On Perfect Powers in $k$-Generalized Pell–Lucas Sequence

Let $k\geq 2$, and let $(Q_{n}^{(k)})_{n\geq 2-k}$ be the $k$-generalized Pell sequence defined by
\begin{equation*} Q_{n}^{(k)}=2Q_{n-1}^{(k)}+Q_{n-2}^{(k)}+\cdots +Q_{n-k}^{(k)} \end{equation*}
for $n\geq 2$ with the initial conditions
\begin{equation*} Q_{-(k-2)}^{(k)}=Q_{-(k-3)}^{(k)}=\cdots =Q_{-1}^{(k)}=0,\qquad Q_{0}^{(k)}=2,Q_{1}^{(k)}=2. \end{equation*}
In this paper, we solve the Diophantine equation
\begin{equation*} Q_{n}^{(k)}=y^{m} \end{equation*}
in positive integers $n,m,y,k$ with $m,y,k\geq 2$. We show that all solutions $(n,m,y)$ of this equation in positive integers $n,m,y,k$ such that $2\leq y\leq 100$ are given by $(n,m,y)=(3,2,4),(3,4,2)$ for $k\geq 3$. Namely, $Q_{3}^{(k)}=16=2^{4}=4^{2}$ for $k\geq 3$.