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.^[1]

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

Largest known bi-twin chains

Largest known bi-twin chains of length k + 1 (as of 22 January 2014^[2])

k	n	Digits	Year	Discoverer
0	3756801695685×2^666669	200700	2011	Timothy D. Winslow, PrimeGrid
1	7317540034×5011#	2155	2012	Dirk Augustin
2	1329861957×937#×2^3	399	2006	Dirk Augustin
3	223818083×409#×2^6	177	2006	Dirk Augustin
4	657713606161972650207961798852923689759436009073516446064261314615375779503143112×149#	138	2014	Primecoin (block 479357)
5	386727562407905441323542867468313504832835283009085268004408453725770596763660073×61#×2^45	118	2014	Primecoin (block 476538)
6	227339007428723056795583×13#×2	29	2004	Torbjörn Alm & Jens Kruse Andersen
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 chains

• Cunningham chain

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.

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