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.