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 ${\left(U_n\right)}_{n\ge 0}$ and the second Lucas sequence ${\left(V_n\right)}_{n\ge 0}$ are defined respectively by $U_{n+2}=a{U}_{n+1}+b{U}_n,U_0=0,U_1=1$ and $V_{n+2}=a{V}_{n+1}+b{V}_n,V_0=2,V_1=a$, where $n\ge 0$. 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 $(U_n{)}_{n\ge 0}$, denoted by $\tau (m)$, i.e., $\tau (m):=\min \{k\ge 1:m\mid U_k\}$. 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 $\tau (U_n{U}_{n+p}{U}_{n+2p})$, thus generalizing the results of Irmak and Ray.

Revised the abstract to better reflect the updated scope of the work and key conclusions. This update refines the mathematical conclusions of Theorem 1.1 and Theorem 1.2