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.Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2010, page 249.
The special case, when the four numbers are all primes, they are called bi-twin primes,[http://www.primenumbers.net/Henri/fr-us/BiTwinRec.htm BiTwin records] such n values are
:6, 30, 660, 810, 2130, 2550, 3330, 3390, 5850, 6270, 10530, 33180, 41610, 44130, 53550, 55440, 57330, 63840, 65100, 70380, 70980, 72270, 74100, 74760, 78780, 80670, 81930, 87540, 93240, … {{OEIS|A066388}}
Except 6, all of these numbers are divisible by 30.
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
class="wikitable"
|+ Largest known bi-twin chains of length k + 1 ({{As of|2025|01|22|lc=y}}Henri Lifchitz, [http://www.primenumbers.net/Henri/fr-us/BiTwinRec.htm BiTwin records]. Retrieved on 2014-01-22.) | ||||
k | n | Digits | Year | Discoverer |
---|---|---|---|---|
0 | 2996863034895×21290000 | align="right" | 388342 | 2016 | Timothy D. Winslow, PrimeGrid |
1 | 117864619517*6907# | align="right" | 2971 | 2017 | Dirk Augustin |
2 | 1329861957×937#×23 | align="right" | 399 | 2006 | Dirk Augustin |
3 | 223818083×409#×26 | align="right" | 177 | 2006 | Dirk Augustin |
4 | 657713606161972650207961798852923689759436009073516446064261314615375779503143112×149# | align="right" | 138 | 2014 | Primecoin ([https://primes.zone/#records block 479357]) |
5 | 386727562407905441323542867468313504832835283009085268004408453725770596763660073×61#×245 | align="right" | 118 | 2014 | Primecoin ([https://primes.zone/#records block 476538]) |
6 | 263840027547344796978150255669961451691187241066024387240377964639380278103523328×47# | align="right" | 99 | 2015 | Primecoin ([https://primes.zone/#records block 942208]) |
7 | 10739718035045524715×13# | align="right" | 24 | 2008 | Jaroslaw Wroblewski |
8 | 1873321386459914635×13#×2 | align="right" | 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 such that is also prime.
- Safe prime is a prime such that is also prime.
Notes and references
{{reflist}}
- {{CCBYSASource|sourcepath=http://number.subwiki.org/wiki/Bitwin_chain|sourcearticle=Bitwin chain|revision=566970742}}
{{Prime number classes}}