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 is a Sophie Germain prime - that is, pi+1 = 2 pi + 1. 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 relatively prime 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.
