Lucas sequences whose 8th term is a square
Andrew Bremner, Nikos Tzanakis
Abstract
Let P and Q be non-zero integers. The Lucas sequence Un(P,Q), n=0,1,2,... is defined by U0=0, U1=1, Un= P Un-1-Q Un-2 for n>1. For each positive integer n<8 we describe all Lucas sequences with (P,Q)=1 having the property that Un(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 U8(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.
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