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In [[mathematics]], '''cousin primes''' are [[prime numbers]] that differ by four.<ref>{{MathWorld|urlname=CousinPrimes|title=Cousin Primes}}</ref>  Compare this with [[twin prime]]s, pairs of prime numbers that differ by two, and [[sexy prime]]s, pairs of prime numbers that differ by six.

The cousin primes (sequences {{OEIS2C|id=A023200}} and {{OEIS2C|id=A046132}} in [[OEIS]]) below 1000 are:

:(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281), (307, 311), (313, 317), (349, 353), (379, 383), (397, 401), (439, 443), (457, 461), (463,467), (487, 491), (499, 503), (613, 617), (643, 647), (673, 677), (739, 743), (757, 761), (769, 773), (823, 827), (853, 857), (859, 863), (877, 881), (883, 887), (907, 911), (937, 941), (967, 971)

== Properties ==

The only prime belonging to two pairs of cousin primes is 7. One of the numbers ''n'',&nbsp;''n''+4,&nbsp;''n''+8 will always be divisible by 3, so ''n'' = 3 is the only case where all three are primes.

{{As of|2009|5}} the largest known cousin prime was (''p'',&nbsp;''p''&nbsp;+&nbsp;4) for

:''p'' = (311778476&nbsp;·&nbsp;587502&nbsp;·&nbsp;9001#&nbsp;·&nbsp;(587502&nbsp;·&nbsp;9001#&nbsp;+&nbsp;1)&nbsp;+&nbsp;210)·(587502&nbsp;·&nbsp;9001#&nbsp;&minus;&nbsp;1)/35&nbsp;+&nbsp;1

where 9001# is a [[primorial]]. It was found by Ken Davis and has 11594 digits.<ref>{{cite mailing list |url=http://tech.groups.yahoo.com/group/primenumbers/message/20235 |title=11594 digit cousin prime pair |date=2009-05-08 |accessdate=2009-05-09 |mailinglist=primenumbers |last=Davis |first=Ken |authorlink= }}</ref>

The largest known cousin [[probable prime]] is
:474435381 · 2<sup>98394</sup> − 1
:474435381 · 2<sup>98394</sup> − 5.
It has 29629 digits and was found by Angel, Jobling and Augustin.<ref>[http://primes.utm.edu/primes/page.php?id=60270 474435381 · 2<sup>98394</sup> − 1]. [[Prime pages]].</ref> While the first of these numbers has been proven prime, there is no known [[primality test]] to easily determine whether the second number is prime.

It follows from the first [[Hardy–Littlewood conjecture]] that cousin primes have the same asymptotic density as [[twin prime]]s. An analogue of [[Brun's constant]] for twin primes can be defined for cousin primes, called '''Brun's constant for cousin primes''', with the initial term (3, 7) omitted, by the convergent sum:<ref>{{cite journal | last=Segal | first=B. | title=Generalisation du théorème de Brun | language=Russian | jfm=57.1363.06 | journal=C. R. Acad. Sci. URSS | volume=1930 | pages=501–507 | year=1930 }}</ref>

:<math>B_4 = \left(\frac{1}{7} + \frac{1}{11}\right) + \left(\frac{1}{13} + \frac{1}{17}\right) + \left(\frac{1}{19} + \frac{1}{23}\right) + \cdots.</math>

Using cousin primes up to 2<sup>42</sup>, the value of ''B''<sub>4</sub> was estimated by Marek Wolf in 1996 as

:''B''<sub>4</sub> &asymp; 1.1970449.<ref>Marek Wolf (1996), [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.1864 ''On the Twin and Cousin Primes''].</ref>

This constant should not be confused with Brun's constant for [[prime quadruplet]]s, which is also denoted ''B''<sub>4</sub>.

The [[Skewes's number#Equivalent for prime k-tuples|Skewes number]] for cousin primes is <math>5206837</math> ({{harvtxt|Tóth|2019}}).

== Notes ==
{{reflist}}

== References ==
* {{cite book | title=Prime Numbers: The Most Mysterious Figures in Math | first=David | last=Wells | publisher=John Wiley & Sons | year=2011 | isbn=1118045718 | page=33 }}
* {{cite book | title=Number theory: an introduction via the distribution of primes | first1=Benjamin | last1=Fine | first2=Gerhard | last2=Rosenberger | publisher=Birkhäuser | year=2007 | isbn=0817644725 | pages=206 }}
*{{citation|first=László|last=Tóth|title=On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood|journal=Computational Methods in Science and Technology|volume=25|year=2019|issue=3|url=http://cmst.eu/wp-content/uploads/files/10.12921_cmst.2019.0000033_TOTH.pdf}}.
* {{cite journal|last1=Wolf|first1=Marek|title=Random walk on the prime numbers|journal=Physica A: Statistical Mechanics and its Applications|date=February 1998|volume=250|issue=1-4|pages=335–344|doi=10.1016/s0378-4371(97)00661-4}}

{{Prime number classes}}

[[Category:Classes of prime numbers]]
[[Category:Unsolved problems in mathematics]]