Sexy prime

In mathematics, sexy primes are prime numbers that differ from each other by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6. 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.

The term "sexy prime" stems from the Latin word for six: sex.

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 May 2009^[update] the largest known sexy prime was found by Ken Davis and has 11,593 digits. The primes are (p, p+6) for

p = (117924851 × 587502 × 9001# × (587502 × 9001# + 1) + 210) × (587502 × 9001# − 1)/35 + 5.^[1]

9001# = 2×3×5×...×9001 is a primorial, i.e., the product of primes ≤ 9001.

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 1000 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).

As of 2013^[update] the largest known sexy prime triplet, found by Ken Davis had 5132 digits:

p = (84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1) / 35 + 1.^[2]

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 1002 digits:

p = 411784973 · 2347# + 3301.^[3]

In September 2010 Ken Davis announced a 1004-digit quadruplet with p = 2³³³³ + 1582534968299.^[4]

Sexy prime quintuplets

In an arithmetic progression of five terms with common difference 6, one of the terms must be divisible by 5, because 6>5 and the two numbers are relatively prime. Thus, the only sexy prime quintuplet is (5,11,17,23,29); no longer sequence of sexy primes is possible.

References

^ Ken Davis, "11,593 digit sexy prime pair". Retrieved 2009-05-06.
^ Jens K. Andersen, "The largest known CPAP-3". Retrieved 2014-06-13.
^ Jens K. Andersen, "Gigantic sexy and cousin primes". Retrieved 2009-01-27.
^ Ken Davis, "1004 sexy prime quadruplet". Retrieved 2010-09-02.

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

Grime, James. "Sexy Primes (and the only sexy prime quintuplet)". Numberphile. Brady Haran.

[1] Ken Davis, "11,593 digit sexy prime pair". Retrieved 2009-05-06.

[2] Jens K. Andersen, "The largest known CPAP-3". Retrieved 2014-06-13.

[sexycousin-3] Jens K. Andersen, "Gigantic sexy and cousin primes". Retrieved 2009-01-27.

[4] Ken Davis, "1004 sexy prime quadruplet". Retrieved 2010-09-02.

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