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

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 $(a=6,b=-1)$.

Keywords

• lucas sequence
• lucas number
• fibonacci number
• exact divisibility
• <i>p</i>-adic valuation