Exact divisibility by powers of the integers in the Lucas sequence of the first kind

Kritkhajohn Onphaeng; Prapanpong Pongsriiam; Kritkhajohn Onphaeng; Prapanpong Pongsriiam

doi:10.3934/math.2020433

AIMS Mathematics

2020, Volume 5, Issue 6: 6739-6748. doi: 10.3934/math.2020433

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Research article

Exact divisibility by powers of the integers in the Lucas sequence of the first kind

Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand

Received: 07 July 2020 Accepted: 14 August 2020 Published: 31 August 2020
MSC : 11B39, 11B37, 11A05

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 $(a = 6, b = -1)$.
- Lucas sequence,
- Lucas number,
- Fibonacci number,
- exact divisibility,
- p-adic valuation
Citation: Kritkhajohn Onphaeng, Prapanpong Pongsriiam. Exact divisibility by powers of the integers in the Lucas sequence of the first kind[J]. AIMS Mathematics, 2020, 5(6): 6739-6748. doi: 10.3934/math.2020433

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Abstract

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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