2004/08/26 by Andrew Bremner, Bremner, Andrew, Nikos Tzanakis +1
Computer Science · Mathematics · #11B39 (primary) #11D25 (secondary) #11G05 #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Topological and Geometric Data Analysis #advanced mathematical theories #math.NT #msc:11B39 #msc:11D25 #msc:11G05
paper · pdf · doi:10.48550/arxiv.math/0408371
21 pages + appendix of 23 pages with computational information
openalex publication_date 2004/08/26 · arxiv created 2004/08/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.