sum_{i=0}^{r}n_{i}p^{i}$, with…

For an integer \( k \geq 2 \), the sequence of \( k \)-generalized Lucas numbers is defined by the recurrence relation \( L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \) for all \( n \geq 2 \), with initial conditions \( L_0^{(k)} = 2…