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Date: 24-7-2020
900
Date: 5-2-2016
1513
Date: 29-11-2020
1010
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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."
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