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 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, 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
Further information: Primorial § Definition for natural numbers

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 October 2019, the largest-known pair of sexy primes was found by P. Kaiser and has 50,539 digits. The primes are:

p = (520461 × 255931+1) × (98569639289 × (520461 × 255931-1)2-3)-1
p+6 = (520461 × 255931+1) × (98569639289 × (520461 × 255931-1)2-3)+5[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 May 2019, Peter Kaiser set a record for the largest-known sexy prime triplet with 6,031 digits:

p = 10409207693×220000−1.[3]

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

p = 20730011943×14221#+344231.[4]

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[5]

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

p = 22582235875×222224+1.[6]

Gerd Lamprecht & Norman Luhn improved the record to 10,602 digits in December 2019:

p = 2683143625525x235176+1.[7]

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.[8]

In September 2010 Ken Davis announced a 1,004 digit quadruplet with p = 23333 + 1582534968299.[9]

In May 2019 Marek Hubal announced a 1,138 digit quadruplet with p = 1567237911 × 2677# + 3301.[10][11]

In June 2019 Peter Kaiser announced a 1,534 digit quadruplet with p = 19299420002127 × 25050 + 17233.[12]

In October 2019 Gerd Lamprecht and Norman Luhn announced a 3,025 digit quadruplet with p = 121152729080 × 7019#/1729 + 1.[13]

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.

See also

References
  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.
  2. Batalov, S. "Let's find some large sexy prime pair[s]". mersenneforum.org. Retrieved 3 October 2019.
  3. Kaiser, Peter (May 2019). "sexy prime triplet". Mersenne forum. Retrieved 13 May 2019.
  4. Andersen, Jens Kruse. "The largest known CPAP's". primerecords.dk. Retrieved 19 August 2019.
  5. Davis, Ken. "Brillhart-Lehmer-Selfridge provable triplet Oct 2019". primenumbers yahoo group. Retrieved 2 October 2019.
  6. Andersen, Jens Kruse. "The largest known CPAP's". primerecords.dk. Retrieved 13 October 2019.
  7. Lamprecht, Gerd; Luhn, Norman. "Gerd Lamprecht & Norman Luhn, Dec 2019". Mersenne forum.
  8. Andersen, Jens K. (November 2005). "Gigantic sexy and cousin primes". primenumbers yahoo group. Retrieved 27 January 2009.
  9. Davis, Ken (September 2010). "1004 sexy prime quadruplet". primenumbers yahoo group. Retrieved 2 September 2010.
  10. Hubal, Marek (May 2019). "CPAP's sexy prime". primenumbers yahoo group. Retrieved 10 May 2019.
  11. Andersen, Jens Kruse (May 2019). "Re: CPAP's sexy prime". primenumbers yahoo group. Retrieved 19 September 2019.
  12. Kaiser, Peter (June 2019). "Let's find some large sexy prime pair (and, perhaps, a triplet)". Mersenne forum. Retrieved 18 August 2019.
  13. Lamprecht, Gerd; Luhn, Norman (October 2019). "CPAP's sexy prime". primenumbers yahoo group. Retrieved 13 October 2019.
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