number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, cousin primes are

prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

that differ by four. Compare this with

twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime' ...

s, pairs of prime numbers that differ by two, and

sexy prime In number theory, sexy primes are prime numbers that differ from each other by . For example, the numbers and are a pair of sexy primes, because both are prime and 11 - 5 = 6. The term "sexy prime" is a pun stemming from the Latin word for six ...

s, pairs of prime numbers that differ by six. The cousin primes (sequences and in

OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...

) 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

The only prime belonging to two pairs of cousin primes is 7. One of the numbers will always be divisible by 3, so is the only case where all three are primes. An example of a large proven cousin prime pair is for :

p = 4111286921397 \times 2^ + 1

which has 20008 digits. In fact, this is part of a prime triple since is also a

(because is also a proven prime). , the largest-known pair of cousin primes was found by S. Batalov and has 86,138 digits. The primes are: :

p = (29571282950 \times (2^ - 1) + 4) \times 2^ - 1

p+4 = (29571282950 \times (2^ - 1) + 4) \times 2^ + 3

If the first Hardy–Littlewood conjecture holds, then cousin primes have the same asymptotic density as

s. 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: :

B_4 = \left(\frac + \frac\right) + \left(\frac + \frac\right) + \left(\frac + \frac\right) + \cdots.

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

B_4 \approx 1.1970449

Marek Wolf (1996)
''On the Twin and Cousin Primes''
This constant should not be confused with Brun's constant for prime quadruplets, which is also denoted . The Skewes number for cousin primes is 5206837 ().

Notes

References

* * *. * {{Prime number classes Classes of prime numbers Unsolved problems in mathematics