Abstract
Let P be an odd integer and \((V_{n})\) denote the generalized Lucas sequence defined by \(V_{0}=2,\) \(V_{1}=P,\) and \(V_{n+1}=PV_{n}+V_{n-1}\) for \(n\ge 1.\) In this study, we solve the equations \(V_{n}=5kx^{2},\) \(V_{n}=7kx^{2},\) \(V_{n}=5kx^{2}\pm 1,\) and \(V_{n}=7kx^{2}\pm 1\) when k|P with \(k>1.\) Moreover, applying some of the results, we obtain complete solutions to the equations \(V_{n}=\sigma x^{2},\) \(\sigma \in \left\{ 15,21,35\right\} \).
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Karaatlı, O. On the Lucas sequence equations \(V_{n}(P,1)=wkx^{2},\) \(w\in \left\{ 5,7\right\} \) . Period Math Hung 73, 73–82 (2016). https://doi.org/10.1007/s10998-016-0130-7
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DOI: https://doi.org/10.1007/s10998-016-0130-7