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Warianty tytułu
Języki publikacji
Abstrakty
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.
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
287-298
Opis fizyczny
Daty
wydano
2015
Twórcy
autor
- Department of Mathematics, Shippensburg University, Shippensburg, PA 17257, U.S.A.
Bibliografia
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-aa170-3-5
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