On square classes in generalized Fibonacci sequences
Volume 174 / 2016
Acta Arithmetica 174 (2016), 277-295
MSC: 11B37, 11B39.
DOI: 10.4064/aa8436-4-2016
Published online: 22 June 2016
Abstract
Let $P$ and $Q$ be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: $U_{0}=0,U_{1}=1,$ $% V_{0}=2,V_{1}=P$ and $U_{n+1}=PU_{n}+QU_{n-1},$ $V_{n+1}=PV_{n}+QV_{n-1}$ for $n\geq 1$. In this paper, when $w\in \{ 1,2,3,6\} ,$ for all odd relatively prime values of $P$ and $Q$ such that $P\geq 1$ and $P^{2}+4Q \gt 0,$ we determine all $n$ and $m$ satisfying the equation $U_{n}=wU_{m}x^{2}.$ In particular, when $k\,|\,P$ and $k \gt 1$, we solve the equations $U_{n}=kx^{2}$ and $U_{n}=2kx^{2}.$ As a result, we determine all $n$ such that $U_{n}=6x^{2}.$
