Sexy prime

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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

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.

Further information: Primorial § Definition for natural numbers

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 × (2172486 - 286243) + 286245 - 5
p+6 = 11922002779 × (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 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

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

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.

See also

References

  1. 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.
  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".

External links

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