On perfect powers in $k$-generalized Pell sequence

Şiar, Zafer ; Keskin, Refik ; Öztaş, Elif Segah

Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 507-518.

Résumé

Let $k\geq 2$ and let $(P_{n}^{(k)})_{n\geq 2-k}$ 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 $n\geq 2$ 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 $k,y\geq 2,$ and give an upper bound on $n.$ Also, we will show that the equation $P_{n}^{(k)}=y^{m}$ with $2\leq y\leq 1000$ has only one solution given by $P_{7}^{(2)}=13^{2}.$

Détail

DOI : 10.21136/MB.2022.0033-22

Classification : 11B39, 11D61, 11J86
Keywords: Fibonacci and Lucas numbers; exponential Diophantine equation; linear forms in logarithms; Baker's method