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

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