Mathematica Bohemica, Vol. 148, No. 4, pp. 507-518, 2023
On perfect powers in $k$-generalized Pell sequence
Zafer Şiar, Refik Keskin, Elif Segah Öztaş
Received March 6, 2022. Published online September 29, 2022.
Abstract: Let $k\geq2$ and let $(P_n^{(k)})_{n\geq2-k}$ be the $k$-generalized Pell sequence defined by $P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots+P_{n-k}^{(k)}$ for $n\geq2$ with initial conditions $P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdots=P_{-1}^{(k)}=P_0^{(k)}=0,P_1^{(k)}=1$. 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
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