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,…,2kn−1,2kn+1

in which every number is prime.[1]

The special case, when the four numbers n−1,n+1,2n−1,2n+1 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 n−1,2n−1,…,2kn−1 form a Cunningham chain of the first kind of length k+1, while n+1,2n+1,…,2kn+1 forms a Cunningham chain of the second kind. Each of the pairs 2in−1,2in+1 is a pair of twin primes. Each of the primes 2in−1 for 0≤i≤k−1 is a Sophie Germain prime and each of the primes 2in−1 for 1≤i≤k 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 22 January 2025[update][3])  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 (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 p such that 2p+1 is also prime.
* Safe prime is a prime p such that (p−1)/2 is also prime.

## Notes and references

1. ↑ Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2010, page 249.
2. ↑ BiTwin records
3. ↑ 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.

* v
* t
* e

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

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Bi-twin chain. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Bi-twin_chain&oldid=3991561"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template