Prime Triplet

A prime triplet is a prime constellation of the form (, , ), (, , ), 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 (, , ) and (, , ).

triplet	Sloane	first member
(, , )	A022004	5, 11, 17, 41, 101, 107, ...
(, , )	A046134	3, 5, 11, 29, 59, 71, 101, ...
(, , )	A046135	5, 11, 17, 29, 41, 59, 71, ...
(, , )	A022005	7, 13, 37, 67, 97, 103, ...
(, , )	A046136	3, 7, 13, 19, 37, 43, 79, ...
(, , )	A046137	7, 19, 67, 97, 127, 229, ...
(, , )	A046138	5, 11, 23, 53, 101, 131, ...
(, , )	A046139	7, 13, 31, 37, 61, 73, 97, ...
(, , )	A023241	5, 7, 11, 17, 31, 41, 47, ...
(, , )	A046141	5, 11, 29, 59, 71, 89, 101, ...

As of Apr. 2019, the largest known prime triplet of the form  has smallest member

and each of its three members has  decimal digits.

REFERENCES:

Forbes, T. "Prime -Tuplets." https://anthony.d.forbes.googlepages.com/ktuplets.htm.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Rivera, C. "Problems & Puzzles: Puzzle 034-Prime Triplets in Arithmetic Progression." https://www.primepuzzles.net/puzzles/puzz_034.htm.

Sloane, N. J. A. Sequences A022004, A022005, A023241, A046134, A046135, A046136, A046137, A046138, A046139, and A046141in "The On-Line Encyclopedia of Integer Sequences."