On squares in Lucas sequences

Abstract

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= P*U_{n-1}-Q*U_{n-2} for n >1. 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)=square, are given by U_8(1,-4)=21^2, U_8(4,-17)=620^2, and U_12(1,-1)=12^2.