Bi-twin chain

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In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers

${\displaystyle n-1,n+1,2n-1,2n+1,\dots ,2^{k}n-1,2^{k}n+1\,}$

in which every number is prime.

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

Video Bi-twin chain

Largest known bi-twin chains

q# denotes the primorial 2×3×5×7×...×q.

As of 2014, the longest known bi-twin chain is of length 8.

Maps Bi-twin chain

Relation with other properties

Related chains

• Cunningham chain

Related properties of primes/pairs of primes

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

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Notes and references

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Source of the article : Wikipedia