The general case on the order of appearance of product of consecutive Lucas numbers

Abstract

Let $F_n$ and $L_n$ be the nth 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 $F_k$. The formula for $z(L_nL_{n+1}L_{n+2}\cdots L_{n+k})$ has been recently obtained by Marques for $1\leq k \leq 3$ and by Marques and Trojovsky for $k = 4$. In this article, we extend the results to the cases $k = 5$ and $k = 6$. Our method gives a general idea on how to obtain the formulas of $z(L_{n}L_{n+1}\cdots L_{n+k})$ for every $k\geq 1$.