Cousin prime

In mathematics, cousin primes are prime numbers that differ by four;^[1] 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)

Properties

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.^[2]

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.^[3] 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:

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

Using cousin primes up to 2⁴², the value of B₄ was estimated by Marek Wolf in 1996 as

B₄ ≈ 1.1970449.^[4]

This constant should not be confused with Brun's constant for prime quadruplets, which is also denoted B₄.

Cousin prime triplets

In an arithmetic progression of three terms with common difference 4, because 4 > 3 and the two numbers are relatively prime, one of the terms must be divisible by 3. Thus, the only cousin prime triplet is (3, 7, 11) with no longer sequence of cousin primes possible.

References

1. ^ Weisstein, Eric W., "Cousin Primes" from MathWorld.
2. ^ Davis, Ken (2009-05-08). "11594 digit cousin prime pair". primenumbers mailing list. http://tech.groups.yahoo.com/group/primenumbers/message/20235. Retrieved 2009-05-09. 
3. ^ 474435381 · 2⁹⁸³⁹⁴ − 1. Prime pages.
4. ^ Marek Wolf, On the Twin and Cousin Primes (PostScript file).