Prime triplet

A prime number Drilling or prime triplet is a set of primes of the form p, p 2, p 6 or p, p 4, p 6. According to the definition contained in a prime drilling two prime numbers always a prime twin.

Note: The most obvious definition would be p, p 2, p 4, of which there are only (3, 5, 7 ), because of three consecutive odd numbers is always one through three is divisible.

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 ), ( 1091, 1093, 1097 ), ( 1277 1279, 1283), ( 1297, 1301, 1303), (1301, 1303, 1307 ), ( 1423, 1427, 1429), (1427, 1429, 1433 ), ( 1447, 1451, 1453), ( 1481, 1483, 1487 ), ( 1483, 1487, 1489 ), ( 1487, 1489, 1493 ), (1607, 1609, 1613 ), ( 1663, 1667, 1669), ( 1693, 1697, 1699), (1783, 1787, 1789 ), ( 1867, 1871, 1873), (1871, 1873, 1877), (1873, 1877, 1879), (1993, 1997, 1999 ), ( 1997, 1999, 2003)

Is a prime number contained in three prime triplets as 103 (97, 101, 103 ), ( 101, 103, 107) and (103, 107, 109), is referred to at the five prime numbers involved a prime quintuplet. A prime quadruplet is composed of two overlapping triplets primary ( P, P 2, 6 p, p 8)

It is believed that there are infinitely many prime triplets. Maynard and Tao showed in 2013 that there are infinitely many triples of primes whose elements are spaced apart by more than 400,000. Their proof uses results from the work of Zhang to twin primes. To prove the existence of infinitely many actual prime triplets, this limit should be reduced to 6.

2013, the biggest ever prime triplet was found with 16737 decimal places. It is 6521953289619 255555 × d with d = -5, -1, 1