# Bi-twin chain

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.[1]

The special case, when the four numbers ${\displaystyle n-1,n+1,2n-1,2n+1}$ are all primes, they are called bi-twin primes,[2] such n 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, … (sequence A066388 in the OEIS)

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

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.

## Largest known bi-twin chains

Largest known bi-twin chains of length k + 1 (as of 22 January 2014[3])
k n Digits Year Discoverer
0 3756801695685×2666669 200700 2011 Timothy D. Winslow, PrimeGrid
1 7317540034×5011# 2155 2012 Dirk Augustin
2 1329861957×937#×23 399 2006 Dirk Augustin
3 223818083×409#×26 177 2006 Dirk Augustin
4 657713606161972650207961798852923689759436009073516446064261314615375779503143112×149# 138 2014 Primecoin (block 479357)
5 386727562407905441323542867468313504832835283009085268004408453725770596763660073×61#×245 118 2014 Primecoin (block 476538)
6 263840027547344796978150255669961451691187241066024387240377964639380278103523328×47# 99 2015 Primecoin (block 942208)
7 10739718035045524715×13# 24 2008 Jaroslaw Wroblewski
8 1873321386459914635×13#×2 24 2008 Jaroslaw Wroblewski

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

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

## Relation with other properties

• 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.

## Notes and references

1. ^ Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2010, page 249.
2. ^ BiTwin records
3. ^ Henri Lifchitz, BiTwin records. Retrieved on 2014-01-22.