Lucas Sequences Whose 12th or 9th Term Is a Square

The sequence {Un(1,−1)} is the familiar Fibonacci sequence, and it was proved by Cohn [12] in 1964 that the only perfect square greater than 1 in this sequence is U12 = 144. The question arises, for which parameters P , Q, can Un(P,Q) be a perfect square? This has been studied by several authors: see for example Cohn [13] [14] [15], Ljunggren [22], and Robbins [25]. Using Baker’s method on linear forms in logarithms, work of Shorey & Tijdeman [26] implies that there can only be finitely many squares in the sequence {Un(P,Q)}. Ribenboim and McDaniel [23] with only elementary methods show that when P and Q are odd, and P 2 − 4Q > 0, then Un can be square only for n = 0, 1, 2, 3, 6 or 12; and that there are at most two indices greater than 1 for which Un can be square. They characterize fully the instances when Un = 2, for n = 2, 3, 6, and observe that U12 = 2 if and only if there is a solution to the Diophantine system