Research article

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

  • 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 $(a = 6, b = -1)$.

    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

    Related Papers:

  • 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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  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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