Generalized Lucas numbers of the form $5kx^{2}$ and $7kx^{2}$
Bull. Korean Math. Soc. 2015 Vol. 52, No. 5, 1467-1480
https://doi.org/10.4134/BKMS.2015.52.5.1467
Printed September 1, 2015
Olcay Karaatlı and Refik Keskin
Sakarya University Faculty of Arts and Science, Faculty of Arts and Science
Abstract : Generalized Fibonacci and Lucas sequences $(U_{n})$ and $(V_{n})$ are defined by the recurrence relations $U_{n+1}=PU_{n}+QU_{n-1}$ and $ V_{n+1}=PV_{n}+QV_{n-1},$ $n\geq 1,$ with initial conditions $U_{0}=0,$ $ U_{1}=1$ and $V_{0}=2,$ $V_{1}=P.$ This paper deals with Fibonacci and Lucas numbers of the form $U_{n}(P,Q)$ and $V_{n}(P,Q)$ with the special consideration that $P\geq 3$ is odd and $Q=-1.$ Under these consideration, we solve the equations $V_{n}=5kx^{2}$, $V_{n}=7kx^{2}$, $V_{n}=5kx^{2}\pm 1,$ and $V_{n}=7kx^{2}\pm 1$ when $k\mid P$ with $k>1.$ Moreover, we solve the equations $V_{n}=5x^{2}\pm 1$ and $V_{n}=7x^{2}\pm 1.$
Keywords : generalized Fibonacci numbers, generalized Lucas numbers, congruences
MSC numbers : 11B37, 11B39, 11B50
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