Solutions of Some Diophantine Equations in terms of Generalized Fibonacci and Lucas Numbers

Authors : Bahar DEMİRTÜRK BİTİM, Refik KESKİN

Pages : 451-459

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Publication Date : 2019-04-01

Article Type : Research Paper

Abstract :In this study, we present some identities involving generalized Fibonacci sequence $\leftinsert ignore into journalissuearticles values(U_{n}\right);$ and generalized Lucas sequence $\leftinsert ignore into journalissuearticles values(V_{n}\right);$. Then we give all solutions of the Diophantine equations $x^{2}-V_{n}xy+insert ignore into journalissuearticles values(-1);^{n}y^{2}=\pm insert ignore into journalissuearticles values(p^{2}+4);U_{n}^{2},$ $x^{2}-V_{n}xy+insert ignore into journalissuearticles values(-1);^{n}y^{2}=\pm U_{n}^{2},$ $x^{2}-insert ignore into journalissuearticles values(p^{2}+4);U_{n}xy-insert ignore into journalissuearticles values(p^{2}+4);insert ignore into journalissuearticles values(-1);^{n}y^{2}=\pm V_{n}^{2},$ $x^{2}-V_{n}xy\pm y^{2}=\pm 1,$ $x^{2}-insert ignore into journalissuearticles values(p^{2}+4);U_{n}xy-insert ignore into journalissuearticles values(p^{2}+4);insert ignore into journalissuearticles values(-1);^{n}y^{2}=1,$ $x^{2}-V_{n}xy+insert ignore into journalissuearticles values(-1);^{n}y^{2}=\pm insert ignore into journalissuearticles values(p^{2}+4);$, $x^{2}-V_{2n}xy+y^{2}=\pminsert ignore into journalissuearticles values(p^{2}+4);V_{n}^{2}$, $x^{2}-V_{2n}xy+y^{2}=insert ignore into journalissuearticles values(p^{2}+4);U_{n}^{2}$ and $x^{2}-V_{2n}xy+y^{2}=\pm V_{n}^{2}$ in terms of the sequences $\leftinsert ignore into journalissuearticles values( U_{n}\right); $ and $\leftinsert ignore into journalissuearticles values( V_{n}\right); $ with $p\geq 1$ and $p^{2}+4$ squarefree.
Keywords : generalized Fibonacci and Lucas sequences, Diophantine equations