Sexy primes

In number theory, sexy primes are prime numbers that differ from another prime by $6$ . For example, the numbers $5$ and $11$ are a pair of 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.

Sexy prime conjecture

The sexy prime conjecture is a weaker analogue of the twin prime conjecture. It asserts that there exist infinitely many pairs of prime numbers that differ by 6, known as sexy primes.

Let $p n$ denote the $n$ th prime number, and define

H_{m}=\liminf _{n\to \infty }(p_{n+m}-p_{n})

The quantity $H m$ is the smallest gap that occurs infinitely often between primes separated by $m$ positions in the sequence of prime numbers. In particular, the sexy prime conjecture is equivalent to the statement

H_{1}\leq 6

,

which asserts that prime gaps of size at most 6 occur infinitely often. Since every prime gap greater than 2 is even, this is equivalent to the existence of infinitely many sexy prime pairs.

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]. Thus, the proof of the sexy prime conjecture is equivalent to the proof of the generalized Elliott-Halberstam conjecture.

Examples

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).

The only sexy primes quintuplet is (5,11,17,23,29).

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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