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 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 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 [1]. 7879# is a primorial.

As of January 2006 the largest known cousin probable prime is (630062 · 2³⁷⁵⁵⁵ + 3, 630062 · 2³⁷⁵⁵⁵ + 7). It has 11311 digits and was found by Donovan Johnson in 2004. There is no known primality test to easily determine whether these numbers are primes.

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:

$B_4 = \left(\frac{1}{7} + \frac{1}{11}\right) + \left(\frac{1}{13} + \frac{1}{17}\right) + \left(\frac{1}{19} + \frac{1}{23}\right) + \cdots$