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 .
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.
As of April 2022[update], the largest-known pair of sexy primes was found by S. Batalov and has 51,934 digits. The primes are:
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 ):
Ken Davis further improved the record with a 6,180 digit Brillhart-Lehmer-Selfridge provable triplet in October 2019: