Bulletin of the
Korean Mathematical Society BKMS

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