Primefree shifted Lucas sequences

Lenny Jones

Acta Arithmetica (2015)

  • Volume: 170, Issue: 3, page 287-298
  • ISSN: 0065-1036

Abstract

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We say a sequence = ( s ₙ ) n ≥ 0 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, a = ( u ₙ ) n ≥ 0 , 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 a ± k are simultaneously primefree. This result extends previous work of the author for the single shifted sequence a - k 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 a ± k has at least two distinct prime factors.

How to cite

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Lenny Jones. "Primefree shifted Lucas sequences." Acta Arithmetica 170.3 (2015): 287-298. <http://eudml.org/doc/279388>.

@article{LennyJones2015,
abstract = {We say a sequence $ = (sₙ)_\{n≥0\}$ 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, $\{\}_a=(uₙ)_\{n≥0\}$, 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 $_a ± k$ are simultaneously primefree. This result extends previous work of the author for the single shifted sequence $_a - k$ 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 $_a ± k$ has at least two distinct prime factors.},
author = {Lenny Jones},
journal = {Acta Arithmetica},
keywords = {Lucas sequences; primefree; coverings},
language = {eng},
number = {3},
pages = {287-298},
title = {Primefree shifted Lucas sequences},
url = {http://eudml.org/doc/279388},
volume = {170},
year = {2015},
}

TY - JOUR
AU - Lenny Jones
TI - Primefree shifted Lucas sequences
JO - Acta Arithmetica
PY - 2015
VL - 170
IS - 3
SP - 287
EP - 298
AB - We say a sequence $ = (sₙ)_{n≥0}$ 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, ${}_a=(uₙ)_{n≥0}$, 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 $_a ± k$ are simultaneously primefree. This result extends previous work of the author for the single shifted sequence $_a - k$ 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 $_a ± k$ has at least two distinct prime factors.
LA - eng
KW - Lucas sequences; primefree; coverings
UR - http://eudml.org/doc/279388
ER -

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