Prime triplet
In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6)Script error: No such module "Check for unknown parameters". or (p, p + 4, p + 6)Script error: No such module "Check for unknown parameters"..[1] With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself).
Examples
The first prime triplets (sequence A098420 in the OEIS) 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
A prime triplet contains a single pair of:
- Twin primes: (p, p + 2)Script error: No such module "Check for unknown parameters". or (p + 4, p + 6)Script error: No such module "Check for unknown parameters".;
- Cousin primes: (p, p + 4)Script error: No such module "Check for unknown parameters". or (p + 2, p + 6)Script error: No such module "Check for unknown parameters".; and
- Sexy primes: (p, p + 6)Script error: No such module "Check for unknown parameters"..
Higher-order versions
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 a prime quintuplet.
A prime quadruplet (p, p + 2, p + 6, p + 8)Script error: No such module "Check for unknown parameters". contains two overlapping prime triplets, (p, p + 2, p + 6)Script error: No such module "Check for unknown parameters". and (p + 2, p + 6, p + 8)Script error: No such module "Check for unknown parameters"..
Conjecture on prime triplets
Similarly to the twin prime conjecture, it is conjectured that there are infinitely many prime triplets. The first known gigantic prime triplet was found in 2008 by Norman Luhn and François Morain. The primes are (p, p + 2, p + 6)Script error: No such module "Check for unknown parameters". with Template:Mvar = 2072644824759 × 233333 − 1. As of October 2020[update]Script error: No such module "Check for unknown parameters". the largest known proven prime triplet contains primes with 20008 digits, namely the primes (p, p + 2, p + 6)Script error: No such module "Check for unknown parameters". with Template:Mvar = 4111286921397 × 266420 − 1.[2]
The Skewes number for the triplet (p, p + 2, p + 6)Script error: No such module "Check for unknown parameters". is 87613571, and for the triplet (p, p + 4, p + 6)Script error: No such module "Check for unknown parameters". it is 337867.[3]
References
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- ↑ Chris Caldwell. The Prime Glossary: prime triple from the Prime Pages. Retrieved on 2010-03-22.
- ↑ The Top Twenty: Triplet from the Prime Pages. Retrieved on 2013-05-06.
- ↑ Script error: No such module "Citation/CS1".
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External links
- Script error: No such module "Template wrapper".
- Template:OEIS el
- Template:OEIS el