Generalized Fibonacci and Lucas numbers of the form $wx^{2}$ and $wx^{2}\mp 1$
Bull. Korean Math. Soc. 2014 Vol. 51, No. 4, 1041-1054
https://doi.org/10.4134/BKMS.2014.51.4.1041
Published online July 1, 2014
Refik Keskin
Sakarya University
Abstract : Let $P\geq 3$ be an integer and let $(U_{n})$ and $(V_{n})$ denote generalized Fibonacci and Lucas sequences defined by $U_{0}=0,U_{1}=1$; $ V_{0}=2,V_{1}=P,$ and $U_{n+1}=PU_{n}-U_{n-1}$, $V_{n+1}=PV_{n}-V_{n-1}$ for $n\geq 1.$ In this study, when $P$ is odd, we solve the equations $ V_{n}=kx^{2}$ and $V_{n}=2kx^{2}$ with $k~|~P$ and $k>1.$ Then, when $k~|~P$ and $k>1,$ we solve some other equations such as $ U_{n}=kx^{2},U_{n}=2kx^{2},U_{n}=3kx^{2},V_{n}=kx^{2}\mp 1,V_{n}=2kx^{2}\mp 1,$ and $U_{n}=kx^{2}\mp 1.$ Moreover, when $P$ is odd, we solve the equations $V_{n}=wx^{2}+1$ and $V_{n}=wx^{2}-1$ for $w=2,3,6.$ After that, we solve some Diophantine equations.
Keywords : generalized Fibonacci numbers, generalized Lucas numbers, congruences, Diophantine equation
MSC numbers : 11B37, 11B39, 11B50, 11B99, 11D41
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