Prime twin bi-chain

In number theory , a prime twin bi-chain of length is a prime number sequence of form ${\displaystyle k+1}$

${\displaystyle (n-1,n+1),(2n-1,2n+1),\ldots (2^{k}\cdot n-1,2^{k}\cdot n+1)}$

(The term comes from the English bi-twin chain or Bitwin chain ).

Examples

• The smallest , which generate a prime number twin bi-chain of length 2 (i.e. lead to the two pairs ), are the following: ${\displaystyle n}$ ${\displaystyle (n-1,n+1),(2n-1,2n+1)}$

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, 102300, 115470, 124770, 133980, 136950, 156420, ... ( continuation A066388 in OEIS )

• The smallest prime number twin bi-chains of length are the following (where is the product of all prime numbers up to ( prime faculty )): ${\displaystyle k+1}$ ${\displaystyle n\#=2\cdot 3\cdot 5\cdot 7\ldots \cdot n}$ ${\displaystyle n}$

${\displaystyle k+1}$	smallest known prime number twin bi-chain of length (as of June 20, 2017) ${\displaystyle k+1}$	Decimal places	Discovery date	Explorer
${\displaystyle 1}$	${\displaystyle 2\#\cdot 2\pm 1}$(so ) ${\displaystyle (3,5)}$	${\displaystyle 1}$	---	---
${\displaystyle 2}$	${\displaystyle 3\#\cdot 2^{n}\pm 1}$with (also ) ${\displaystyle n=0,1}$ ${\displaystyle (5,7),(11,13)}$	${\displaystyle 1}$ and ${\displaystyle 2}$	---	---
${\displaystyle 3}$	${\displaystyle 1005\cdot 7\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=0,1,2}$	${\displaystyle 6}$	September 1998	Henri Lifchitz
${\displaystyle 4}$	${\displaystyle 151\cdot 7\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=3,4,5,6}$	${\displaystyle 6}$ to ${\displaystyle 7}$	September 1998	Henri Lifchitz
${\displaystyle 5}$	${\displaystyle 1394847\cdot 13\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=0,1,2,3,4}$	${\displaystyle 11}$ to ${\displaystyle 12}$	December 1998	Jack Burn
${\displaystyle 6}$	${\displaystyle 1228253271\cdot 13\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=1,2,3,4,5,6}$	${\displaystyle 14}$ to ${\displaystyle 16}$	December 1998	Jack Burn
${\displaystyle 7}$	${\displaystyle 11228462199623\cdot 13\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=0,1,2,3,4,5,6}$	${\displaystyle 18}$ to ${\displaystyle 20}$	October 1999	Paul Jobling
${\displaystyle 8}$	${\displaystyle 21033215071024191\cdot 13\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=0,1,2,3,4,5,6,7}$	${\displaystyle 21}$ to ${\displaystyle 23}$	February 2002	Paul Jobling, Phil Carmody
${\displaystyle 9}$	${\displaystyle 1873321386459914635\cdot 13\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=1,2,3,4,5,6,7,8,9}$	${\displaystyle 24}$ to ${\displaystyle 26}$	December 2008	Jaroslaw Wroblewski

• The largest prime twin bi-chains of length are as follows: ${\displaystyle k+1}$

${\displaystyle k+1}$	largest known prime number twin bi-chain of length (as of June 20, 2017) ${\displaystyle k+1}$	Decimal places	Discovery date	Explorer	source
${\displaystyle 1}$	${\displaystyle 2996863034895\cdot 2^{1290000}\pm 1}$	${\displaystyle 388342}$	September 2016	Tom Greer
${\displaystyle 2}$	${\displaystyle 117864619517\cdot 6907\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=1,2}$	${\displaystyle 2971}$ and ${\displaystyle 2972}$	June 2017	Oscar Ostlin
${\displaystyle 3}$	${\displaystyle 204035674219\cdot 1609\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=0,1,2}$	${\displaystyle 695,695}$ and ${\displaystyle 695}$	July 2016	Didier Boivin
${\displaystyle 4}$	${\displaystyle 9185178409925\cdot 487\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=0,1,2,3}$	${\displaystyle 214,214,215}$ and ${\displaystyle 215}$	February 2017	Didier Boivin
${\displaystyle 5}$	${\displaystyle 4423253751167971164005\cdot 307\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=15,16,17,18,19}$	${\displaystyle 149,150,150,150}$ and ${\displaystyle 151}$	April 2015	Andrey Balyakin
${\displaystyle 6}$	${\displaystyle 386727562407905441323542867468313504832835283009085268004408453725770596763660073\cdot 61\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=45,46,47,48,49,50}$	${\displaystyle 118,118,118,119,}$ ${\displaystyle 119}$ and ${\displaystyle 119}$	April 2014	Primecoin
${\displaystyle 7}$	${\displaystyle 1165654412459516061933\cdot 73\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=7,8,9,10,11,12,13}$	${\displaystyle 52,53,53,53,}$ ${\displaystyle 53,54}$ and ${\displaystyle 54}$	April 2015	Andrey Balyakin
${\displaystyle 8}$	${\displaystyle 10739718035045524715\cdot 13\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=0,1,2,3,4,5,6,7}$	${\displaystyle 24,24,25,25,25,}$ ${\displaystyle 26,26}$ and ${\displaystyle 26}$	December 2008	Jaroslaw Wroblewski
${\displaystyle 9}$	${\displaystyle 1873321386459914635\cdot 13\#\cdot 2^{n}\pm 1}$ With ${\displaystyle n=1,2,3,4,5,6,7,8,9}$	${\displaystyle 24,24,24,24,25,}$ ${\displaystyle 25,25,26}$ and ${\displaystyle 26}$	December 2008	Jaroslaw Wroblewski

The prime number twin bi-chain of length 9 is currently (as of June 20, 2017) the longest known chain. It is also the only known chain of this length.

properties

• A prime twin bi-chain of length 1 has the form . They are called prime twins . ${\displaystyle (n-1,n+1)}$
• Each of the pairs with is a prime twin. ${\displaystyle (2^{i}\cdot n-1,2^{i}\cdot n+1)}$ ${\displaystyle 0\leq i\leq k}$
• The numbers form a Cunningham chain of the first kind with links. ${\displaystyle n-1,2n-1,\ldots ,2^{k}n-1}$ ${\displaystyle k+1}$
• The numbers form a Cunningham chain of the second type with links. ${\displaystyle n+1,2n+1,\ldots ,2^{k}n+1}$ ${\displaystyle k+1}$
• Every prime of the form with is a Sophie Germain prime . ${\displaystyle 2^{i}\cdot n-1}$ ${\displaystyle 0\leq i\leq k-1}$
• Any prime of the form with is a safe prime . ${\displaystyle 2^{i}\cdot n-1}$ ${\displaystyle 1\leq i\leq k}$
• Be with , so that at least one prime twin-Bi-chain is the length. 2 Then: ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n>6}$ ${\displaystyle (n-1,n+1),(2n-1,2n+1)}$

${\displaystyle n=30k}$ With ${\displaystyle k\in \mathbb {N} }$

generalization

A generalized prime twin bi-chain of length is a prime number sequence of the form ${\displaystyle k+1}$

${\displaystyle (n-1,n+1),(b\cdot n-1,b\cdot n+1),\ldots (b^{k}\cdot n-1,b^{k}\cdot n+1)}$ With ${\displaystyle b\in \mathbb {N} }$

Examples

• The largest generalized prime twin bi-chains of length are as follows: ${\displaystyle k+1}$

${\displaystyle k+1}$	largest known generalized twin prime bi-chain of length (as of June 20, 2017) ${\displaystyle k+1}$	Decimal places	Discovery date	Explorer
${\displaystyle 1}$	${\displaystyle 570323880\cdot {\frac {16500\#}{673949}}\cdot b^{n}\pm 1}$with and ${\displaystyle b=8087388}$ ${\displaystyle n=1,2}$	${\displaystyle 7120}$ and ${\displaystyle 7127}$	September 2004	Phil Carmody, Jens K. Andersen
${\displaystyle 2}$	${\displaystyle 2203\#\cdot 2^{161}\cdot 3^{66}\cdot 5^{69}\cdot 7^{73}\cdot 11^{23}\cdot 13^{23}\cdot 17^{26}\cdot 19^{23}\cdot 23^{22}\cdot 29^{22}\cdot 31^{22}\cdot 37^{25}\cdot 41^{23}\cdot 43^{25}\cdot 47^{25}\cdot 53^{26}\cdot 59^{25}\cdot 61^{23}\cdot b^{n}\pm 1}$ with and ${\displaystyle b=2^{21}\cdot 3^{12}\cdot 5^{11}\cdot 7^{9}\cdot 11^{3}\cdot 13^{3}\cdot 17^{2}\cdot 19^{3}\cdot 23^{4}\cdot 29^{4}\cdot 31^{4}\cdot 37^{3}\cdot 41^{3}\cdot 43^{3}\cdot 47^{3}\cdot 53^{2}\cdot 59^{3}\cdot 61^{3}}$ ${\displaystyle n=1,2,3}$	${\displaystyle 1702,1793}$ and ${\displaystyle 1884}$	October 2004	Ralph Twain
${\displaystyle 3}$	${\displaystyle {\frac {66\cdot 630^{5}\cdot 22218733^{2}\cdot 19\#\cdot 317\#}{561857}}\cdot b^{n}\pm 1}$with and ${\displaystyle b={\frac {8084448001600\cdot (13\#)^{3}\cdot 197\#}{12863477}}}$ ${\displaystyle n=1,2,3,4}$	${\displaystyle 261,360,459}$ and ${\displaystyle 558}$	August 2004	Jens K. Andersen
${\displaystyle 4}$	${\displaystyle (1250561\cdot 151\#[phone])\cdot b^{n}\pm 1}$with and ${\displaystyle b=4}$ ${\displaystyle n=1,2,3,4,5}$	${\displaystyle 67,67,68,68}$ and ${\displaystyle 69}$	August 2004	Jens K. Andersen
${\displaystyle 5}$	${\displaystyle {\frac {526583\cdot 83\#[phone]\cdot 19\#}{4}}\cdot b^{n}\pm 1}$with and ${\displaystyle b=4}$ ${\displaystyle n=1,2,3,4,5,6}$	${\displaystyle 39,39,40,40,41}$ and ${\displaystyle 42}$	August 2004	Jens K. Andersen
${\displaystyle 6}$	${\displaystyle 1394855870347655081\cdot 13\#\cdot b^{n}\pm 1}$with and ${\displaystyle b=4}$ ${\displaystyle n=2,3,4,5,6,7,8}$	${\displaystyle 24,25,26,26,27,27}$ and ${\displaystyle 28}$	August 2004	Jens K. Andersen

Individual evidence

1. ^ Eric W. Weisstein : CRC Concise Ennyclopedia of Mathematics. Chapman & Hall / CRC, 2015, p. 249 , accessed July 4, 2018 . 
2. ^ ^{A b c} Henri Lifchitz: BiTwin records. 2017, accessed July 4, 2018 . 
3. ↑ Chris K. Caldwell: The Top Twenty: Twin Primes. Prime Pages, accessed July 4, 2018 . 
4. ↑ 2996863034895 • 2 ^1290000 - 1 on Prime Pages
5. ↑ 2996863034895 • 2 ^1290000 + 1 on Prime Pages
6. ^ Neil Sloane : Numbers n such that n and 2n are both between a pair of twin primes. (Comments). OEIS , 2018, accessed July 5, 2018 . 

Web links

• Eric W. Weisstein : Bitwin Chain . In: MathWorld (English).