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 $10^7$.

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): $554,688,278,429$