On perfect powers in $k$-generalized Pell sequence

Abstract

Let $k\geq2$ and let $(P_n^{(k)})_{n\geq2-k}$ be the $k$-generalized Pell sequence defined by \begin{equation*} P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots+P_{n-k}^{(k)} \end{equation*}for $n\geq2$ with initial conditions \begin{equation*} P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdots=P_{-1}^{(k)}=P_0^{(k)}=0,P_1^{(k)}=1. \end{equation*}In this study, we handle the equation $P_n^{(k)}=y^m$ in positive integers $n$, $m$, $y$, $k$ such that $k,y\geq2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_n^{(k)}=y^m$ with $2\leq y\leq1000$ has only one solution given by $P_7^{(2)}=13^2.$

Keywords

• fibonacci and lucas numbers
• exponential diophantine equation
• linear forms in logarithms
• baker's method