Definition:Cunningham Chain/Second Kind

Definition

A Cunningham chain of the second kind is a (finite) sequence $\tuple {p_1, p_2, \ldots, p_n}$ such that:

$(1): \quad \forall i \in \set {1, 2, \ldots, n - 1}: p_{i + 1} = 2 p_i - 1$

$(2): \quad p_i$ is prime for all $i \in \set {1, 2, \ldots, n - 1}$

$(3): \quad n$ is not prime such that $2 n - 1 = p_1$

$(4): \quad 2 p_n - 1$ is not prime.

Also see

• Definition:Cunningham Chain of the First Kind

• Results about Cunningham chains can be found here.

Source of Name

This entry was named for Allan Joseph Champneys Cunningham.

Historical Note

Cunningham chains of the first kind were investigated by Derrick Norman Lehmer, who determined that there are only $3$ such chains of $7$ primes with the first term less than $10^7$.