Vipul: Created page with '==Definition== Let $k$ be a natural number. A '''Cunningham chain of the first kind''' of length $k$ is a sequence of primes $q_1 < q_2 < \dots < q_k<...'$

2009-04-20T18:01:59Z

Created page with '==Definition== Let <math>k</math> be a natural number. A '''Cunningham chain of the first kind''' of length <math>k</math> is a sequence of primes <math>q_1 < q_2 < \dots < q_k<...'

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==Definition==

Let <math>k</math> be a natural number. A '''Cunningham chain of the first kind''' of length <math>k</math> is a sequence of primes <math>q_1 < q_2 < \dots < q_k</math> such that <math>q_{i+1} = 2q_i + 1</math> for each <math>1 \le i \le k-1</math>.

A '''complete Cunningham chain of the first kind''' is a Cunningham chain of the first kind that cannot be extended further in either direction.

Given a Cunningham chain of the first kind of length <math>2</math>, the first prime in the chain is a [[Sophie Germain prime]] and the second prime in the chain is a [[safe prime]]. More generally, in any Cunningham chain of length <math>k</math>, the first <math>k-1</math> primes are Sophie Germain primes and the last <math>k-1</math> primes are safe primes.

==Related facts and conjectures==

* [[Conjecture on existence of Cunningham chains of the first kind of arbitrary length]]

==Relation with other properties==

* [[Cunningham chain of the second kind]]
* [[Bitwin chain]] is a combination of Cunningham chains of both kinds and [[twin primes]].

==Testing/listing==

{{oeis|A005602}}

This lists, for every <math>k</math>, the smallest prime beginning a complete Cunningham chain of length <math>k</math>.

Cunningham chain of the first kind - Revision history

Vipul: Created page with '==Definition== Let k be a natural number. A '''Cunningham chain of the first kind''' of length k is a sequence of primes q_1 < q_2 < \dots < q_k<...'

Vipul: Created page with '==Definition== Let $k$ be a natural number. A '''Cunningham chain of the first kind''' of length $k$ is a sequence of primes $q_1 < q_2 < \dots < q_k<...'$