Sexy prime

Sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because 11 - 5 = 6.

The term "sexy prime" is a pun stemming from the Latin word for six: sex.

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.

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 and 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 April 2019, the largest known pair of sexy primes was found by S. Batalov and has 31,002 digits. The primes are:

(p, p+6) = (187983281 × 251478 + 4)×(5 × 251478 - 1) + 2 ${\displaystyle \pm }$  3.[1]

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).

In May 2019, Peter Kaiser set a record for the largest known sexy prime triplet with 6031 digits:

p = 10409207693×220000−1.[2]

Gerd Lamprecht improved the record to 6116 digits in August 2019:

p = 20730011943×14221#+344231.[3]

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).

In November 2005 the largest known sexy prime quadruplet, found by Jens Kruse Andersen had 1002 digits:

p = 411784973 · 2347# + 3301.[4]

In September 2010 Ken Davis announced a 1004-digit quadruplet with p = 23333 + 1582534968299.[5]

In May 2019 Marek Hubal announced a 1138-digit quadruplet with p = 1567237911 × 2677# + 3301.[6] [7]

In June 2019 Peter Kaiser announced a 1534-digit quadruplet with p = 19299420002127 × 25050 + 17233.[8]

Sexy prime quintuplets

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