On the Largest Prime factor of the $k$-generalized Pell numbers

Abstract

Let $k \ge 2$ be an integer and consider the $k$-generalized Pell sequence $\{P_n^{(k)}\}_{n \ge 2-k}$, defined by the initial values $0, \ldots, 0, 0, 1$ (a total of $k$ terms), and the recurrence $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \cdots + P_{n-k}^{(k)}$, for all $n\ge 2$. For any integer $m$, let $\mathcal{P}(m)$ denote the largest prime factor of $m$, with the convention $\mathcal{P}(0) = \mathcal{P}(\pm1) = 1$. In this paper, we prove that for $n \ge 4$, the inequality $\mathcal{P}(P_n^{(k)}) > (1/104) \log \log n$ holds. Additionally, we find all $k$-generalized Pell numbers $P_n^{(k)}$, whose largest prime factor does not exceed $7$.