2011/04/08 by Lenny Jones, Jones, Lenny
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1104.1579
arxiv created 2011/04/08 · arxiv updated 2011/04/11
Let ε∈ \-1,1\. A sequence of prime numbers p1, p2, p3, ..., such that pi=2pi-1+ε for all i, is called a \it 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. 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 f1(x), f2(x), ..., such that fi(x)∈ \Z[x], f1(x) has positive leading coefficient, fi(x) is irreducible in \Q[x], and fi(x)=xfi-1(x)+ε for all i, is defined to be a \it 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 over \Q, then we say the chain has length k. In this article, for chains of each kind, we explicitly give infinitely many polynomials f1(x), such that fk+1(x) is the only term in the sequence \fi(x)\i=1∞ 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.