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

## Largest known bi-twin chains

Largest known bi-twin chains of length k + 1 ((As of January 2014)[2])
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 $\displaystyle{ p }$ 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. Henri Lifchitz, BiTwin records. Retrieved on 2014-01-22.