Related papers: Lucas sequences whose nth term is a square or an a…

A positive integer $n$ is called a balancing number if there exists a positive integer $r$ such that $1 + 2 + \cdots + (n-1) = (n+1) + (n+2) + \cdots + (n+r)$. The corresponding value $r$ is known as the balancer of $n$. If $n$ is a…

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 $(L_n^{(k)})_{n\geq 2-k}$ 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 afterwards is the sum of the preceding $k$ terms. In this paper, we…

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…

Harmonic numbers $H_k=\sum_{0<j\le k}1/j (k=0,1,2,...)$ arise naturally in many fields of mathematics. In this paper we initiate the study of congruences involving both harmonic numbers and Lucas sequences. One of our three theorems is as…

This paper explores the intricate relationships between Lucas numbers and Diophantine equations, offering significant contributions to the field of number theory. We first establish that the equation regarding Lucas number $L_n = 3x^2$ has…

For integers $k \geq 2$ and $n \neq 0$, let $v_k(n)$ denotes the greatest nonnegative integer $e$ such that $k^e$ divides $n$. Moreover, let $u_n$ be a nondegenerate Lucas sequence satisfying $u_0 = 0$, $u_1 = 1$, and $u_{n + 2} = a u_{n +…

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…

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…

Let $(U_n)_{n\geq 0}$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $\ell$, let $\ell \cdot 2^{\ell} + 1$ be a Cullen number. Recently in \cite{bmt}, generalized Cullen numbers in…

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…

The van der Laan-Padovan sequence $P_n ~ (n=0, 1, \ldots)$ is defined by $P_0=1, P_1=P_2=0$, and $P_{n+3}=P_{n+1}+P_n$ for $n=0, 1, \ldots$. We determine all pairs $(P_m, P_n)$ satisfying $P_m^b=2^{g_1} 3^{g_2} 5^{g_3} 7^{g_4} P_n^a$ for…

Let $u_n$ be a nondegenerate Lucas sequence. We generalize the results of Bugeaud, Mignotte, and Siksek, 2006 to give a systematic approach towards the problem of determining all perfect powers in any particular Lucas sequence. We then…

Let $\mathbf{u} = (u_n)_{n \geq 0}$ be a Lucas sequence, that is, a sequence of integers satisfying $u_0 = 0$, $u_1 = 1$, and $u_n = a_1 u_{n - 1} + a_2 u_{n - 2}$ for every integer $n \geq 2$, where $a_1$ and $a_2$ are fixed nonzero…

An $(n,k)$ sequence covering array is a set of permutations of $[n]$ such that each sequence of $k$ distinct elements of $[n]$ is a subsequence of at least one of the permutations. An $(n,k)$ sequence covering array is perfect if there is a…

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…

The Lucas sequences are integers defined by a homogeneous recurrence relation. They include the well-known Fibonacci numbers, which appear abundantly in nature. The complementary Lucas numbers, defined by the same recurrence relation, are…

We show that the $Kn$--smooth part of $a^n-1$ for an integer $a>1$ is $a^{o(n)}$ for most positive integers $n$.

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…

This article gives an alternative proof of the fact that N_{Q(zeta)/Q}(1-zeta)=p where p is an odd prime number and zeta is a primitive p-th root of unity, and uses it to prove that N_{Q(zeta)/Q}(1+zeta-zeta^2)=L(p) the p-th Lucas number.…