Cunningham chain

Consider the sequence of primes 2, 5, 11, 23, 47. Each is twice the previous one plus 1.

When, in a sequence of primes p1,pk each pn=2pn-1+1 for 1<nk (or alternatively, each pn=pn-1-12 for 0<n<k), the sequence is called a Cunningham chainMathworldPlanetmath of the first kind. In a Cunningham chain of the first kind, all primes except the largest are Sophie Germain primesMathworldPlanetmath, while all primes except the smallest are safe primes. The primes in a Cunningham chain are related to the Mersenne numbers thus: pn2n-1mod2n, where n is the prime’s position in the Cunningham chain (except in the case of the chain starting with 2, which is a special case).

In a Cunningham chain of the second kind, the relation among primes is 2pn-1-1. An example of a Cunningham chain of the second kind is 1531, 3061, 6121, 12241, 24481.

It is strongly believed that there are infinitely many Cunningham chains of either kind, but this remains to be proven.

External links’s listing of Cunningham chains

Title Cunningham chain
Canonical name CunninghamChain
Date of creation 2013-03-22 16:04:22
Last modified on 2013-03-22 16:04:22
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Definition
Classification msc 11N05