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 (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)

Like twin primes, 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; the sexy prime triplets (sequences A046118, A046119 and A046120 in OEIS) below 1000 are:

(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 (sequences A046121, A046122, A046123 and A046124 in OEIS) 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)

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).

External links

• MathWorld: Sexy Primes
• Sexy Primes