On the GCD-s of k consecutive terms of Lucas sequences
详细信息 查看全文
- 作者：L. Hajdu ; M. Szikszai
- 关键词：Lucas sequences ; Greatest common divisor ; Divisibility sequences ; Pillai?s problem
- 刊名：Journal of Number Theory
- 出版年：2012
- 出版时间：December, 2012
- 年：2012
- 卷：132
- 期：12
- 页码：3056-3069
- 全文大小：223 K
文摘
Let be a Lucas sequence, that is a binary linear recurrence sequence of integers with initial terms and . We show that if k is large enough then one can find k consecutive terms of u such that none of them is relatively prime to all the others. We even give the exact values and for each u such that the above property first holds with ; and that it holds for all , respectively. We prove similar results for Lehmer sequences as well, and also a generalization for linear recurrence divisibility sequences of arbitrarily large order. On our way to prove our main results, we provide a positive answer to a question of Beukers from 1980, concerning the sums of the multiplicities of 1 and ?1 values in non-degenerate Lucas sequences. Our results yield an extension of a problem of Pillai from integers to recurrence sequences, as well.