On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence {Un(P,Q)} is defined by U0=0, U1=1, Un=PUn−1−QUn−2 (n⩾2). The question of when Un(P,Q) can be a perfect square has generated interest in the literature. We show that for n=2,…,7, Un is a square for infinitely many pairs (P,Q) with gcd(P,Q)=1; further, for n=8,…,12, the only non-degenerate sequences where gcd(P,Q)=1 and Un(P,Q)=□, are given by U8(1,−4)=212, U8(4,−17)=6202, and U12(1,−1)=122.