Cousin prime
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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 A023200 (http://www.research.att.com/projects/OEIS?Anum=A023200) and A046132 (http://www.research.att.com/projects/OEIS?Anum=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)
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:
- <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 242, the value of B4 was estimated by Marek Wolf in 1996 as
- B4 ≈ 1.1970449
This constant should not be confused with Brun's constant for prime quadruplets, which is also denoted B4.
External links
- MathWorld: Cousin Primes (http://mathworld.wolfram.com/CousinPrimes.html)