Prime triplet
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In mathematics, a prime triplet is a set of three prime numbers on the form (p, p+2, p+6) or (p, p+4, p+6). 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 (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)
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), .
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 2006 the largest known prime triplet contains primes with 5132 digits.[1] It was found by Ken Davis and has the form (p, p+4, p+6) with
p = (99241437759×205881×4001# × (205881×4001# + 1) + 210) × (205881×4001# − 1) / 35 + 1
where 4001# is the primorial 2×3×5×...×4001.
