bi-twin chain

In number theory, a bi-twin chain of length k + 1 is a sequence of natural numbers

: n-1,n+1,2n-1,2n+1, \dots, 2^k n - 1, 2^k n + 1 \,

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 n-1,n+1,2n-1,2n+1 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 n-1, 2n-1, \dots, 2^kn - 1 form a Cunningham chain of the first kind of length k + 1, while n+1, 2n + 1, \dots, 2^kn + 1 forms a Cunningham chain of the second kind. Each of the pairs 2^in - 1, 2^in+ 1 is a pair of twin primes. Each of the primes 2^in - 1 for 0 \le i \le k - 1 is a Sophie Germain prime and each of the primes 2^in - 1 for 1 \le i \le k 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.)

knDigitsYearDiscoverer
02996863034895×21290000align="right" | 3883422016Timothy D. Winslow, PrimeGrid
1117864619517*6907#align="right" | 29712017Dirk Augustin
21329861957×937#×23align="right" | 3992006Dirk Augustin
3223818083×409#×26align="right" | 1772006Dirk Augustin
4657713606161972650207961798852923689759436009073516446064261314615375779503143112×149#align="right" | 1382014Primecoin ([https://primes.zone/#records block 479357])
5386727562407905441323542867468313504832835283009085268004408453725770596763660073×61#×245align="right" | 1182014Primecoin ([https://primes.zone/#records block 476538])
6263840027547344796978150255669961451691187241066024387240377964639380278103523328×47#align="right" | 992015Primecoin ([https://primes.zone/#records block 942208])
710739718035045524715×13#align="right" | 242008Jaroslaw Wroblewski
81873321386459914635×13#×2align="right" | 242008Jaroslaw 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 =

Notes and references

{{reflist}}

  • {{CCBYSASource|sourcepath=http://number.subwiki.org/wiki/Bitwin_chain|sourcearticle=Bitwin chain|revision=566970742}}

Category:Prime numbers

{{Prime number classes}}