In mathematics, cousin primes are 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 and 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)
As of May 2009[update] 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.[1]
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.[1] 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, 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.[2]
This constant should not be confused with Brun's constant for prime quadruplets, which is also denoted B4.
References
- ^ Davis, Ken (2009-05-08). "11594 digit cousin prime pair". primenumbers mailing list. http://tech.groups.yahoo.com/group/primenumbers/message/20235. Retrieved on 2009-05-09.
- ^ Marek Wolf, On the Twin and Cousin Primes (PostScript file).
- Weisstein, Eric W., "Cousin Primes" from MathWorld.
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