Lucas sequences whose 12th or 9th term is a square
数论
2007-05-23 v1
摘要
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 . 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.
引用
@article{arxiv.math/0405306,
title = {Lucas sequences whose 12th or 9th term is a square},
author = {Andrew Bremner and Nikos Tzanakis},
journal= {arXiv preprint arXiv:math/0405306},
year = {2007}
}
备注
13 pages