Convex position
In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers
in which every number is prime.[1]
The numbers form a Cunningham chain of the first kind of length , while forms a Cunningham chain of the second kind. Each of the pairs is a pair of twin primes. Each of the primes for is a Sophie Germain prime and each of the primes for is a safe prime.
Largest known bi-twin chains
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
- Twin primes
- Sophie Germain prime is a prime such that is also prime.
- Safe prime is a prime such that is also prime.
Notes and references
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- ↑ Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2010, page 249.
- ↑ Henri Lifchitz, BiTwin records. Retrieved on 2014-01-22.