Sexy primes
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In number theory, sexy primes are prime numbers that differ from another prime by Template:Math. For example, the numbers Template:Math and Template:Math are a pair of sexy primes, because both are prime and .
The term "sexy prime" is a pun stemming from the Latin word for six: Template:Lang.
If Template:Math or Template:Math (where Template:Mvar 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 Template:Math denote the Template:Mathth prime number, and define
The quantity Template:Math is the smallest gap that occurs infinitely often between primes separated by Template:Math positions in the sequence of prime numbers. In particular, the sexy prime conjecture is equivalent to the statement
- ,
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 Template:OEIS2C and Template:OEIS2C 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).