Cousin prime

Encyclopedia : C : CO : COU : 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] 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, 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 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 [link]. 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:

[B_4 = \left(\frac + \frac\right) + \left(\frac + \frac\right) + \left(\frac + \frac\right) + \cdots]

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]

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