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) = 2, ar