Bi-twin chain

Summary

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 edit

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 657713606161972650207961798852923689759436009073516446064261314615375779503143112×149# 138 2014 Primecoin (block 479357)
5 386727562407905441323542867468313504832835283009085268004408453725770596763660073×61#×245 118 2014 Primecoin (block 476538)
6 263840027547344796978150255669961451691187241066024387240377964639380278103523328×47# 99 2015 Primecoin (block 942208)
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 edit

Related chains edit

Related properties of primes/pairs of primes edit

  • 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 edit

  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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