# Cunningham chain

In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham.

A Cunningham chain of the first kind is a sequence of prime numbers (p1,...,pn) such that for all 1 ≤ i < n, pi+1 = 2 pi + 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). Similarly, a Cunningham chain of the second kind is a sequence of prime numbers (p1,...,pn) such that for all 1 ≤ i < n, pi+1 = 2 pi - 1.

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

A Cunningham chain is called complete if it cannot be further extended, i.e., if the next term in the chain would not be a prime number anymore.

According to the strong prime k-tuple conjecture, which is widely believed to be true, for every k there are infinitely many Cunningham chains of length k. There are, however, no known direct methods of generating such chains.

As of August 2005, the longest Cunningham chain of either kind found is of length 16. Such a chain of the second kind was discovered by Tony Forbes in 1997, starting with 3203000719597029781. A chain of the first kind was discovered by Phil Carmody and Paul Jobling in 2002, starting with 810433818265726529159.

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