# Definition:Cunningham Chain

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## 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 $10^7$.

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1,122,659$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $554,688,278,429$