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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.17954 |
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Table of Contents:
- 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 $τ(m)$, i.e., $τ(m):=\min\{k\geq1:m\mid U_k\}$. When $a>0$ and $Δ=a^2+4b>0$, we give explicit formulae for $τ(U_m V_n), τ(U_m U_n)$, $τ(V_m V_n)$ and $τ(U_nU_{n+p}U_{n+2p})$, thus generalizing the results of Irmak and Ray.