Polynomial Cunningham Chains

TL;DR

This paper introduces polynomial Cunningham chains, extending the classical prime chains to polynomial sequences, and proves the existence of infinitely many such chains with specific reducibility properties.

Contribution

It constructs explicit polynomial Cunningham chains of any finite length and proves the existence of infinitely many chains of infinite length for both kinds.

Findings

Existence of infinitely many polynomial Cunningham chains of each finite length.

Existence of infinitely many polynomial Cunningham chains of infinite length.

Explicit construction of initial polynomials for these 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 $f_i(x)\in \Z[x]$, $f_1(x)$ has positive leading coefficient, $f_i(x)$ is irreducible in $\Q[x]$, and $f_i(x)=xf_{i-1}(x)+\epsilon$ 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.