Cunningham chain
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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.
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 Cunningham 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 Cunningham Chains of the second kind with length n, for 1 <= n <= 15de:Cunningham-Kette