Related papers: On a divisibility relation for Lucas sequences

Let $\{U_n\}_{n\geq 0}$ be a Lucas sequence. Then the equation $$|U_n|=m_1!m_2!\cdots m_k!$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in \{1,2, 3, 4, 6, 8, 12\}$. Further the equation $$|U_n|=D_{m_1}D_{m_2}\cdots D_{m_k}, \qquad…

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…

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…

(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 s and t be variables. Define polynomials {n} in s, t by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial…

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…

We consider the family of Lucas sequences uniquely determined by $U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values $U_0(k)=0$ and $U_1(k)=1$ and $k\ge 1$ an arbitrary integer. For any integer $n\ge 1$ the discriminator function…

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…

For $A,B\in\mathbb Z$, the Lucas sequence $u_n(A,B)\ (n=0,1,2,\ldots)$ are defined by $u_0(A,B)=0$, $u_1(A,B)=1$, and $u_{n+1}(A,B) = Au_n(A,B)-Bu_{n-1}(A,B)$ $(n=1,2,3,\ldots).$ For any odd prime $p$ and positive integer $n$, we establish…

Let $U = (U_n)_{n \geq 0}$ be a Lucas sequence and, for every prime number $p$, let $\rho_U(p)$ be the rank of appearance of $p$ in $U$, that is, the smallest positive integer $k$ such that $p$ divides $U_k$, whenever it exists.…

Let $(u_n)_{n \geq 0}$ be a non-degenerate Lucas sequence, given by the relation $u_n=a_1 u_{n-1}+a_2 u_{n-2}$. Let $\ell_u(m)=lcm(m, z_u(m))$, for $(m,a_2)=1$, where $z_u(m)$ is the rank of appearance of $m$ in $u_n$. We prove that…

In this paper, we provide some novel results concerning the behavior of $\frac{U_{kn}}{U_k}$ modulo ${U_n}$, where $(U_n)_{n\in\mathbb{N}}$ is the Lucas sequence of the first kind. As a consequence, we obtain some primality criteria which…

Here, we show that if $\{U_n\}_{n\ge 0}$ is a Lucas sequence, then the largest $n$ such that $|U_n|=m_1!m_2!\cdots m_k!$ with $1<m_1\le m_2\le \cdots\le m_k$ satisfies $n<3\times 10^5$. We also give better bounds in case the roots of the…

Let $(L_n)_{n \geq 1}$ be the sequence of Lucas numbers, defined recursively by $L_1 := 1$, $L_2 := 3$, and $L_{n + 2} := L_{n + 1} + L_n$, for every integer $n \geq 1$. We determine the asymptotic behavior of $\log \operatorname{lcm} (L_1…

We study divisibility properties of a set $\{f_1(\mathbf{U}_n^{(s)}),\ldots,f_m(\mathbf{U}_n^{(s)})\}$, where $f_1,\ldots,f_m$ are polynomials in $s$ variables over $\mathbb{Z}$ and $\mathbf{U}_n^{(s)}$ is a point picked uniformly at random…

Let $(L_n^{(k)})_{n\geq 2-k}$ be the sequence of $k$-generalized Lucas numbers for some fixed integer $k\ge 2$, whose first $k$ terms are $0,\;\ldots\;,\;0,\;2,\;1$ and each term afterward is the sum of the preceding $k$ terms. In this…

Let $a$ and $b$ be relatively prime integers. Then the first Lucas sequence $\left(U_n\right)_{n\geq0}$ and the second Lucas sequence $\left(V_n\right)_{n\geq0}$ are defined respectively by $U_{n+2}=aU_{n+1}+bU_{n},\, U_0=0,\,U_1=1$ and…

We study linear divisibility sequences of order 4, providing a characterization by means of their characteristic polynomials and finding their factorization as a product of linear divisibility sequences of order 2. Moreover, we show a new…

Let $r\ge 1$ be an integer and ${\bf U}:=(U_{n})_{n\ge 0} $ be the Lucas sequence given by $U_0=0$, $U_1=1, $ and $U_{n+2}=rU_{n+1}+U_n$, for all $ n\ge 0 $. In this paper, we show that there are no positive integers $r\ge 3,~x\ne 2,~n\ge…

Lucas polynomials are polynomials in $s_1$ and $s_2$ defined recursively by $\{0\}=0$, $\{1\}=1$, and $\{m\}=s_1\{m-1\}+s_2\{m-2\}$ for $m \geq 2$. We generalize Lucas polynomials from 2-variable polynomials to multivariable polynomials.…