Sexy prime

Short description: Prime numbers which differ by 6

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 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. 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.^[1]

Primorial 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 OEIS: A023201 and OEIS: A046117 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 2022), the largest-known pair of sexy primes was found by S. Batalov and has 51,934 digits. The primes are:

p =

11922002779 x (2¹⁷²⁴⁸⁶ - 2⁸⁶²⁴³) + 2⁸⁶²⁴⁵ - 5

p +6 =

11922002779 x (2¹⁷²⁴⁸⁶ - 2⁸⁶²⁴³) + 2⁸⁶²⁴⁵ + 1^[2]

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 (OEIS: A046118, OEIS: A046119, OEIS: A046120):

(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 January 2005 Ken Davis set a record for the largest-known sexy prime triplet with 5132 digits:

p =

(84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1) / 35 + 1. ^[3]

In May 2019, Peter Kaiser improved this record to 6,031 digits:

p =

10409207693×2²⁰⁰⁰⁰−1.^[4]

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

p =

20730011943×14221#+344231.^[5]

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

p =

(72865897*809857*4801#*(809857*4801#+1)+210)*(809857*4801#-1)/35+1^[6]

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

p =

22582235875×2²²²²⁴+1.^[7]

Serge Batalov improved the record to 15,004 digits in April 2022:

p =

2494779036241x2⁴⁹⁸⁰⁰+1.^[8]

Sexy prime quadruplets

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 (OEIS: A023271, OEIS: A046122, OEIS: A046123, OEIS: A046124):

(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.^[9]

In September 2010 Ken Davis announced a 1,004 digit quadruplet with $p =$ 2³³³³ + 1582534968299.^[10]

In May 2019 Marek Hubal announced a 1,138 digit quadruplet with $p =$ 1567237911 × 2677# + 3301.^[11]^[12]

In June 2019 Peter Kaiser announced a 1,534 digit quadruplet with $p =$ 19299420002127 × 2⁵⁰⁵⁰ + 17233.^[13]

In October 2019 Gerd Lamprecht and Norman Luhn announced a 3,025 digit quadruplet with $p =$ 121152729080 × 7019#/1729 + 1.^[14]

In July 2023 Ken Davis announced a 3,207 digit quadruplet with $p =$ (1021328211729*2521#*(483*2521#+1)+2310)*(483*2521#-1)/210+1.^[15]

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. However, all multiples of 5 (except itself) cannot be prime numbers. Thus, the only sexy prime quintuplet is (5,11,17,23,29); no longer sequence of sexy primes is possible, since adding 6 to the last number in the set of sexy prime quintuplets (29) equals 35, which is a composite number.

References

↑ D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences 1 (12). doi:10.1186/s40687-014-0012-7.
↑ Batalov, S.. "Let's find some large sexy prime pair[s"]. https://www.mersenneforum.org/showpost.php?p=527198&postcount=37.
↑ Davis, Ken (January 2023). "sexy prime triplet". https://primes.utm.edu/primes/page.php?id=77460.
↑ Kaiser, Peter (May 2019). "sexy prime triplet". https://mersenneforum.org/showpost.php?p=516604&postcount=31.
↑ Andersen, Jens Kruse. "The largest known CPAP's". http://primerecords.dk/cpap.htm.
↑ Davis, Ken. "Brillhart-Lehmer-Selfridge provable triplet Oct 2019". https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/26291.
↑ Andersen, Jens Kruse. "The largest known CPAP's". http://primerecords.dk/cpap.htm#sexy.
↑ Batalov, Serge. "Consecutive primes arithmetic progression (d=6), Apr 2022". https://primes.utm.edu/primes/page.php?id=133846.
↑ Andersen, Jens K. (November 2005). "Gigantic sexy and cousin primes". https://groups.yahoo.com/group/primeform/message/6637.
↑ Davis, Ken (September 2010). "1004 sexy prime quadruplet". http://tech.groups.yahoo.com/group/primenumbers/message/21783.
↑ Hubal, Marek (May 2019). "CPAP's sexy prime". https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/26239. ^{[|permanent dead link|dead link}}]}
↑ Andersen, Jens Kruse (May 2019). "Re: CPAP's sexy prime". https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/26240. ^{[|permanent dead link|dead link}}]}
↑ Kaiser, Peter (June 2019). "Let's find some large sexy prime pair (and, perhaps, a triplet)". https://www.mersenneforum.org/showpost.php?p=520001&postcount=34.
↑ Lamprecht, Gerd; Luhn, Norman (October 2019). "CPAP's sexy prime". https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/26239. ^{[|permanent dead link|dead link}}]}
↑ "PrimePage Primes: (1021328211729 · 2521# · (483 · 2521# + 1) + 2310) · (483 · 2521# - 1)/210 + 1". https://t5k.org/primes/page.php?id=136218.

Weisstein, Eric W.. "Sexy Primes". http://mathworld.wolfram.com/SexyPrimes.html. Retrieved on 2007-02-28 (requires composite $p$ +18 in a sexy prime triplet, but no other similar restrictions)

External links

Grime, James. "Sexy Primes (and the only sexy prime quintuplet)". in Brady Haran. http://www.numberphile.com/videos/sexy_primes.html.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Sexy prime. Read more

[1] D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences 1 (12). doi:10.1186/s40687-014-0012-7.

[2] Batalov, S.. "Let's find some large sexy prime pair[s"]. https://www.mersenneforum.org/showpost.php?p=527198&postcount=37.

[3] Davis, Ken (January 2023). "sexy prime triplet". https://primes.utm.edu/primes/page.php?id=77460.

[4] Kaiser, Peter (May 2019). "sexy prime triplet". https://mersenneforum.org/showpost.php?p=516604&postcount=31.

[5] Andersen, Jens Kruse. "The largest known CPAP's". http://primerecords.dk/cpap.htm.

[6] Davis, Ken. "Brillhart-Lehmer-Selfridge provable triplet Oct 2019". https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/26291.

[7] Andersen, Jens Kruse. "The largest known CPAP's". http://primerecords.dk/cpap.htm#sexy.

[8] Batalov, Serge. "Consecutive primes arithmetic progression (d=6), Apr 2022". https://primes.utm.edu/primes/page.php?id=133846.

[sexycousin-9] Andersen, Jens K. (November 2005). "Gigantic sexy and cousin primes". https://groups.yahoo.com/group/primeform/message/6637.

[10] Davis, Ken (September 2010). "1004 sexy prime quadruplet". http://tech.groups.yahoo.com/group/primenumbers/message/21783.

[11] Hubal, Marek (May 2019). "CPAP's sexy prime". https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/26239. ^{[|permanent dead link|dead link}}]}

[12] Andersen, Jens Kruse (May 2019). "Re: CPAP's sexy prime". https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/26240. ^{[|permanent dead link|dead link}}]}

[13] Kaiser, Peter (June 2019). "Let's find some large sexy prime pair (and, perhaps, a triplet)". https://www.mersenneforum.org/showpost.php?p=520001&postcount=34.

[14] Lamprecht, Gerd; Luhn, Norman (October 2019). "CPAP's sexy prime". https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/26239. ^{[|permanent dead link|dead link}}]}

[15] "PrimePage Primes: (1021328211729 · 2521# · (483 · 2521# + 1) + 2310) · (483 · 2521# - 1)/210 + 1". https://t5k.org/primes/page.php?id=136218.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Safe ( $p, (p - 1)/2$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers

Anonymous

Search

Sexy prime

Namespaces

More

Page actions

Contents

Primorial n# notation

Types of groupings

Sexy prime pairs

Sexy prime triplets

Sexy prime quadruplets

Sexy prime quintuplets

See also

References

External links

Navigation

Navigation

Help

Translate

Wiki tools

Wiki tools

Anonymous

Search

Sexy prime

Primorial n# notation

Types of groupings

Sexy prime pairs

Sexy prime triplets

Sexy prime quadruplets

Sexy prime quintuplets

See also

References

External links

Navigation

Wiki tools

Page tools

Other projects

Categories