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.
The term "sexy prime" is a pun stemming from the Latin word for six: sex.
If + 2 or + 4 is also prime, then the sexy prime is part of a prime triplet.

Primorial ''n''# notation

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

Types of groupings

Sexy prime pairs

The sexy primes below 500 are:
, the largest-known pair of sexy primes was found by P. Kaiser and has 50,539 digits. The primes are:

Sexy prime triplets

Sexy primes can be extended to larger constellations. Triplets of primes such that +18 is composite are called sexy prime triplets. Those below 1,000 are :
In May 2019, Peter Kaiser set a record for the largest-known sexy prime triplet with 6,031 digits:
Gerd Lamprecht improved the record to 6,116 digits in August 2019:
Ken Davis further improved the record with a 6,180 digit Brillhart-Lehmer-Selfridge provable triplet in October 2019:
Norman Luhn & Gerd Lamprecht improved the record to 6,701 digits in October 2019:
Gerd Lamprecht & Norman Luhn improved the record to 10,602 digits in December 2019:

Sexy prime quadruplets

Sexy prime quadruplets can only begin with primes ending in a 1 in their decimal representation. The sexy prime quadruplets below 1000 are :
In November 2005 the largest-known sexy prime quadruplet, found by Jens Kruse Andersen had 1,002 digits:
In September 2010 Ken Davis announced a 1,004 digit quadruplet with 23333 + 1582534968299.
In May 2019 Marek Hubal announced a 1,138 digit quadruplet with 1567237911 × 2677# + 3301.
In June 2019 Peter Kaiser announced a 1,534 digit quadruplet with 19299420002127 × 25050 + 17233.
In October 2019 Gerd Lamprecht and Norman Luhn announced a 3,025 digit quadruplet with 121152729080 × 7019#/1729 + 1.

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 ; no longer sequence of sexy primes is possible.