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