Cousin Primes

Pairs of primes of the form (, ) are called cousin primes. The first few are (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), ... (OEIS A023200 and A046132).

A large pair of cousin (proven) primes start with

(1)

where  is a primorial. These primes have 10154 digits and were found by T. Alm, M. Fleuren, and J. K. Andersen (Andersen 2005).

As of Jan. 2006, the largest known pair of cousin (probable) primes are

(2)

which have 11311 digits and were found by D. Johnson in May 2004.

According to the first Hardy-Littlewood conjecture, the cousin primes have the same asymptotic density as the twin primes,

			(3)
			(4)

where  (OEIS A114907) is the twin primes constant.

An analogy to Brun's constant, the constant

(5)

(omitting the initial term ) can be defined. Using cousin primes up to , the value of  is estimated as

(6)

REFERENCES:

Andersen, J. K. "Gigantic Sexy and Cousin Primes." Post to primeform user forum. Nov. 3, 2005. https://groups.yahoo.com/group/primeform/message/6637.

Sloane, N. J. A. Sequences A023200, A046132, A114907 in "The On-Line Encyclopedia of Integer Sequences."

Wolf, M. "On Twin and Cousin Primes." https://www.ift.uni.wroc.pl/~mwolf/.