Bi-twin chain

Definition

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

Relation with other properties

Related chains

• Cunningham chain

Related properties of primes/pairs of primes

• Twin primes
• Sophie Germain prime is a prime $p$ such that $2p+1$ is also prime.
• Safe prime is a prime $p$ such that $(p-1)/2$ is also prime.