Bitwin chain: Difference between revisions

Latest revision as of 02:20, 2 May 2010

A bitwin chain of length $k+1$ is defined as a collection of natural numbers:

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

such that all the numbers in the chain are prime.

Note that the numbers $n-1,2n-1,\cdots 2^{k}n-1$ forms a Cunningham chain of the first kind of length $k+1$ , while $n+1,2n+1,\dots ,2^{k}n+1$ 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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 ==Definition==
-A '''bitwin chain''' of length <math>k</math> is defined as a collection of natural numbers:
+A '''bitwin chain''' of length <math>k + 1</math> is defined as a collection of natural numbers:
 <math>(n-1,n+1,2n-1,2n+1, \dots 2^k\cdot n - 1, 2^k \cdot n + 1)</math>
@@ Line 7: / Line 7: @@
 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==