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Encyclopedia > 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 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, 891), (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:

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. Results from FactBites:

 Euler prime - Wikipedia, the free encyclopedia (73 words) In number theory, Euler primes or symmetric primes are primes that are the same distance from a given number. For example 3 and 13 are both 5 units from the number 8, hence are symmetric primes. All twin primes, cousin primes, and sexy primes are symmetric primes.
 Sexy prime - Wikipedia, the free encyclopedia (259 words) In mathematics, a sexy prime is a pair of prime numbers that differ by six; compare this with twin primes, pairs of prime numbers that differ by two, and cousin primes, pairs of prime numbers that differ by four. Like twin primes, sexy primes can be extended to larger constellations. Sexy prime quadruplets can only begin with primes ending in a 1 in their decimal representation (apart from 5); the sexy prime quadruplets (sequences A046121, A046122, A046123 and A046124 in OEIS) below 1000 are:
More results at FactBites »

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