Lucas sequences whose nth term is a square or an almost square

Abstract

(Below, \Box means "perfect square") Let $P$ and $Q$ be non-zero integers. The Lucas sequence $\{U_n(P,Q)\}$ is defined by $U_0=0$, $U_1=1$, $U_n=P U_{n-1}-Q U_{n-2}$, $(n \geq 2)$. Historically, there has been much interest in when the terms of such sequences are perfect squares (or higher powers). Here, we summarize results on this problem, and investigate for fixed $k$ solutions of $U_n(P,Q)= k\Box$, $(P,Q)=1$. We show finiteness of the number of solutions, and under certain hypotheses on $n$, describe explicit methods for finding solutions. These involve solving finitely many Thue-Mahler equations. As an illustration of the methods, we find all solutions to $U_n(P,Q)=k\Box$ where $k=\pm1,\pm2$, and $n$ is a power of 2.