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

Abstract

Let $a$ and $b$ be relatively prime integers. Then the first Lucas sequence $\left(U_n\right)_{n\geq0}$ and the second Lucas sequence $\left(V_n\right)_{n\geq0}$ are defined respectively by $U_{n+2}=aU_{n+1}+bU_{n},\, U_0=0,\,U_1=1$ and $V_{n+2}=aV_{n+1}+bV_{n},\, V_0=2,\,V_1=a$, where $n\geq0$. Let $m$ be an integer with $\gcd(m,\,b)=1$. Then the smallest positive integer $k$ satisfying $m\mid U_k$ is called the order of appearance of $m$ in the first Lucas sequence $(U_n)_{n\geq0}$, denoted by $\tau(m)$, i.e., $\tau(m):=\min\{k\geq1:m\mid U_k\}$. When $a>0$ and $\Delta=a^2+4b>0$, we give explicit formulae for $\tau(U_m V_n), \tau(U_m U_n)$, $\tau(V_m V_n)$ and $\tau(U_nU_{n+p}U_{n+2p})$, thus generalizing the results of Irmak and Ray.