# Bi-twin chain

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

$n-1,n+1,2n-1,2n+1,\dots ,2^{k}n-1,2^{k}n+1\,$ in which every number is prime.

The numbers $n-1,2n-1,\dots ,2^{k}n-1$ form 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.

## Largest known bi-twin chains

Largest known bi-twin chains of length k + 1 (as of 22 January 2014)
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

### 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.

## Notes and references

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