On second order linear sequences of composite numbers

Abstract

In this paper we present a new proof of the following 2010 result of Dubickas, Novikas, and Siurys: Let $(a,b)\in {\mathbb{Z}}^2$ and let $(x_n{)}_{n\ge 0}$ be the sequence defined by some initial values $x_0$ and $x_1$ and the second order linear recurrence \begin{equation*} x_{n+1}=ax_n+bx_{n-1} \end{equation*} for $n\ge 1$. Suppose that $b\ne 0$ and $(a,b)\ne (2,-1),(-2,-1)$. Then there exist two relatively prime positive integers $x_0$, $x_1$ such that $|x_n|$ is a composite integer for all $n\in \mathbb{N}$. The above theorem extends a result of Graham who solved the problem when $(a,b)=(1,1)$.