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Prime Triplet


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).

tripletSloanefirst member
(p, p+2, p+6)A0220045, 11, 17, 41, 101, 107, ...
(p, p+2, p+8)A0461343, 5, 11, 29, 59, 71, 101, ...
(p, p+2, p+12)A0461355, 11, 17, 29, 41, 59, 71, ...
(p, p+4, p+6)A0220057, 13, 37, 67, 97, 103, ...
(p, p+4, p+10)A0461363, 7, 13, 19, 37, 43, 79, ...
(p, p+4, p+12)A0461377, 19, 67, 97, 127, 229, ...
(p, p+6, p+8)A0461385, 11, 23, 53, 101, 131, ...
(p, p+6, p+10)A0461397, 13, 31, 37, 61, 73, 97, ...
(p, p+6, p+12)A0232415, 7, 11, 17, 31, 41, 47, ...
(p, p+8, p+12)A0461415, 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.


See also

Integer Sequence Primes, Prime Constellation, Prime Quadruplet, Twin Primes

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References

Forbes, T. "Prime k-Tuplets." http://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." http://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."

Referenced on Wolfram|Alpha

Prime Triplet

Cite this as:

Weisstein, Eric W. "Prime Triplet." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimeTriplet.html

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