- Hacettepe Journal of Mathematics and Statistics
- Vol: 48 Issue: 2
- Solutions of Some Diophantine Equations in terms of Generalized Fibonacci and Lucas Numbers
Solutions of Some Diophantine Equations in terms of Generalized Fibonacci and Lucas Numbers
Authors : Bahar Demirtürk Bitim, Refik Keskin
Pages : 451-459
View : 12 | Download : 4
Publication Date : 2019-04-01
Article Type : Research
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.Keywords : generalized Fibonacci and Lucas sequences, Diophantine equations