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 ...

, 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 generally refers to the creation of one or more groups, or to the groups themselves. More specifically, grouping may refer to: * Shot grouping in shooting sports and other uses of firearms * the use of symbols of grouping in mathemati ...

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 divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...

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 or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime' ...

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 OE ...

s: or ; 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: .

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 contains two overlapping prime triplets, and .

Conjecture on prime triplets

Similarly to the

twin prime 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 or In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' ...

, it is conjectured that there are infinitely many prime triplets. The first known

gigantic prime A megaprime is a prime number with at least one million decimal digits. Other terms for large primes include "titanic prime", coined by Samuel Yates in the 1980s for a prime with at least 1000 digits (of which the smallest is 10999+7), and "gigan ...

triplet was found in 2008 by Norman Luhn and François Morain. The primes are with . the largest known proven 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