Category:Definitions/Cunningham Chains

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This category contains definitions related to Cunningham Chains.
Related results can be found in Category:Cunningham Chains.


There are $2$ types of Cunningham chain:


First Kind

A Cunningham chain of the first kind is a (finite) sequence $\tuple {p_1, p_2, \ldots, p_n}$ such that:

$(1): \quad \forall i \in \set {1, 2, \ldots, n - 1}: p_{i + 1} = 2 p_i + 1$
$(2): \quad p_i$ is prime for all $i \in \set {1, 2, \ldots, n - 1}$
$(3): \quad n$ is not prime such that $2 n + 1 = p_1$
$(4): \quad 2 p_n + 1$ is not prime.

Thus:

each term except the last is a Sophie Germain prime
each term except the first is a safe prime.


Second Kind

A Cunningham chain of the second kind is a (finite) sequence $\tuple {p_1, p_2, \ldots, p_n}$ such that:

$(1): \quad \forall i \in \set {1, 2, \ldots, n - 1}: p_{i + 1} = 2 p_i - 1$
$(2): \quad p_i$ is prime for all $i \in \set {1, 2, \ldots, n - 1}$
$(3): \quad n$ is not prime such that $2 n - 1 = p_1$
$(4): \quad 2 p_n - 1$ is not prime.