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.

Largest known bi-twin chains

Largest known bi-twin chains of length k + 1 ((As of January 2014)^[2])
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), 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

↑ Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2010, page 249.
↑ Henri Lifchitz, BiTwin records. Retrieved on 2014-01-22.

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[1] Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2010, page 249.

[records-2] Henri Lifchitz, BiTwin records. Retrieved on 2014-01-22.

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v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
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 $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) 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 ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Safe ( $p, (p - 1)/2$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) 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