Exact divisibility by powers of the integers in the Lucas sequence of the first kind
Lucas sequence of the first kind is an integer sequence $(U_n)_{n\geq0}$ which depends on parameters $a,b\in\mathbb{Z}$ and is defined by the recurrence relation $U_0=0$, $U_1=1$, and $U_n=aU_{n-1}+bU_{n-2}$ for $n\geq2$. In this article, we obtain exact divisibility results concerning $U_n^k$ for a...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2020-09-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/10.3934/math.2020433/fulltext.html |
| Summary: | Lucas sequence of the first kind is an integer sequence $(U_n)_{n\geq0}$ which depends on parameters $a,b\in\mathbb{Z}$ and is defined by the recurrence relation $U_0=0$, $U_1=1$, and $U_n=aU_{n-1}+bU_{n-2}$ for $n\geq2$. In this article, we obtain exact divisibility results concerning $U_n^k$ for all positive integers $n$ and $k$. This extends many results in the literature from 1970 to 2020 which dealt only with the classical Fibonacci and Lucas numbers $(a=b=1)$ and the balancing and Lucas-balancing numbers $(a=6,b=-1)$. |
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| ISSN: | 2473-6988 |