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

Abstract

<abstract><p>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.</p></abstract>