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Consider the sequence of primes 2, 5, 11, 23, 47. Each is twice the previous one plus 1.
When, in a sequence of primes $p_1, \ldots p_k$ each $p_n = 2p_{n - 1} + 1$ for $1 < n \le k$ (or alternatively, each $p_n = {{p_{n - 1} - 1} \over 2}$ for $0 < n < k$ ), the sequence is called a Cunningham chain of the first kind. In a Cunningham chain of the first kind, all primes except the largest are Sophie Germain primes, while all primes except the smallest are safe primes. The primes in a Cunningham chain are related to the Mersenne numbers thus: $p_n \equiv 2^n - 1 \mod 2^n$ , where $n$ is the
prime's position in the Cunningham chain (except in the case of the chain starting with 2, which is a special case).
In a Cunningham chain of the second kind, the relation among primes is $2p_{n - 1} - 1$ . An example of a Cunningham chain of the second kind is 1531, 3061, 6121, 12241, 24481.
It is strongly believed that there are infinitely many Cunningham chains of either kind, but this remains to be proven.
External links
PrimeFan's listing of Cunningham chains
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