Smallest Cunningham Chain of the First Kind of Length 6

Theorem
The smallest Cunningham chain of length $6$ is:
 * $\left({89, 179, 359, 719, 1439, 2879}\right)$

Proof
By definition, a Cunningham chain is a sequence of prime numbers $\left({p_1, p_2, \ldots, p_n}\right)$ such that:
 * $p_{k + 1} = 2 p_k + 1$
 * $\dfrac {p_1 - 1} 2$ is not prime
 * $2 p_n + 1$ is not prime.

Thus each term except the last is a Sophie Germain prime.

Let $P: \Z \to \Z$ be the mapping defined as:
 * $P \left({n}\right) = 2 n + 1$

Applying $P$ iteratively to each of the smallest Sophie Germain primes in turn:

Thus $\left({2, 5, 11, 23, 47}\right)$ is a Cunningham chain of length $5$.

It is noted that $\dfrac {89 - 1} 2 = 44$ which is not prime.

Hence the sequence of $6$:
 * $\left({89, 179, 359, 719, 1439, 2879}\right)$