Prime triplet

In mathematics, a prime triplet is a set of three prime numbers of the form (p, p + 2, p + 6) or (p, p + 4, p + 6). With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime.

Examples

The first prime triplets (sequence A098420 in OEIS) 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 pair of twin primes (p and p + 2, or p + 4 and p + 6), a pair of cousin primes (p and p + 4, or p + 2 and p + 6), and a pair of sexy primes (p and p + 6).

Higher-order versions

A prime can be a member of up to three prime triplets - for example, 103 is a member of (97, 101, 103), (101, 103, 107) and (103, 107, 109). When this happens, the five involved primes form a prime quintuplet.

A prime quadruplet (p, p + 2, p + 6, p + 8) contains two overlapping prime triplets, (p, p + 2, p + 6) and (p + 2, p + 6, p + 8).

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 (p, p + 2, p + 6) with p = 2072644824759 × 2³³³³³ − 1. As of May 2013 the largest known prime triplet contains primes with 16737 digits and was found by Peter Kaiser. The primes are (p, p + 4, p + 6) with p = 6521953289619 × 2⁵⁵⁵⁵⁵ − 5.

External links

• Weisstein, Eric W., "Prime Triplet", MathWorld.
•  A022004 in OEIS
•  A022005