Exact divisibility by powers of the integers in the Lucas sequences of the first and second kinds

Abstract

Lucas sequences of the first and second kinds are, respectively, the integer sequences $ (U_n)_{n\geq0} $ and $ (V_n)_{n\geq0} $ depending on parameters $ a, b\in\mathbb{Z} $ and defined by the recurrence relations $ U_0 = 0 $, $ U_1 = 1 $, and $ U_n = aU_{n-1}+bU_{n-2} $ for $ n\geq2 $, $ V_0 = 2 $, $ V_1 = a $, and $ V_n = aV_{n-1}+bV_{n-2} $ for $ n\geq2 $. In this article, we obtain exact divisibility results concerning $ U_n^k $ and $ V_n^k $ for all positive integers $ n $ and $ k $. This and our previous article extend many results in the literature and complete a long investigation on this problem from 1970 to 2021.

Keywords

• lucas sequence
• lucas number
• fibonacci number
• exact divisibility
• p-adic valuation