Primefree shifted Lucas sequences
Tom 170 / 2015
Streszczenie
We say a sequence ${{\mathcal S}}=(s_n)_{n\ge 0}$ is primefree if $|s_n|$ is not prime for all $n\ge 0$, and to rule out trivial situations, we require that no single prime divides all terms of ${{\mathcal S}}$. In this article, we focus on the particular Lucas sequences of the first kind, ${\mathcal U}_a=(u_n)_{n\ge 0}$, defined by $$u_0=0,\hskip 1em u_1=1, \hskip 1em \hbox {and} \hskip 1em u_n=au_{n-1}+u_{n-2} \hskip 1em \hbox {for $n\ge 2$},$$ 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 ${\mathcal U}_a\pm k$ are simultaneously primefree. This result extends previous work of the author for the single shifted sequence ${\mathcal U}_a-k$ 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 ${\mathcal U}_a\pm k$ has at least two distinct prime factors.
