Inclusion–exclusion principle: Difference between revisions
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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'', ''n''+4, ''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'', ''p'' + 4) for | |||
:''p'' = (311778476 · 587502 · 9001# · (587502 · 9001# + 1) + 210)·(587502 · 9001# − 1)/35 + 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> ≈ 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]] | |||
Revision as of 13:20, 29 January 2014
In mathematics, cousin primes are prime numbers that differ by four.[1] Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six.
The cousin primes (sequences Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. and Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. 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, n+4, n+8 will always be divisible by 3, so n = 3 is the only case where all three are primes.
Template:As of the largest known cousin prime was (p, p + 4) for
- p = (311778476 · 587502 · 9001# · (587502 · 9001# + 1) + 210)·(587502 · 9001# − 1)/35 + 1
where 9001# is a primorial. It was found by Ken Davis and has 11594 digits.[2]
The largest known cousin probable prime is
- 474435381 · 298394 − 1
- 474435381 · 298394 − 5.
It has 29629 digits and was found by Angel, Jobling and Augustin.[3] 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 primes. 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:
Using cousin primes up to 242, the value of B4 was estimated by Marek Wolf in 1996 as
- B4 ≈ 1.1970449.[4]
This constant should not be confused with Brun's constant for prime quadruplets, which is also denoted B4.
References
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- ↑
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- ↑ 474435381 · 298394 − 1. Prime pages.
- ↑ Marek Wolf, On the Twin and Cousin Primes (PostScript file).