number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

, sexy primes are

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s 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 A pun, also known as paronomasia, is a form of word play that exploits multiple meanings of a term, or of similar-sounding words, for an intended humorous or rhetorical effect. These ambiguities can arise from the intentional use of homophoni ...

stemming from the

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

word for six: . If or (where is the lower prime) is also prime, then the sexy prime is part of a

prime triplet In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or . With the exceptions of and , this is the closest possible grouping of t ...

. In August 2014 the

Polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...

group, seeking the proof of the

twin prime conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...

, showed that if the

generalized Elliott–Halberstam conjecture A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...

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 Twins are two offspring produced by the same pregnancy.MedicineNet > Definition of TwinLast Editorial Review: 19 June 2000 Twins can be either ''monozygotic'' ('identical'), meaning that they develop from one zygote, which splits and forms two em ...

cousin Most generally, in the lineal kinship system used in the English-speaking world, a cousin is a type of familial relationship in which two relatives are two or more familial generations away from their most recent common ancestor. Commonly, ...

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 The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...

) 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 Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...

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 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 An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...

of five terms with common difference 6, one of the terms must be divisible by 5, because 5 and 6 are

relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

. 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