Cunningham chain

A Cunningham chain ( according to Allan Joseph Champneys Cunningham ) of the first sort a sequence of primes of the form:

So p, 2p 1, 2 (2p 1) 1, 2 (2 (2p 1) 1) 1, ...

All primes of such a sequence, except the last prime number, are Sophie Germain primes.

The first Cunningham chain is the sequence: 2, 5, 11, 23, 47 It results for and can be explicitly described as follows: an = 3 · 2n - 1 for n = 0, 1, 2, 3, 4.

Cunningham a chain of the second type is a sequence of prime numbers of the form:

Two examples of Cunningham chains of the second type are the sequence 2, 3, 5, and the sequence 1531, 3061, 6121, 12241, 24481st

The longest known Cunningham chain of any type is of the first type has a length of 17 and starts with 2759832934171386593519. Found it was in March 2008. The first chain of length 16 was found in 1997.

Cryptography

Cunningham chains are studied in cryptography, as they provide the framework for an implementation of the ElGamal cryptosystem.

Tables with Cunningham chains

Cunningham chain of the first kind with k links

K = 5

K = 6:

Cunningham chains of the second kind with k links

K = 5

K = 7

A generalized Cunningham chain

A sequence of prime numbers of the form: P, ap b, A ( AP b) b, ... with a fixed and solid B are prime to each other, it is called a generalized Cunningham chain.

• Examples of generalized Cunningham chains with the element number k = 5

K = 5