The order of appearance of the product of the first and second Lucas numbers

Abstract: Let $\left(U_n\right){n\geq0}$ and $\left(V_n\right){n\geq0}$ be the first and second Lucas sequences, respectively. Let $m$ be a positive integer. Then the order of appearance of $m$ in the first Lucas sequence is defined as the smallest positive integer $k$ such that $m$ divides $U_k$ and denoted by $\tau(m)$. In this paper, we give explicit formulae for the terms $\tau(U_m V_n)$, $\tau(U_m U_n)$, $\tau(V_m V_n)$ and $\tau(U_nU_{n+p}U_{n+2p})$, where $p\geq3$ is a prime number.