Cunningham chain - Definition and Overview

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,...,p_n) such that for all 1 ≤ i < n, p_i+1 = 2 p_i + 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,...,p_n) such that for all 1 ≤ i < n, p_i+1 = 2 p_i - 1.

Cunningham chains are also sometimes generalized to sequences of prime numbers (p₁,...,p_n) such that for all 1 ≤ i < n, p_i+1 = ap_i + 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

• The Prime Glossary: Cunningham chain (http://primes.utm.edu/glossary/page.php?sort=CunninghamChain)
• PrimeLinks++: Cunningham chain (http://primes.utm.edu/links/theory/special_forms/Cunningham_chains/)
• Sequence A005602 (http://www.research.att.com/projects/OEIS?Anum=A005602) in OEIS: the first term of the lowest complete Cunnigham Chains of the first kind of length n, for 1 <= n <= 14
• Sequence A005603 (http://www.research.att.com/projects/OEIS?Anum=A005603) in OEIS: the first term of the lowest complete Cunnigham Chains of the second kind with length n, for 1 <= n <= 15