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Cousin prime
*** Shopping-Tip: Cousin prime
In
mathematics, a '''cousin prime''' is a pair of
prime numbers that differ by four; 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 [http://www.research.att.com/projects/OEIS?Anum=A023200 A023200] and [http://www.research.att.com/projects/OEIS?Anum=A046132 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, 441), (457, 461), (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)
As of 2005 As of November 2005 the largest known cousin prime is (''p'', ''p''+4) for
''p'' = (9771919142 · ((53238 · 7879#)
2 - 1) + 2310) · 53238 · 7879#/385 + 1
It has 10154 digits and was found by Torbjörn Alm, Micha Fleuren
and Jens Kruse Andersen [http://groups.yahoo.com/group/primeform/message/6637].
7879# is a
primorial.
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, with the initial term (3, 7) omitted:
:
Using cousin primes up to 2
42, the value of ''B''
4 was estimated by
Marek Wolf in
1996 as
:''B''
4 ≈ 1.1970449
This constant should not be confused with Brun's constant for
prime quadruplets, which is also denoted ''B''
4.
External links
-
MathWorld: Cousin Primes
Category:Prime numbers
fr:Nombres premiers cousins
