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2013-07-27T23:41:24Z

matousek also sources the vertices of convex hull def

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In [[number theory]], a '''bi-twin chain''' of length ''k'' + 1 is a sequence of natural numbers

: <math> n-1,n+1,2n-1,2n+1, \dots, 2^k n - 1, 2^k n + 1 \,</math>

in which every number is [[prime number|prime]].<ref>[[Eric W. Weisstein]], ''CRC Concise Encyclopedia of Mathematics'', CRC Press, 2010, page 249.</ref>

The numbers <math>n-1, 2n-1, \dots, 2^kn - 1</math> form a [[Cunningham chain]] of the first kind of length <math>k + 1</math>, while <math>n+1, 2n + 1, \dots, 2^kn + 1</math> forms a Cunningham chain of the second kind. Each of the pairs <math>2^in - 1, 2^in+ 1</math> is a pair of [[twin primes]]. Each of the primes <math>2^in - 1</math> for <math>0 \le i \le k - 1</math> is a [[Sophie Germain prime]] and each of the primes <math>2^in - 1</math> for <math>1 \le i \le k</math> is a [[safe prime]].

== Largest known bi-twin chains ==

{| class="wikitable"
|+ Largest known bi-twin chains of length ''k'' + 1 (as of 22 January 2014<ref name="records">Henri Lifchitz, [http://www.primenumbers.net/Henri/fr-us/BiTwinRec.htm ''BiTwin records'']. Retrieved on 2014-01-22.</ref>)
|-
! ''k'' !! ''n'' !! Digits !! Year !! Discoverer
|-
| 0 || 3756801695685×2666669 || align="right" | 200700 || 2011 || Timothy D. Winslow, [[PrimeGrid]]
|-
| 1 || 7317540034×5011# || align="right" | 2155 || 2012 || Dirk Augustin
|-
| 2 || 1329861957×937#×23 || align="right" | 399 || 2006 || Dirk Augustin
|-
| 3 || 223818083×409#×26 || align="right" | 177 || 2006 || Dirk Augustin
|-
| 4 || 39027761902802007714618528725397363585108921377235848032440823132447464787653697269×139# || align="right" | 138 || 2013 || [[Primecoin]] ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=85429 block 85429])
|-
| 5 || 21011322942641319956617296739603541408400161540614555944797832220749394306836551×67#×27 || align="right" | 107 || 2014 || Primecoin ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=383918 block 383918])
|-
| 6 || 227339007428723056795583×13#×2 || align="right" | 29 || 2004 || Torbjörn Alm & Jens Kruse Andersen
|-
| 7 || 10739718035045524715×13# || align="right" | 24 || 2008 || Jaroslaw Wroblewski
|-
| 8 || 1873321386459914635×13#×2 || align="right" | 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>p</math> such that <math>2p + 1</math> is also prime.
* [[Safe prime]] is a prime <math>p</math> such that <math>(p-1)/2</math> is also prime.

== Notes and references ==
{{reflist}}
*{{CCBYSASource|sourcepath=http://number.subwiki.org/wiki/Bitwin_chain|sourcearticle=Bitwin chain|revision=566970742}}
[[Category:Prime numbers]]

{{Prime number classes}}

{{numtheory-stub}}

Convex position - Revision history

en>David Eppstein: matousek also sources the vertices of convex hull def