Editing Prime quadruplet (section)

== Prime quadruplets ==
The first eight prime quadruplets are:

{{nowrap|{[[5 (number)|5]], [[7 (number)|7]], [[11 (number)|11]], [[13 (number)|13]]},}} {{nowrap|{11, 13, [[17 (number)|17]], [[19 (number)|19]]},}} {{nowrap|{[[101 (number)|101]], [[103 (number)|103]], [[107 (number)|107]], [[109 (number)|109]]},}} {{nowrap|{[[191 (number)|191]], [[193 (number)|193]], [[197 (number)|197]], [[199 (number)|199]]},}} {{nowrap|{821, 823, 827, 829},}} {{nowrap|{1481, 1483, 1487, 1489},}} {{nowrap|{1871, 1873, 1877, 1879},}} {{nowrap|{2081, 2083, 2087, 2089} }}{{OEIS|id=A007530}}

All prime quadruplets except {{nowrap|{5, 7, 11, 13} }} are of the form {{math|{30''n'' + 11, 30''n'' + 13, 30''n'' + 17, 30''n'' + 19} }} for some integer {{mvar|n}}.  This structure is necessary to ensure that none of the four primes are divisible by 2,&nbsp;3 or 5.  The first prime of all such quadruplets end with the digit ''1'' and the last prime ends with the digit ''9'', in base 10.  Thus prime quadruplet of this form is called a '''prime decade'''.

A prime quadruplet can be described as a consecutive pair of [[twin prime]]s, two overlapping sets of [[prime triplet]]s, or two intermixed pairs of [[sexy prime]]s.  These "quad" primes can also  form the core of ''prime quintuplets'' and ''prime sextuplets'' when adding or subtracting 8 from their centers yields a prime.

All prime decades starting above 5 have centers of form 210n + 15, 210n + 105, or 210n + 195, since the centers must be −1, 0, or 1, modulo 7.  The +15 form may also give rise to a (high) prime quintuplet; the +195 form can also give rise to a (low) quintuplet; while the +105 form can yield both types of quintuplets and possibly prime sextuplets.  It is no accident that each prime in a prime decade is displaced from its center by a power of 2, since all centers are odd and divisible by both 3 and 5.

It is not known if there are infinitely many prime quadruplets.  A proof that there are infinitely many would imply the [[twin prime conjecture]], but it is consistent with current knowledge that there may be infinitely many pairs of twin primes and only finitely many prime quadruplets. The number of prime quadruplets with {{mvar|n}} digits in base 10 for {{nowrap|1={{mvar|n}} = 2, 3, 4, ...}} is 
:1, 3, 7, 27, 128, 733, 3869, 23620, 152141, 1028789, 7188960, 51672312, 381226246, 2873279651 {{OEIS|id=A120120}}.

{{As of|2019|February}} the largest known prime quadruplet has 10132 digits.<ref>[http://primes.utm.edu/top20/page.php?id=55 ''The Top Twenty: Quadruplet''] at The [[Prime Pages]]. Retrieved on 2019-02-28.</ref> It starts with {{nowrap|1={{mvar|p}} = 667674063382677 × 2<sup>33608</sup> − 1}}, found by Peter Kaiser.

The constant representing the sum of the reciprocals of all prime quadruplets, [[Brun's constant]] for prime quadruplets, denoted by {{math|''B''<sub>4</sub>}}, is the sum of the reciprocals of all prime quadruplets:

<math display=block>B_4 = \left(\frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13}\right)
+ \left(\frac{1}{11} + \frac{1}{13} + \frac{1}{17} + \frac{1}{19}\right)
+ \left(\frac{1}{101} + \frac{1}{103} + \frac{1}{107} + \frac{1}{109}\right) + \cdots</math>

with value:

:{{math|''B''<sub>4</sub>}} = 0.87058 83800 ± 0.00000 00005. 

This constant should not be confused with the '''Brun's constant for [[cousin prime]]s''', prime pairs of the form {{math|(''p'', ''p'' + 4)}}, which is also written as {{math|''B''<sub>4</sub>}}.

The prime quadruplet {11, 13, 17, 19} is alleged to appear on the [[Ishango bone]] although this is disputed.

Excluding the first prime quadruplet, the shortest possible distance between two quadruplets {{math|{''p'', ''p'' + 2, ''p'' + 6, ''p'' + 8} }} and {{math|{''q'', ''q'' + 2, ''q'' + 6, ''q'' + 8} }} is  {{math|''q'' − ''p''}}&nbsp;=&nbsp;30. The first occurrences of this are for {{mvar|p}} = 1006301, 2594951, 3919211, 9600551, 10531061, ... ({{OEIS2C|id=A059925}}).

The [[Skewes's number#Equivalent for prime k-tuples|Skewes number]] for prime quadruplets {{math|{''p'', ''p'' + 2, ''p'' + 6, ''p'' + 8} }} is 1172531 ({{harvtxt|Tóth|2019}}).