2007/01/09 by A. Bremner N. Tzanakis, Tzanakis, A. Bremner N.
Mathematics · Physics and Astronomy · #11B39 #11D41 #11D59 #11G30 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B39 #msc:11D41 #msc:11D59 #msc:11G30
paper · pdf · doi:10.48550/arxiv.math/0701252
24 pages (double spaced). To appear in Acta Arithmetica
arxiv created 2007/01/09 · openalex publication_date 2007/01/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
(Below, \Box means "perfect square") 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, (n ≥ 2). Historically, there has been much interest in when the terms of such sequences are perfect squares (or higher powers). Here, we summarize results on this problem, and investigate for fixed k solutions of Un(P,Q)= k\Box, (P,Q)=1. We show finiteness of the number of solutions, and under certain hypotheses on n, describe explicit methods for finding solutions. These involve solving finitely many Thue-Mahler equations. As an illustration of the methods, we find all solutions to Un(P,Q)=k\Box where k=±1,±2, and n is a power of 2.