On second order linear sequences of composite numbers

TL;DR

This paper provides a new proof for a 2010 result showing that certain second order linear recurrence sequences can be constructed with all terms being composite numbers, extending previous work by Graham.

Contribution

The paper introduces a novel proof technique for the existence of second order linear sequences of composite numbers, generalizing prior results to broader parameter values.

Findings

Existence of sequences with all composite terms for specified parameters

Extension of Graham's result to more general recurrence relations

New proof method for the composite sequence problem

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 $b\neq 0$ and $(a,b)\neq (2,-1), (-2, -1)$. 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 $(a,b)=(1,1)$.