Polynomial Cunningham Chains

Abstract

Let $\epsilon \in \{-1,1\}$. A sequence of prime numbers $p_1,p_2,p_3,...$, such that $p_i=2p_{i-1}+\epsilon$ for all $i$, is called a {\it Cunningham chain} of the first or second kind, depending on whether $\epsilon =1$ or -1 respectively. If $k$ is the smallest positive integer such that $2p_k+\epsilon$ is composite, then we say the chain has length $k$. Although such chains are necessarily finite, it is conjectured that for every positive integer $k$, there are infinitely many Cunningham chains of length $k$. A sequence of polynomials $f_1(x),f_2(x),...$, such that $f_i(x)\in \mathbb{Z}[x]$, $f_1(x)$ has positive leading coefficient, $f_i(x)$ is irreducible in $\Q[x]$, and $f_i(x)=x{f}_{i-1}(x)+\epsilon$ for all $i$, is defined to be a {\it polynomial Cunningham chain} of the first or second kind, depending on whether $\epsilon =1$ or -1 respectively. If $k$ is the least positive integer such that $f_{k+1}(x)$ is reducible over $\Q$, then we say the chain has length $k$. In this article, for chains of each kind, we explicitly give infinitely many polynomials $f_1(x)$, such that $f_{k+1}(x)$ is the only term in the sequence $\{f_i(x){\}}_{i=1}^{\infty }$ that is reducible. As a first corollary, we deduce that there exist infinitely many polynomial Cunningham chains of length $k$ of both kinds, and as a second corollary, we have that, unlike the situation in the integers, there exist infinitely many polynomial Cunningham chains of infinite length of both kinds.