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

TL;DR

This paper derives explicit formulas for the order of appearance of products of Lucas numbers, generalizing previous results and providing new insights into the divisibility properties of Lucas sequences.

Contribution

It introduces explicit formulas for the order of appearance of products of Lucas numbers, extending prior work to more general Lucas sequences and their products.

Findings

Explicit formulas for τ(U_m V_n), τ(U_m U_n), τ(V_m V_n), τ(U_n U_{n+p} U_{n+2p})

Generalization of previous results by Irmak and Ray

Enhanced understanding of divisibility in Lucas sequences

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.