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

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 $0,\dots ,0,0,1$ (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 $\mathcal{P}(0)=\mathcal{P}(\pm 1)=1$. In this paper, we prove that for $n\ge 4$, the inequality $\mathcal{P}(P_n^{(k)})>(1\mathrm{/}104)\log \log n$ holds. Additionally, we find all $k$-generalized Pell numbers $P_n^{(k)}$, whose largest prime factor does not exceed $7$.