How many prime triplets are there?
Answers
Answer:
A prime triplet is a prime constellation of the form (p, p+2, p+6), (p, p+4, p+6), etc. Hardy and Wright (1979, p. 5) conjecture, and it seems almost certain to be true, that there are infinitely many prime triplets of the form (p, p+2, p+6) and (p, p+4, p+6).
triplet Sloane first member
(p, p+2, p+6) A022004 5, 11, 17, 41, 101, 107, ...
(p, p+2, p+8) A046134 3, 5, 11, 29, 59, 71, 101, ...
(p, p+2, p+12) A046135 5, 11, 17, 29, 41, 59, 71, ...
(p, p+4, p+6) A022005 7, 13, 37, 67, 97, 103, ...
(p, p+4, p+10) A046136 3, 7, 13, 19, 37, 43, 79, ...
(p, p+4, p+12) A046137 7, 19, 67, 97, 127, 229, ...
(p, p+6, p+8) A046138 5, 11, 23, 53, 101, 131, ...
(p, p+6, p+10) A046139 7, 13, 31, 37, 61, 73, 97, ...
(p, p+6, p+12) A023241 5, 7, 11, 17, 31, 41, 47, ...
(p, p+8, p+12) A046141 5, 11, 29, 59, 71, 89, 101, ...
As of Apr. 2019, the largest known prime triplet of the form (p,p+2,p+6) has smallest member
p=4111286921397·2^(66420)-1,
and each of its three members has 20008 decimal digits.
16 are primitive trimplets with hypotenuse less than 100.