On squares in Lucas sequences

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 = PUn-l - QU(n-2) (n >= 2). The question of when U-n(P, Q) can be a perfect square has generated interest in the literature. We show that for n = 2, . . ., 7, U-n 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 U-n(P, Q) = rectangle, are given by U-8(1, -4) = 21(2), U-8(4, -17) = 620(2), and U-12(1, - 1) = 12(2). (C) 2006 Elsevier Inc. All rights reserved.