In
number theory, 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 only prime belonging to two pairs of cousin primes is 7. One of the numbers n, n + 4, n + 8 will always be divisible by 3, so n = 3 is the only case where all three are primes.
An example of a large
proven cousin prime pair is (p, p + 4) for
which has 20008 digits. In fact, this is part of a
prime triple since p is also a
twin prime (because p – 2 is also a proven prime).
As of April 2022[update], the largest-known pair of cousin primes was found by S. Batalov and has 51,934 digits. The primes are:
If the first
Hardy–Littlewood conjecture holds, then cousin primes have the same asymptotic density as
twin primes. An analogue of
Brun's constant for twin primes can be defined for cousin primes, called Brun's constant for cousin primes, with the initial term (3, 7) omitted, by the convergent sum:[3]
Using cousin primes up to 242, the value of B4 was estimated by Marek Wolf in 1996 as