Primefree shifted Lucas sequences of the second kind

JOURNAL OF NUMBER THEORY

卷:173

Article

Ismailescu, Dan¹  Jones, Lenny²  Phillips, Tristan² 

[1] Hofstra Univ, Dept Math, Hempstead, NY 11550 USA

[2] Shippensburg Univ, Dept Math, Shippensburg, PA 17257 USA

关键词: Lucas sequences;    Primefree;    Coverings;   

DOI  :  10.1016/j.jnt.2016.09.016

来源: Elsevier

【 摘 要 】

We say a sequence S = (s(n))(n >= 0) is primefree if vertical bar s(n)vertical bar is not prime for all n >= 0 arid, to rule out trivial situations, we require that no single prime divides all terms of S. Recently, the second author showed that there exist infinitely many integers k such that both of the shifted sequences u +/- k are simultaneously primefree, where U is a particular Lucas sequence of the first kind. In this article, we prove an analogous result for the Lucas sequences v(a) = (v(n))(n >= 0) of the second kind, defined by v(0) = 2, v(1) = a, and v(n) = av(n-1)+v(n-2), for n >= 2, where a is a fixed integer. More precisely, we show that for any integer a, there exist infinitely many integers k such that both of the shifted sequences V-a +/- k are simultaneously primefree. This result provides additional evidence to support a conjecture of Ismailescu and Shim. Moreover, we show that there are infinitely many values of k such that every term of both of the shifted sequences V-a +/- k has at least two distinct prime factors. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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