Prime triplet

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). [Chris Caldwell. [http://primes.utm.edu/glossary/page.php?sort=Triplet The Prime Glossary: triplet] from the Prime Pages. Retrieved on 2007-06-15.] With the trivial exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since every third odd number greater than 3 is divisible by 3, and hence not prime.

The first prime triplets OEIS|id=A098420 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)

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).

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).

Similarly to the twin prime conjecture, it is conjectured that there are infinitely many prime triplets. As of 2008 the largest known prime triplet contains primes with 6223 digits. [ [http://primes.utm.edu/primes/lists/short.txt "The Largest Known Primes"; Samuel Yates and Chris Caldwell] . Retrieved on 2008-01-04.] It was found by Norman Luhn and has the form ("p", "p"+2, "p"+6) with "p" = 5612052289×14489#/5−1 where 14489# is the primorial 2×3×5×...×14489.

References


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