\begin{definition}[Definition:Cunningham Chain/Second Kind]
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.
\end{definition}