2004/05/15 by Andrew Bremner, Bremner, Andrew, Nikos Tzanakis +1
Mathematics · Physics and Astronomy · #11B39 #11G05 #11G07 #11Y50 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B39 #msc:11G05 #msc:11G07 #msc:11Y50
paper · pdf · doi:10.48550/arxiv.math/0405306
13 pages
arxiv created 2004/05/15 · openalex publication_date 2004/05/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let P and Q be non-zero relatively prime 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 sequence Un(1,-1) is the familiar Fibonacci sequence, and it was proved by Cohn that the only perfect square greater than 1 in this sequence is U12=144. The question arises, for which parameters P, Q, can Un(P,Q) be a perfect square? In this paper, we complete recent results of Ribenboim and MacDaniel. Under the only restriction GCD(P,Q)=1 we determine all Lucas sequences Un(P,Q) with U12= square. It turns out that the Fibonacci sequence provides the only example. Moreover, we also determine all Lucas sequences Un(P,Q) with U9= square.