Bi-twin chain

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

Largest known bi-twin chains of length k + 1 (as of 22 January 2014[2])
knDigitsYearDiscoverer
03756801695685×26666692007002011Timothy D. Winslow, PrimeGrid
17317540034×5011#21552012Dirk Augustin
21329861957×937#×233992006Dirk Augustin
3223818083×409#×261772006Dirk Augustin
4657713606161972650207961798852923689759436009073516446064261314615375779503143112×149#1382014Primecoin (block 479357)
5386727562407905441323542867468313504832835283009085268004408453725770596763660073×61#×2451182014Primecoin (block 476538)
6263840027547344796978150255669961451691187241066024387240377964639380278103523328×47#992015Primecoin (block 942208)
710739718035045524715×13#242008Jaroslaw Wroblewski
81873321386459914635×13#×2242008Jaroslaw 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

  • 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

  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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