Primefree shifted Lucas sequences

• Autorzy: Lenny Jones
• Języki publikacji: EN

Abstrakty

EN

We say a sequence $𝓢 = (sₙ)_{n≥0}$ 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 𝓢. In this article, we focus on the particular Lucas sequences of the first kind, ${𝓤}_a=(uₙ)_{n≥0}$, defined by
u₀ = 0, u₁ = 1, and uₙ = au_{n-1} + u_{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 $𝓤_a ± k$ are simultaneously primefree. This result extends previous work of the author for the single shifted sequence $𝓤_a - k$ 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 $𝓤_a ± k$ has at least two distinct prime factors.

Kategorie tematyczne

• 11B37: Recurrences
• 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
• 11B83: Special sequences and polynomials

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 170
• Numer: 3
• Strony: 287-298

Daty

wydano

2015

Twórcy

autor

Lenny Jones

• Department of Mathematics, Shippensburg University, Shippensburg, PA 17257, U.S.A.

Identyfikatory

DOI

10.4064/aa170-3-5