Sexy prime
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In mathematics, a sexy prime is a pair (p, p + 6) of prime numbers that differ by six. The name "sexy prime" stems from the Latin word for six: sex. If p + 2 or p + 4 is also prime, then the sexy prime is part of a prime triplet.
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[edit] Examples
The sexy primes (sequences A023201 and A046117 in OEIS) below 500 are:
- (5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (31,37), (37,43), (41,47), (47,53), (53,59), (61,67), (67,73), (73,79), (83,89), (97,103), (101,107), (103,109), (107,113), (131,137), (151,157), (157,163), (167,173), (173,179), (191,197), (193,199), (223,229), (227,233), (233,239), (251,257), (263,269), (271,277), (277,283), (307,313), (311,317), (331,337), (347,353), (353,359), (367,373), (373,379), (383,389), (433,439), (443,449), (457,463), (461,467)
As of November 2005 the largest known sexy prime is (p, p+6) for
- p = (48011837012 · ((53238 · 7879#)2 - 1) + 2310) · 53238 · 7879#/385 + 1, where 7879# is a primorial
It has 10154 digits and was found by Torbjörn Alm, Micha Fleuren and Jens Kruse Andersen. [1]
[edit] Sexy prime triplets
Sexy primes can be extended to larger constellations. Triplets of primes (p, p + 6, p + 12) such that p + 18 is composite are called sexy prime triplets. Those below 1000 are (A046118, A046119, [[OEIS:A046120]|A046120]]]:
- (7,13,19), (17,23,29), (31,37,43), (47,53,59), (67,73,79), (97,103,109), (151,157,163), (167,173,179), (227,233,239), (257,263,269), (271,277,283), (347,353,359), (367,373,379), (557,563,569), (587,593,599), (607,613,619), (647,653,659), (727,733,739), (941,947,953), (971,977,983)
As of April 2006 the largest known sexy prime triplet is by Ken Davis and has 5132 digits: [2]
- p = (84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1) / 35 + 1
[edit] Sexy prime quadruplets
Sexy prime quadruplets (p, p + 6, p + 12, p + 18) can only begin with primes ending in a 1 in their decimal representation (except for the quadruplet with p = 5). The sexy prime quadruplets below 1000 are (A046121, A046122, A046123, A046124):
- (5,11,17,23), (11,17,23,29), (41,47,53,59), (61,67,73,79), (251,257,263,269), (601,607,613,619), (641,647,653,659)
As of November 2005 the largest known sexy prime quadruplet was found by Jens Kruse Andersen and has 1002 digits: [3]
- p = 411784973 · 2347# + 3301
Some term in an arithmetic progression of five terms or longer with common difference 6 is divisible by 5, so the only sexy prime quintuplet is (5,11,17,23,29). No longer sequence of sexy primes is possible.
[edit] See also
- Twin prime (two primes that differ by 2)
- Cousin prime (two primes that differ by 4)
[edit] References
- Eric W. Weisstein, Sexy Primes at MathWorld. Retrieved on February 28, 2007 (requires composite p+18 in a sexy prime triplet, but no other similar restrictions)
