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.

## Contents

* 1 Largest known bi-twin chains
* 2 Relation with other properties
* 2.1 Related chains
* 2.2 Related properties of primes/pairs of primes
* 3 Notes and references

## Largest known bi-twin chains

Largest known bi-twin chains of length k + 1 ((As of January 2014)[2])  k | n | Digits | Year | Discoverer

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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

* Cunningham chain

### 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

1. ↑ Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2010, page 249.
2. ↑ 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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Prime number classes

By formula|

* 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)
* Carol (2n − 1)2 − 2
* Kynea (2n \+ 1)2 − 2
* Leyland (xy \+ yx)
* Thabit (3·2n − 1)
* Williams ((b−1)·bn − 1)
* Mills (⌊A3n⌋)

By integer sequence|

* Fibonacci
* Lucas
* Pell
* Newman–Shanks–Williams
* Perrin
* Partitions
* Bell
* Motzkin

By property|

* Wieferich (pair)
* Wall–Sun–Sun
* Wolstenholme
* Wilson
* Lucky
* Fortunate
* Ramanujan
* Pillai
* Regular
* Strong
* Stern
* Supersingular (elliptic curve)
* Supersingular (moonshine theory)
* Good
* Super
* Higgs
* Highly cototient

Base-dependent|

* Happy
* Dihedral
* Palindromic
* Emirp
* Repunit (10n − 1)/9
* Permutable
* Circular
* Truncatable
* Strobogrammatic
* Minimal
* Weakly
* Full reptend
* Unique
* Primeval
* Self
* Smarandache–Wellin
* Tetradic

Patterns|

* Twin (p, p \+ 2)
* Bi-twin chain (n − 1, n \+ 1, 2n − 1, 2n \+ 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 (p, 2p + 1)
* Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
* Safe (p, (p − 1)/2)
* Arithmetic progression (p \+ a·n, n = 0, 1, 2, 3, ...)
* Balanced (consecutive p − n, p, p \+ n)

By size|

* Titanic (1,000+ digits)
* Gigantic (10,000+ digits)
* Mega (1,000,000+ digits)
* Largest known

Complex numbers|

* Eisenstein prime
* Gaussian prime

Composite numbers|

* Pseudoprime
* Catalan
* Elliptic
* Euler
* Euler–Jacobi
* Fermat
* Frobenius
* Lucas
* Somer–Lucas
* Strong
* Carmichael number
* Almost prime
* Semiprime
* Interprime
* Pernicious

Related topics|

* Probable prime
* Industrial-grade prime
* Illegal prime
* Formula for primes
* Prime gap

First 60 primes|

* 2
* 3
* 5
* 7
* 11
* 13
* 17
* 19
* 23
* 29
* 31
* 37
* 41
* 43
* 47
* 53
* 59
* 61
* 67
* 71
* 73
* 79
* 83
* 89
* 97
* 101
* 103
* 107
* 109
* 113
* 127
* 131
* 137
* 139
* 149
* 151
* 157
* 163
* 167
* 173
* 179
* 181
* 191
* 193
* 197
* 199
* 211
* 223
* 227
* 229
* 233
* 239
* 241
* 251
* 257
* 263
* 269
* 271
* 277
* 281

List of prime numbers

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