Sexy prime

In mathematics, a sexy prime is a pair ("p", "p" + 6) of prime numbers that differ by six. For example, the numbers 5 and 11 are both prime numbers which together have a difference of 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".

Types of groupings

Sexy prime pairs

The sexy primes (sequences OEIS2C|id=A023201 and OEIS2C|id=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), (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).

As of November 2005 the largest known sexy prime, found by Jens Kruse Andersen had 10154 digits. The number was ("p", "p"+6) for:"p" = (48011837012 · ((53238 · 7879#)² - 1) + 2310) · 53238 · 7879#/385 + 1, where 7879# is a primorial. [http://groups.yahoo.com/group/primeform/message/6637 - Retrieved: November 2005.]

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 (OEIS2C|id=A046118, OEIS2C|id=A046119, OEIS2C|id=A046120):

:(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).

As of April 2006 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. [http://hjem.get2net.dk/jka/math/cpap.htm#k3 - Retrieved: April 2006]

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 (OEIS2C|id=A023271, OEIS2C|id=A046122, OEIS2C|id=A046123, OEIS2C|id=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).

As of November 2005 the largest known sexy prime quadruplet, found by Jens Kruse Andersen had 1002 digits: : "p" = 411784973 · 2347# + 3301. [http://groups.yahoo.com/group/primeform/message/6637 - Retrieved: November 2005]

Sexy prime quintuplets

In an arithmetic progression of five terms with common difference 6, 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.

See also

*Twin prime (two primes that differ by 2)
*Cousin prime (two primes that differ by 4)

References

* Retrieved on 2007-02-28 (requires composite "p"+18 in a sexy prime triplet, but no other similar restrictions)