Bi-twin chain
In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers
- [math]\displaystyle{ n-1,n+1,2n-1,2n+1, \dots, 2^k n - 1, 2^k n + 1 \, }[/math]
in which every number is prime.[1]
The numbers [math]\displaystyle{ n-1, 2n-1, \dots, 2^kn - 1 }[/math] form a Cunningham chain of the first kind of length [math]\displaystyle{ k + 1 }[/math], while [math]\displaystyle{ n+1, 2n + 1, \dots, 2^kn + 1 }[/math] forms a Cunningham chain of the second kind. Each of the pairs [math]\displaystyle{ 2^in - 1, 2^in+ 1 }[/math] is a pair of twin primes. Each of the primes [math]\displaystyle{ 2^in - 1 }[/math] for [math]\displaystyle{ 0 \le i \le k - 1 }[/math] is a Sophie Germain prime and each of the primes [math]\displaystyle{ 2^in - 1 }[/math] for [math]\displaystyle{ 1 \le i \le k }[/math] 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 | 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
Related chains
Related properties of primes/pairs of primes
- Twin primes
- Sophie Germain prime is a prime [math]\displaystyle{ p }[/math] such that [math]\displaystyle{ 2p + 1 }[/math] is also prime.
- Safe prime is a prime [math]\displaystyle{ p }[/math] such that [math]\displaystyle{ (p-1)/2 }[/math] is also prime.
Notes and references
- ↑ Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2010, page 249.
- ↑ Henri Lifchitz, BiTwin records. Retrieved on 2014-01-22.
- As of this edit, this article uses content from "Bitwin chain", which is licensed in a way that permits reuse under the Creative Commons Attribution-ShareAlike 3.0 Unported License, but not under the GFDL. All relevant terms must be followed.
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