Then we give all solutions of the Diophantine equations $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm (p^{2}+4)U_{n}^{2},$ $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm U_{n}^{2},$ $x^{2}-(p^{2}+4)U_{n}xy-(p^{2}+4)(-1)^{n}y^{2}=\pm V_{n}^{2},$ $x^{2}-V_{n}xy\pm y^{2}=\pm 1,$ $x^{2}-(p^{2}+4)U_{n}xy-(p^{2}+4)(-1)^{n}y^{2}=1,$ $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm (p^{2}+4)$, $x^{2}-V_{2n}xy+y^{2}=\pm(p^{2}+4)V_{n}^{2}$, $x^{2}-V_{2n}xy+y^{2}=(p^{2}+4)U_{n}^{2}$ and $x^{2}-V_{2n}xy+y^{2}=\pm V_{n}^{2}$ in terms of the sequences $\left( U_{n}\right) $ and $\left( V_{n}\right) $ with $p\geq 1$ and $p^{2}+4$ squarefree.
📌 3 Key Takeaways
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Objective & Hypothesis:In this study, we present some identities involving generalized Fibonacci sequence $\left(U_{n}\right)$ and generalized Lucas sequence $\left(V_{n}\right)$.
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Key Findings & Data:In this study, we present some identities involving generalized Fibonacci sequence $\left(U_{n}\right)$ and generalized Lucas sequence $\left(V_{n}\right)$.
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Conclusion & Impact:Then we give all solutions of the Diophantine equations $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm (p^{2}+4)U_{n}^{2},$ $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm U_{n}^{2},$ $x^{2}-(p^{2}+4)U_{n}xy-(p^{2}+4)(-1)^{n}y^{2}=\pm V_{n}^{2},$ $x^{2}-V_{n}xy\pm y^{2}=\pm 1,$ $x^{2}-(p^{2}+4)U_{n}xy-(p^{2}+4)(-1)^{n}y^{2}=1,$ $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm (p^{2}+4)$, $x^{2}-V_{2n}xy+y^{2}=\pm(p^{2}+4)V_{n}^{2}$, $x^{2}-V_{2n}xy+y^{2}=(p^{2}+4)U_{n}^{2}$ and $x^{2}-V_{2n}xy+y^{2}=\pm V_{n}^{2}$ in terms of the sequences $\left( U_{n}\right) $ and $\left( V_{n}\right) $ with $p\geq 1$ and $p^{2}+4$ squarefree.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:generalized Fibonacci and Lucas sequences · Diophantine equations
Abstract
In this study, we present some identities involving generalized Fibonacci sequence $\left(U_{n}\right)$ and generalized Lucas sequence $\left(V_{n}\right)$. Then we give all solutions of the Diophantine equations $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm (p^{2}+4)U_{n}^{2},$ $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm U_{n}^{2},$ $x^{2}-(p^{2}+4)U_{n}xy-(p^{2}+4)(-1)^{n}y^{2}=\pm V_{n}^{2},$ $x^{2}-V_{n}xy\pm y^{2}=\pm 1,$ $x^{2}-(p^{2}+4)U_{n}xy-(p^{2}+4)(-1)^{n}y^{2}=1,$ $x^{2}-V_{n}xy+(-1)^{n}y^{2}=\pm (p^{2}+4)$, $x^{2}-V_{2n}xy+y^{2}=\pm(p^{2}+4)V_{n}^{2}$, $x^{2}-V_{2n}xy+y^{2}=(p^{2}+4)U_{n}^{2}$ and $x^{2}-V_{2n}xy+y^{2}=\pm V_{n}^{2}$ in terms of the sequences $\left( U_{n}\right) $ and $\left( V_{n}\right) $ with $p\geq 1$ and $p^{2}+4$ squarefree.
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