# 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 October 2019, the largest-known pair of sexy primes was found by P. Kaiser and has 50,539 digits. The primes are:

p = (520461 × 255931+1) × (98569639289 × (520461 × 255931-1)2-3)-1
p+6 = (520461 × 255931+1) × (98569639289 × (520461 × 255931-1)2-3)+5. [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 1,000 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 6,031 digits:

p = 10409207693×220000−1.[2]

Gerd Lamprecht improved the record to 6,116 digits in August 2019:

p = 20730011943×14221#+344231.[3]

Ken Davis further improved the record with a 6,180 digit Brillhart-Lehmer-Selfridge provable triplet in Oct 2019:

p = (72865897*809857*4801#*(809857*4801#+1)+210)*(809857*4801#-1)/35+1 [4]

Norman Luhn & Gerd Lamprecht improved the record to 6,701 digits in Oct 2019:

p = 22582235875×222224+1.[5]

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 1,002 digits:

p = 411784973 · 2347# + 3301.[6]

In September 2010 Ken Davis announced a 1,004-digit quadruplet with p = 23333 + 1582534968299.[7]

In May 2019 Marek Hubal announced a 1,138-digit quadruplet with p = 1567237911 × 2677# + 3301.[8] [9]

In June 2019 Peter Kaiser announced a 1,534-digit quadruplet with p = 19299420002127 × 25050 + 17233.[10]

In October 2019 Gerd Lamprecht and Norman Luhn announced a 3,025-digit quadruplet with p = 121152729080 × 7019#/1729 + 1.[11]

### 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.

## References

1. ^ S. Batalov, "Let's find some large sexy prime pair". mersenneforum.org. Retrieved 2019-10-03.
2. ^ Mersenneforum, [1]. Retrieved 2019-05-13.
3. ^ Andersen, Jens Kruse. "The Largest Known CPAP's". primerecords.dk. Retrieved 2019-08-19.
4. ^ primenumbersyahoogroup, [2]. Retrieved 2019-10-02.
5. ^ Andersen, Jens Kruse. "The Largest Known CPAP's". primerecords.dk. Retrieved 2019-10-13.
6. ^ Jens K. Andersen, "Gigantic sexy and cousin primes". Retrieved 2009-01-27.
7. ^ Ken Davis, "1004 sexy prime quadruplet". Retrieved 2010-09-02.
8. ^ Marek Hubal, "CPAP's sexy prime". Retrieved 2019-05-10.
9. ^ Jens Kruse Andersen, "Re: CPAP's sexy prime". Retrieved 2019-09-19.
10. ^ Kaiser, Peter. "Let's find some large sexy prime pair (and, perhaps, a triplet)". mersenneforum.org. Retrieved 18 August 2019.
11. ^ "CPAP's sexy prime". Retrieved 2019-13-10.
• Retrieved on 2007-02-28 (requires composite p+18 in a sexy prime triplet, but no other similar restrictions)