Lucas sequences whose 12th or 9th term is a square
| dc.creator | Bremner, Andrew | |
| dc.creator | Tzanakis, Nikos | |
| dc.date | 2004-05-15 | |
| dc.date.accessioned | 2026-07-07T05:08:18Z | |
| dc.date.available | 2026-07-07T05:08:18Z | |
| dc.description | Let P and Q be non-zero relatively prime integers. The Lucas sequence {U_n(P,Q) is defined by U_0=0, U_1=1, U_n = P U_{n-1}-Q U_{n-2} for n>1. The sequence {U_n(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 $U_{12}=144$. The question arises, for which parameters P, Q, can U_n(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 {U_n(P,Q)} with U_{12}= square. It turns out that the Fibonacci sequence provides the only example. Moreover, we also determine all Lucas sequences {U_n(P,Q) with U_9= square. | |
| dc.description | 13 pages | |
| dc.identifier | https://arxiv.org/abs/math/0405306 | |
| dc.identifier | http://arxiv.org/abs/math/0405306 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/71210 | |
| dc.subject | Number Theory | |
| dc.subject | 11B39; 11G07; 11G05; 11Y50 | |
| dc.title | Lucas sequences whose 12th or 9th term is a square | |
| dc.type | text |