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

Let \(F_{n}\) and \(L_n\) be the \(n\)th Fibonacci and Lucas number, respectively. For each positive integer \(m\), the order of appearance of \(m\) in the Fibonacci sequence, denoted by \(z(m)\), is the smallest positive integer \(k\) such that \(m\) divides \(F_k\). Recently, D. Marques has obtained a formula for \(z(F_{n}F_{n+1})\), \(z(F_{n}F_{n+1}F_{n+2})\), and \(z(F_{n}F_{n+1}F_{n+2}F_{n+3})\). In this paper, we extend Marques' result to the case \(z(F_{n}F_{n+1}\cdots F_{n+k})\) for every \(4\leq k \leq 6\). We also give a formula for \(z(L_nL_{n+1}\cdots L_{n+k})\) when \(k = 5,6\) which extends the recent result of Marques and Trojovský. Our method gives a general idea on how to obtain the formulas for \(z(F_nF_{n+1}\cdots F_{n+k})\) and \(z(L_nL_{n+1}\cdots L_{n+k})\) for every \(k\geq 1\).