On squares in Lucas sequences
A. Bremner, N. Tzanakis
Abstract
Let P and Q be non-zero integers. The Lucas sequence Un(P,Q) is defined by U0=0, U1=1, Un= P*Un-1-Q*Un-2 for n >1. 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)=square, are given by U8(1,-4)=212, U8(4,-17)=6202, and U12(1,-1)=122.
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