We say a sequence is primefree if |sₙ| is not prime for all n ≥ 0, and to rule out trivial situations, we require that no single prime divides all terms of . In this article, we focus on the particular Lucas sequences of the first kind, , defined by u₀ = 0, u₁ = 1, and uₙ = aun-1 + un-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 are simultaneously primefree. This result extends previous work of the author for the single shifted sequence when a = 1 to all other values of a, and establishes a weaker form of 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 has at least two distinct prime factors.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-3-5,
author = {Lenny Jones},
title = {Primefree shifted Lucas sequences},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {287-298},
zbl = {06477196},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-3-5}
}
Lenny Jones. Primefree shifted Lucas sequences. Acta Arithmetica, Tome 168 (2015) pp. 287-298. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-3-5/