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Keywords:
Fibonacci and Lucas numbers; exponential Diophantine equation; linear forms in logarithms; Baker's method
Fibonacci and Lucas numbers; exponential Diophantine equation; linear forms in logarithms; Baker's method
Summary:
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 $P_{7}^{(2)}=13^{2}.$
References:
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