In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and . The term "sexy prime" is a pun stemming from the Latin word for six: . If or (where is the lower prime) is also prime, then the sexy prime is part of a prime triplet. In August 2014 the Polymath group, seeking the proof of the twin prime conjecture, showed that if the generalized Elliott–Halberstam conjecture is proven, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6 and as such they are either twin, cousin or sexy primes.

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 (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). , the largest-known pair of sexy primes was found by S. Batalov and has 51,934 digits. The primes are: : 11922002779 x (2¹⁷²⁴⁸⁶ - 2⁸⁶²⁴³) + 2⁸⁶²⁴⁵ - 5 : 11922002779 x (2¹⁷²⁴⁸⁶ - 2⁸⁶²⁴³) + 2⁸⁶²⁴⁵ + 1

Sexy prime triplets

Sexy primes can be extended to larger constellations. Triplets of primes (, +6, +12) such that +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: : 10409207693×2²⁰⁰⁰⁰−1. Gerd Lamprecht improved the record to 6,116 digits in August 2019: : 20730011943×14221#+344231. Ken Davis further improved the record with a 6,180 digit Brillhart-Lehmer-Selfridge provable triplet in October 2019: : (72865897*809857*4801#*(809857*4801#+1)+210)*(809857*4801#-1)/35+1 Norman Luhn & Gerd Lamprecht improved the record to 6,701 digits in October 2019: : 22582235875×2²²²²⁴+1. Serge Batalov improved the record to 15,004 digits in April 2022: : 2494779036241x2⁴⁹⁸⁰⁰+1.

Sexy prime quadruplets

Sexy prime quadruplets (, +6, +12, +18) can only begin with primes ending in a 1 in their

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

representation (except for the quadruplet with 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: : 411784973 · 2347# + 3301. 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 (5,11,17,23,29); no longer sequence of sexy primes is possible.

References

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

External links

* {{Prime number classes Classes of prime numbers Unsolved problems in number theory