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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

, a prime triplet is a set of three

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...

s 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 Grouping may refer to: * Muenchian grouping * Principles of grouping * Railways Act 1921, also known as Grouping Act, a reorganisation of the British railway system * Grouping (firearms), the pattern of multiple shots from a sidearm See also ...

of three prime numbers, since one of every three sequential

odd number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...

s is a multiple of three, and hence not prime (except for 3 itself).

Examples

The first prime triplets are (5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647), (821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877, 881, 883), (881, 883, 887)

Subpairs of primes

A prime triplet contains a single pair of: *

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 p ...

s: or ; *

Cousin prime In number theory, cousin primes are prime numbers that differ by four. 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 cousin primes (sequences and in ...

s: or ; 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 ...

s: .

Higher-order versions

A prime can be a member of up to three prime triplets - for example, 103 is a member of , and . When this happens, the five involved primes form a prime quintuplet. A

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 ...

contains two overlapping prime triplets, and .

Conjecture on prime triplets

Similarly to the twin prime conjecture, it is conjectured that there are infinitely many prime triplets. The first known gigantic prime triplet was found in 2008 by Norman Luhn and François Morain. The primes are with . the largest known

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 Churc ...

prime triplet contains primes with 20008 digits, namely the primes with . The Skewes number for the triplet is 87613571, and for the triplet it is 337867.

References

External links

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