Prime triplet
Innumber theory,aprime tripletis a set of threeprime numbersin which the smallest and largest of the three differ by 6. In particular, the sets must have the form(p,p+ 2,p+ 6)or(p,p+ 4,p+ 6).[1]With the exceptions of(2, 3, 5)and(3, 5, 7),this is the closest possiblegroupingof three prime numbers, since one of every three sequentialodd numbersis a multiple of three, and hence not prime (except for 3 itself).
Examples
[edit]The first prime triplets (sequenceA098420in theOEIS) 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)
Subpairs of primes
[edit]A prime triplet contains a single pair of:
- Twin primes:(p,p+ 2)or(p+ 4,p+ 6);
- Cousin primes:(p,p+ 4)or(p+ 2,p+ 6);and
- Sexy primes:(p,p+ 6).
Higher-order versions
[edit]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).When this happens, the five involved primes form aprime quintuplet.
Aprime 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).
Conjecture on prime triplets
[edit]Similarly to thetwin prime conjecture,it is conjectured that there are infinitely many prime triplets. The first knowngigantic primetriplet was found in 2008 by Norman Luhn and François Morain. The primes are(p,p+ 2,p+ 6)withp= 2072644824759 × 233333− 1.As of October 2020[update]the largest knownprovenprime triplet contains primes with 20008 digits, namely the primes(p,p+ 2,p+ 6)withp= 4111286921397 × 266420− 1.[2]
TheSkewes numberfor the triplet(p,p+ 2,p+ 6)is 87613571, and for the triplet(p,p+ 4,p+ 6)it is 337867.[3]
References
[edit]- ^Chris Caldwell.The Prime Glossary: prime triplefrom thePrime Pages.Retrieved on 2010-03-22.
- ^The Top Twenty: Tripletfrom the Prime Pages. Retrieved on 2013-05-06.
- ^Tóth, László (2019)."On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood"(PDF).Computational Methods in Science and Technology.25(3): 143–148.doi:10.12921/cmst.2019.0000033.Retrieved10 November2019.