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Cunningham ChainIn 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. External links
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