Cunningham chain

In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes.

Definition

A Cunningham chain of the first kind of length n is a sequence of prime numbers (p₁, ..., p_n) such that for all 1 ≤ i < n, p_i+1 = 2p_i + 1. (Hence each term of such a chain except the last one is a Sophie Germain prime, and each term except the first is a safe prime).

It follows that

Or, by setting (the number is not part of the sequence and need not be a prime number), we have

Similarly, a Cunningham chain of the second kind of length n is a sequence of prime numbers (p₁,...,p_n) such that for all 1 ≤ i < n, p_i+1 = 2p_i − 1.

It follows that the general term is

Now, by setting , we have .

Cunningham chains are also sometimes generalized to sequences of prime numbers (p₁, ..., p_n) such that for all 1 ≤ i ≤ n, p_i+1 = ap_i + b for fixed coprime integers a, b; the resulting chains are called generalized Cunningham chains.