Sexy prime
In mathematics, a sexy prime is a pair of prime numbers that differ by six; compare this with twin primes, pairs of prime numbers that differ by two, and cousin primes, pairs of prime numbers that differ by four. The name "sexy prime" stems from the Latin word for six, sex.
The sexy primes below 500 are (also see Sloane's A023201 and A046117):
- (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)
Like twin primes, sexy primes can be extended to larger constellations. Triplets of primes (p, p + 6, p + 12) such that p + 18 are called sexy prime triplets; the sexy prime triplets below 1000 are (also see Sloane's A046118, A046119 and 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)
Sexy prime quadruplets can only begin with primes ending in a 1 in their decimal representation (apart from 5); the sexy prime quadruplets below 1000 are (also see Sloane's A046121, A046122, A046123 and 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)
Since every fifth number of the form 6n ± 1 is divisible by 5, only one sexy prime quintuplet exists, namely, (5,11,17,23,29).
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