Lucas sequences whose 8th term is a square
| dc.creator | Bremner, Andrew | |
| dc.creator | Tzanakis, Nikos | |
| dc.date | 2004-08-26 | |
| dc.date | 2004-08-27 | |
| dc.date.accessioned | 2026-07-07T05:11:35Z | |
| dc.date.available | 2026-07-07T05:11:35Z | |
| dc.description | Let P and Q be non-zero integers. The Lucas sequence U_n(P,Q), n=0,1,2,... is defined by U_0=0, U_1=1, U_n= P U_{n-1}-Q U_{n-2} for n>1. For each positive integer n<8 we describe all Lucas sequences with (P,Q)=1 having the property that U_n(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 U_8(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. | |
| dc.description | 21 pages + appendix of 23 pages with computational information | |
| dc.identifier | https://arxiv.org/abs/math/0408371 | |
| dc.identifier | http://arxiv.org/abs/math/0408371 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72295 | |
| dc.subject | Number Theory | |
| dc.subject | 11B39 (primary); 11G05 ; 11D25 (secondary) | |
| dc.title | Lucas sequences whose 8th term is a square | |
| dc.type | text |