Inclusion–exclusion principle: Difference between revisions

Revision as of 17:11, 8 February 2014

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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.
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-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 analogy 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:
-:<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, [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.1864 ''On the Twin and Cousin Primes''] ([[PostScript]] file).</ref>
-This constant should not be confused with Brun's constant for [[prime quadruplet]]s, which is also denoted ''B''<sub>4</sub>.
-== References ==
-{{reflist}}
-{{Prime number classes}}
-[[Category:Classes of prime numbers]]

Inclusion–exclusion principle: Difference between revisions

Revision as of 17:11, 8 February 2014

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