Lucas sequences whose 8th term is a square

Abstract

Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q), n=0,1,2,... is defined by U_0=0, U_1=1, U_n= P U_{n-1}-Q U_{n-2} for n>1. For each positive integer n<8 we describe all Lucas sequences with (P,Q)=1 having the property that U_n(P,Q) is a perfect square. The arguments are elementary. The main part of the paper is devoted to finding all Lucas sequences such that U_8(P,Q) is a perfect square. This reduces to a number of problems of similar type, namely, finding all points on an elliptic curve defined over a quartic number field subject to a ``Q-rationality'' condition on the X-coordinate. This is achieved by p-adic computations (for a suitable prime p) using the formal group of the elliptic curve.