# 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 (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
It has 10154 digits and was found by Torbjörn Alm, Micha Fleuren and Jens Kruse Andersen [1]. 7879# is a primorial.

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)

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

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)

As of November 2005 the largest known sexy prime quadruplet (p, p+6, p+12, p+18) is by Jens Kruse Andersen [3] and has 1002 digits:
p = 411784973 · 2347# + 3301

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), and no larger sequences of sexy primes are possible.