In mathematics, sexy primes are prime numbers that differ from each other by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6. If p + 2 or p + 4 (where p is the lower prime) is also prime, then the sexy prime is part of a prime triplet.
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The term "sexy prime" stems from the Latin word for six: sex.
n# notation
As used in this article, n# stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ n.
Sexy prime pairs
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), (257,263), (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 May 2009 the largest known sexy prime was found by Ken Davis and has 11,593 digits. The primes are (p, p+6) for
p = (117924851 × 587502 × 9001# × (587502 × 9001# + 1) + 210) × (587502 × 9001# − 1)/35 + 5.9001# = 2×3×5×...×9001 is a primorial, i.e., the product of primes ≤ 9001.
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, A046120):
(5,11,17), (7,13,19), (17,23,29), (31,37,43), (47,53,59), (67,73,79), (97,103,109), (101,107,113), (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 2013 the largest known sexy prime triplet, found by Ken Davis had 5132 digits:
p = (84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1) / 35 + 1.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 ( A023271, 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).In November 2005 the largest known sexy prime quadruplet, found by Jens Kruse Andersen had 1002 digits:
p = 411784973 · 2347# + 3301.In September 2010 Ken Davis announced a 1004-digit quadruplet with p = 23333 + 1582534968299.
Sexy prime quintuplets
In an arithmetic progression of five terms with common difference 6, one of the terms must be divisible by 5, because the two numbers are relatively prime. Thus, the only sexy prime quintuplet is (5,11,17,23,29); no longer sequence of sexy primes is possible.