Communications of the
Korean Mathematical Society CKMS

Let $F_{n}$ and $L_n$ be the $n$th Fibonacci number and Lucas number, respectively. The order of appearance of $m$ in the Fibonacci sequence, denoted by $z(m)$, is the smallest positive integer $k$ such that $m$ divides $F_k$. Marques obtained the formula of $z(L_n^k)$ in some cases. In this article, we obtain the formula of $z(L_n^k)$ for all $n, k\geq 1$.

Keywords: Fibonacci number, Lucas number, the order of appearance, the rank of appearance

MSC numbers: Primary 11B39