Innumber theory, abi-twin chain of lengthk + 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 isprime.^[1]

The special case, when the four numbers${\displaystyle n-1,n+1,2n-1,2n+1}$ are all primes, they are calledbi-twin primes,^[2] suchn values are

6, 30, 660, 810, 2130, 2550, 3330, 3390, 5850, 6270, 10530, 33180, 41610, 44130, 53550, 55440, 57330, 63840, 65100, 70380, 70980, 72270, 74100, 74760, 78780, 80670, 81930, 87540, 93240, … (sequenceA066388 in theOEIS)

Except 6, all of these numbers are divisible by 30.

The numbers${\displaystyle n-1,2n-1,\dots ,2^{k}n-1}$ form aCunningham 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 oftwin primes. Each of the primes${\displaystyle 2^{i}n-1}$ for${\displaystyle 0\le i\le k-1}$ is aSophie Germain prime and each of the primes${\displaystyle 2^{i}n-1}$ for${\displaystyle 1\le i\le k}$ is asafe prime.

q# denotes theprimorial 2×3×5×7×...×q.

As of 2014^[update], the longest known bi-twin chain is of length 8.

• Cunningham chain

• 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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• Prime numbers

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