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
![p={9771919142·[(53238·7879#)^2-1]+2310}·53238·7879#/385+1,](/images/equations/CousinPrimes/NumberedEquation1.svg) |
(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)
|
