Abstract
Let \(P\ge 3\) be an integer and let \((U_{n})\) and \((V_{n})\) denote generalized Fibonacci and Lucas sequences defined by \(U_{0}=0,U_{1}=1\); \( V_{0}=2,V_{1}=P,\) and \(U_{n+1}=PU_{n}-U_{n-1}\), \(V_{n+1}=PV_{n}-V_{n-1}\) for \(n\ge 1.\) In this study, when P is odd, we solve the equation \( U_{n}=wx^{2}+1\) for \(w=1,2,3,5,6,7,10.\) After then, we solve some Diophantine equations utilizing solutions of these equations.
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Keskin, R., Öğüt, Ü. Generalized Fibonacci numbers of the form \(wx^{2}+1\) . Period Math Hung 73, 165–178 (2016). https://doi.org/10.1007/s10998-016-0133-4
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DOI: https://doi.org/10.1007/s10998-016-0133-4