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 ${\textstyle 11-5=6}$ .

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]

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).

References

^ 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.

External links

Weisstein, Eric W. "Sexy Primes". MathWorld.
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.

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