Convex position

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

n1,n+1,2n1,2n+1,,2kn1,2kn+1

in which every number is prime.[1]

The numbers n1,2n1,,2kn1 form a Cunningham chain of the first kind of length k+1, while n+1,2n+1,,2kn+1 forms a Cunningham chain of the second kind. Each of the pairs 2in1,2in+1 is a pair of twin primes. Each of the primes 2in1 for 0ik1 is a Sophie Germain prime and each of the primes 2in1 for 1ik 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 3756801695685×2666669 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

Related properties of primes/pairs of primes

Notes and references

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Template:Prime number classes

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