Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.7

Integer Solutions of Some Diophantine Equations via Fibonacci and Lucas Numbers


Bahar Demirtürk and Refik Keskin
Department of Mathematics
Sakarya University
54187 Sakarya
Turkey

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.

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(Concerned with sequences A000032 A000045.)

Received July 17 2009; revised version received December 8 2009. Published in Journal of Integer Sequences, December 8 2009. Minor correction to equation (13) and the line before it, and to the statements of Theorems 15 and 16, December 16 2009.

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