Polynomial Cunningham chains

Abstract

A sequence of prime numbers p 1 , p 2 , p 3 , … , such that p i = 2 p i − 1 + ϵ for all i, is called a Cunningham chain of the first or second kind, depending on whether ϵ = 1 or −1 respectively. If k is the smallest positive integer such that 2 p k + ϵ is composite, then we say the chain has length k. It is conjectured that there are infinitely many Cunningham chains of length k for every positive integer k. A sequence of polynomials f 1 ( x ) , f 2 ( x ) , … in Z [ x ] , such that f 1 ( x ) has positive leading coefficient, each f i ( x ) is irreducible in Q [ x ] and f i ( x ) = x f i − 1 ( x ) + ϵ for all i, is defined to be a polynomial Cunningham chain of the first or second kind, depending on whether ϵ = 1 or −1 respectively. If k is the least positive integer such that f k + 1 ( x ) is reducible in Q [ x ] , then we say the chain has length k. In this article, for polynomial Cunningham chains of both kinds, we prove that there are infinitely many chains of length k and, unlike the situation in the integers, that there are infinitely many chains of infinite length, by explicitly giving infinitely many polynomials f 1 ( x ) , such that f k + 1 ( x ) is the only term in the sequence { f i ( x ) } i = 1 ∞ that is reducible.