Sexy prime

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]

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

Types of groupings[edit]

Sexy prime pairs[edit]

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^[update], the largest-known pair of sexy primes was found by S. Batalov and has 51,934 digits. The primes are:

p =

11922002779 × (2¹⁷²⁴⁸⁶ - 2⁸⁶²⁴³) + 2⁸⁶²⁴⁵ - 5

p +6 =

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

Sexy prime triplets[edit]

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 5,132 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[edit]

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 1,000 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[edit]

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[edit]

^ D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences. 1 (12). arXiv:1407.4897. doi:10.1186/s40687-014-0012-7. MR 3373710. S2CID 119699189.
^ Batalov, S. "Let's find some large sexy prime pair[s]". mersenneforum.org. Retrieved 3 October 2019.
^ Davis, Ken (January 2023). "sexy prime triplet". Prime Pages. Retrieved 24 January 2023.
^ Kaiser, Peter (May 2019). "sexy prime triplet". Mersenne forum. Retrieved 13 May 2019.
^ Andersen, Jens Kruse. "The largest known CPAP's". primerecords.dk. Retrieved 19 August 2019.
^ Davis, Ken. "Brillhart-Lehmer-Selfridge provable triplet Oct 2019". primenumbers yahoo group. Retrieved 2 October 2019.^{[dead link]}
^ Andersen, Jens Kruse. "The largest known CPAP's". primerecords.dk. Retrieved 13 October 2019.
^ Batalov, Serge. "Consecutive primes arithmetic progression (d=6), Apr 2022". Primes.utm.edu.
^ Andersen, Jens K. (November 2005). "Gigantic sexy and cousin primes". primenumbers yahoo group. Archived from the original on 29 May 2012. Retrieved 27 January 2009.
^ Davis, Ken (September 2010). "1004 sexy prime quadruplet". primenumbers yahoo group. Archived from the original on 30 May 2012. Retrieved 2 September 2010.
^ Hubal, Marek (May 2019). "CPAP's sexy prime". primenumbers yahoo group. Retrieved 10 May 2019.^{[dead link]}
^ Andersen, Jens Kruse (May 2019). "Re: CPAP's sexy prime". primenumbers yahoo group. Retrieved 19 September 2019.^{[dead link]}
^ Kaiser, Peter (June 2019). "Let's find some large sexy prime pair (and, perhaps, a triplet)". Mersenne forum. Retrieved 18 August 2019.
^ Lamprecht, Gerd; Luhn, Norman (October 2019). "CPAP's sexy prime". primenumbers yahoo group. Retrieved 13 October 2019.^{[dead link]}
^ "PrimePage Primes: (1021328211729 · 2521# · (483 · 2521# + 1) + 2310) · (483 · 2521# - 1)/210 + 1".

Weisstein, Eric W. "Sexy Primes". MathWorld. Retrieved on 2007-02-28 (requires composite $p$ +18 in a sexy prime triplet, but no other similar restrictions)

External links[edit]

Grime, James. Brady Haran (ed.). "Sexy Primes (and the only sexy prime quintuplet)". Numberphile. Archived from the original on 23 October 2018.

[1] D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences. 1 (12). arXiv:1407.4897. doi:10.1186/s40687-014-0012-7. MR 3373710. S2CID 119699189.

[2] Batalov, S. "Let's find some large sexy prime pair[s]". mersenneforum.org. Retrieved 3 October 2019.

[3] Davis, Ken (January 2023). "sexy prime triplet". Prime Pages. Retrieved 24 January 2023.

[4] Kaiser, Peter (May 2019). "sexy prime triplet". Mersenne forum. Retrieved 13 May 2019.

[5] Andersen, Jens Kruse. "The largest known CPAP's". primerecords.dk. Retrieved 19 August 2019.

[6] Davis, Ken. "Brillhart-Lehmer-Selfridge provable triplet Oct 2019". primenumbers yahoo group. Retrieved 2 October 2019.^{[dead link]}

[7] Andersen, Jens Kruse. "The largest known CPAP's". primerecords.dk. Retrieved 13 October 2019.

[8] Batalov, Serge. "Consecutive primes arithmetic progression (d=6), Apr 2022". Primes.utm.edu.

[sexycousin-9] Andersen, Jens K. (November 2005). "Gigantic sexy and cousin primes". primenumbers yahoo group. Archived from the original on 29 May 2012. Retrieved 27 January 2009.

[10] Davis, Ken (September 2010). "1004 sexy prime quadruplet". primenumbers yahoo group. Archived from the original on 30 May 2012. Retrieved 2 September 2010.

[11] Hubal, Marek (May 2019). "CPAP's sexy prime". primenumbers yahoo group. Retrieved 10 May 2019.^{[dead link]}

[12] Andersen, Jens Kruse (May 2019). "Re: CPAP's sexy prime". primenumbers yahoo group. Retrieved 19 September 2019.^{[dead link]}

[13] Kaiser, Peter (June 2019). "Let's find some large sexy prime pair (and, perhaps, a triplet)". Mersenne forum. Retrieved 18 August 2019.

[14] Lamprecht, Gerd; Luhn, Norman (October 2019). "CPAP's sexy prime". primenumbers yahoo group. Retrieved 13 October 2019.^{[dead link]}

[15] "PrimePage Primes: (1021328211729 · 2521# · (483 · 2521# + 1) + 2310) · (483 · 2521# - 1)/210 + 1".

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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 ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
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Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) 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/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
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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