Bitwin chain: Difference between revisions

Revision as of 02:17, 2 May 2010

A bitwin chain of length $k$ is defined as a collection of natural numbers:

$(n - 1, n + 1, 2 n - 1, 2 n + 1, \dots 2^{k} \cdot n - 1, 2^{k} \cdot n + 1)$

such that all the numbers in the chain are prime.

Note that the numbers $n - 1, 2 n - 1, \dots 2^{k} n - 1$ forms a Cunningham chain of the first kind of length $k + 1$ , while $n + 1, 2 n + 1, \dots, 2^{k} n + 1$ forms a Cunningham chain of the second kind. Each of the pairs $2^{i} n - 1, 2^{i} n + 1$ is a pair of twin primes. Each of the primes $2^{i} n - 1$ for $0 \leq i \leq k - 1$ is a Sophie Germain prime and each of the primes $2^{i} n - 1$ for $1 \leq i \leq k$ is a safe prime.

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 such that all the numbers in the chain are prime.
-Note that the numbers <math>n-1, 2n-1, \cdots 2^kn - 1</math> forms a [[Cunningham chain of the first kind]] of length <math>k + 1</math>, while <math>n+1, 2n + 1, \dots, 2^kn + 1</math> forms a [[Cunninghan chain of the second kind]]. Each of the pairs <math>2^in - 1, 2^in+ 1</math> is a pair of [[twin primes]].
+Note that the numbers <math>n-1, 2n-1, \cdots 2^kn - 1</math> forms a [[defining ingredient::Cunningham chain of the first kind]] of length <math>k + 1</math>, while <math>n+1, 2n + 1, \dots, 2^kn + 1</math> forms a [[defining ingredient::Cunningham chain of the second kind]]. Each of the pairs <math>2^in - 1, 2^in+ 1</math> is a pair of [[defining ingredient::twin primes]]. Each of the primes <math>2^in - 1</math> for <math>0 \le i \le k - 1</math> is a [[defining ingredient::Sophie Germain prime]] and each of the primes <math>2^in - 1</math> for <math>1 \le i \le k</math> is a [[defining ingredient::safe prime]].
 ==Relation with other properties==