Lucas sequences whose nth term is a square or an almost square
| dc.creator | Tzanakis, A. Bremner N. | |
| dc.date | 2007-01-09 | |
| dc.date.accessioned | 2026-07-07T07:39:27Z | |
| dc.date.available | 2026-07-07T07:39:27Z | |
| dc.description | (Below, \Box means "perfect square") Let $P$ and $Q$ be non-zero integers. The Lucas sequence $\{U_n(P,Q)\}$ is defined by $U_0=0$, $U_1=1$, $U_n=P U_{n-1}-Q U_{n-2}$, $(n \geq 2)$. 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 $U_n(P,Q)= k\Box$, $(P,Q)=1$. 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 $U_n(P,Q)=k\Box$ where $k=\pm1,\pm2$, and $n$ is a power of 2. | |
| dc.description | 24 pages (double spaced). To appear in Acta Arithmetica | |
| dc.identifier | https://arxiv.org/abs/math/0701252 | |
| dc.identifier | http://arxiv.org/abs/math/0701252 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/121454 | |
| dc.subject | Number Theory | |
| dc.subject | 11D41; 11B39; 11D59; 11G30 | |
| dc.title | Lucas sequences whose nth term is a square or an almost square | |
| dc.type | text |