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\begin{abstract}
We study the problem of finding all integer solutions of
the Diophantine equations $x^{2}-5F_{n}xy-5\left( -1\right) ^{n}y^{2}=\pm
L_{n}^{2},$ $x^{2}-L_{n}xy+\left( -1\right) ^{n}y^{2}=\pm 5F_{n}^{2},$ and $%
x^{2}-L_{n}xy+\left( -1\right) ^{n}y^{2}=\pm F_{n}^{2}.$  Using these
equations, we also explore all integer solutions of some other Diophantine
equations.
\end{abstract}

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