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.[1]
The special case, when the four numbers are all primes, they are called bi-twin primes,[2] 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, … (sequence A066388 in the OEIS)
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 | 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[update], the longest known bi-twin chain is of length 8.
Relation with other properties
Related chains
- Cunningham chain
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
- 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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Prime number classes
- Fermat (22n + 1)
- Mersenne (2p − 1)
- Double Mersenne (22p−1 − 1)
- Wagstaff (2p + 1)/3
- Proth (k·2n + 1)
- Factorial (n! ± 1)
- Primorial (pn# ± 1)
- Euclid (pn# + 1)
- Pythagorean (4n + 1)
- Pierpont (2m·3n + 1)
- Quartan (x4 + y4)
- Solinas (2m ± 2n ± 1)
- Cullen (n·2n + 1)
- Woodall (n·2n − 1)
- Cuban (x3 − y3)/(x − y)
- Leyland (xy + yx)
- Thabit (3·2n − 1)
- Williams ((b−1)·bn − 1)
- Mills (⌊A3n⌋)
- Fibonacci
- Lucas
- Pell
- Newman–Shanks–Williams
- Perrin
- Palindromic
- Emirp
- Repunit (10n − 1)/9
- Permutable
- Circular
- Truncatable
- Minimal
- Delicate
- Primeval
- Full reptend
- Unique
- Happy
- Self
- Smarandache–Wellin
- Strobogrammatic
- Dihedral
- Tetradic
- Twin (p, p + 2)
- Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
- Triplet (p, p + 2 or p + 4, p + 6)
- Quadruplet (p, p + 2, p + 6, p + 8)
- k-tuple
- Cousin (p, p + 4)
- Sexy (p, p + 6)
- Chen
- Sophie Germain/Safe (p, 2p + 1)
- Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
- Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
- Balanced (consecutive p − n, p, p + n)