number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, 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 (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...

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 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and . The term "sexy prime" is a pun stemming from the Latin word for six: . If o ...

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 the ...

) 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 Proven is a rural village in the Belgian province of West Flanders, and a "deelgemeente" of the municipality Poperinge. The village has about 1400 inhabitants. The church and parish of Proven are named after Saint Victor. The Saint Victor Chur ...

cousin prime pair is for :

p = 4111286921397 \times 2^ + 1

which has 20008 digits. In fact, this is part of a

prime triple In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or . With the exceptions of and , this is the closest possible grouping of ...

since is also a

(because is also a proven prime). , the largest-known pair of cousin primes was found by S. Batalov and has 51,934 digits. The primes are: :

p = 29055814795 \times (2^ - 2^) + 2^ - 3

p+4 = 29055814795 \times (2^ - 2^) + 2^ + 1

It follows from the first

Hardy–Littlewood conjecture 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 (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...

that cousin primes have the same asymptotic density as

s. An analogue of

Brun's constant In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by ''B''2 . Brun's theorem was proved by V ...

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 quadruplet In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. Prim ...

s, 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