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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.17954 |
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| _version_ | 1866909752026464256 |
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| author | Li, Hongjian Xiao, Huiming Yuan, Pingzhi |
| author_facet | Li, Hongjian Xiao, Huiming Yuan, Pingzhi |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The order of appearance of the product of the first and second Lucas numbers Li, Hongjian Xiao, Huiming Yuan, Pingzhi Number Theory 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. |
| title | The order of appearance of the product of the first and second Lucas numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.17954 |