From Wikipedia, the free encyclopedia
In
number theory , a bi-twin chain of length k + 1 is a sequence of natural numbers
n
−
1
,
n
+
1
,
2
n
−
1
,
2
n
+
1
,
…
,
2
k
n
−
1
,
2
k
n
+
1
{\displaystyle n-1,n+1,2n-1,2n+1,\dots ,2^{k}n-1,2^{k}n+1\,}
in which every number is
prime .
[1]
The special case, when the four numbers
n
−
1
,
n
+
1
,
2
n
−
1
,
2
n
+
1
{\displaystyle 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
,
2
n
−
1
,
…
,
2
k
n
−
1
{\displaystyle n-1,2n-1,\dots ,2^{k}n-1}
form a
Cunningham chain of the first kind of length
k
+
1
{\displaystyle k+1}
, while
n
+
1
,
2
n
+
1
,
…
,
2
k
n
+
1
{\displaystyle n+1,2n+1,\dots ,2^{k}n+1}
forms a Cunningham chain of the second kind. Each of the pairs
2
i
n
−
1
,
2
i
n
+
1
{\displaystyle 2^{i}n-1,2^{i}n+1}
is a pair of
twin primes . Each of the primes
2
i
n
−
1
{\displaystyle 2^{i}n-1}
for
0
≤
i
≤
k
−
1
{\displaystyle 0\leq i\leq k-1}
is a
Sophie Germain prime and each of the primes
2
i
n
−
1
{\displaystyle 2^{i}n-1}
for
1
≤
i
≤
k
{\displaystyle 1\leq i\leq k}
is a
safe prime .
Largest known bi-twin chains
Largest known bi-twin chains of length k + 1 (as of 22 January 2014
[update]
[3] )
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
Related properties of primes/pairs of primes
Twin primes
Sophie Germain prime is a prime
p
{\displaystyle p}
such that
2
p
+
1
{\displaystyle 2p+1}
is also prime.
Safe prime is a prime
p
{\displaystyle p}
such that
(
p
−
1
)
/
2
{\displaystyle (p-1)/2}
is also prime.
Notes and references
By formula By integer sequence By property
Base -dependent Patterns
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 )
By size
Complex numbers
Composite numbers Related topics First 60 primes