sexy primes

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.

The term "sexy prime" stems from the Latin word for six: sex.

Sexy prime pairs

The sexy primes 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)

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 :

(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).

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 :

(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).