# Definition:Cunningham Chain

## Definition

There are $2$ types of Cunningham chain:

### First Kind

A Cunningham chain of the first 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.

Thus:

each term except the last is a Sophie Germain prime
each term except the first is a safe prime.

### 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.

## Also see

• 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 $10^7$.