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.
The special case, when the four numbers are all primes, they are called bi-twin primes, such n values are
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
| k | n | Digits | Year | Discoverer |
| 0 | 2996863034895×21290000 | 388342 | 2016 | Timothy D. Winslow, PrimeGrid |
| 1 | 117864619517*6907# | 2971 | 2017 | Dirk Augustin |
| 2 | 1329861957×937#×23 | 399 | 2006 | Dirk Augustin |
| 3 | 223818083×409#×26 | 177 | 2006 | Dirk Augustin |
| 4 | 657713606161972650207961798852923689759436009073516446064261314615375779503143112×149# | 138 | 2014 | Primecoin |
| 5 | 386727562407905441323542867468313504832835283009085268004408453725770596763660073×61#×245 | 118 | 2014 | Primecoin |
| 6 | 263840027547344796978150255669961451691187241066024387240377964639380278103523328×47# | 99 | 2015 | Primecoin |
| 7 | 10739718035045524715×13# | 24 | 2008 | Jaroslaw Wroblewski |
| 8 | 1873321386459914635×13#×2 | 24 | 2008 | Jaroslaw Wroblewski |
q# denotes the primorial 2×3×5×7×...×q.
, 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.