Polynomial Cunningham chains

A sequence of prime numbers p1,p2,p3,…, such that pi=2pi−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 2pk+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 f1(x),f2(x),… in Z[x], such that f1(x) has positive leading coefficient, each fi(x) is irreducible in Q[x] and fi(x)=xfi−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 fk+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 f1(x), such that fk+1(x) is the only term in the sequence that is reducible.