# Sexy

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 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.

## n# notation

As used in this article, n# stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ n.

## Types of groupings

### Sexy prime pairs

The sexy primes (sequences Template:OEIS2C and Template:OEIS2C 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).

Template:As of the largest known sexy prime was found by Ken Davis and has 11593 digits. The primes are (p, p+6) for

p = (117924851 × 587502 × 9001# × (587502 × 9001# + 1) + 210) × (587502 × 9001# − 1)/35 + 5.[1]

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 (Template:OEIS2C, Template:OEIS2C, Template:OEIS2C):

(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).

Template:As of 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.[2]

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 (Template:OEIS2C, Template:OEIS2C, Template:OEIS2C, Template:OEIS2C):

(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.[3]

In September 2010 Ken Davis announced a 1004-digit quadruplet with p = 23333 + 1582534968299.[4]

### Sexy prime quintuplets

In an arithmetic progression of five terms with common difference 6, because 6>5 and the two numbers are relatively prime, one of the terms must be divisible by 5. Thus, the only sexy prime quintuplet is (5,11,17,23,29) with no longer sequence of sexy primes possible.

## References

1. Ken Davis, "11593 digit sexy prime pair". Retrieved 2009-05-06.
2. Jens K. Andersen, "The largest known CPAP-3". Retrieved 2009-01-27.
3. Jens K. Andersen, "Gigantic sexy and cousin primes". Retrieved 2009-01-27.
4. Ken Davis, "1004 sexy prime quadruplet". Retrieved 2010-09-02.
• Template:Mathworld Retrieved on 2007-02-28 (requires composite p+18 in a sexy prime triplet, but no other similar restrictions)