Convex position

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	39027761902802007714618528725397363585108921377235848032440823132447464787653697269×139#	138	2013	Primecoin (block 85429)
5	21011322942641319956617296739603541408400161540614555944797832220749394306836551×67#×27	107	2014	Primecoin (block 383918)
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.

Template:As of, 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

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