You are probably shocked about the title and you will probably think that this is a joke. Well, it’s not. There are really sexy primes and in fact, there are some research about them.
Sexy primes are prime numbers which differ by 6. The pair (5,11) are examples of sexy primes, while (7,13,19) are triplet sexy primes. Quadruplet and quintuplet primes also exist and examples of them are (5,11,17,23) and (5,11,17,23,29) respectively. Sadly, there is only one quintuplet sexy prime.
Just like the search for twin primes, the search for sexy primes are also in progress (especially for single mathematicians). As of this writing, the greatest sexy prime pair found has 11,593 digits (the first number) and the greatest prime triplet has 5132 digits (the first number).
In September 2010, Ken Davis found the largest quadruplet sexy prime yet, a 1004-digit with p = 23333 + 1582534968299.
The proof that there is one quintuplet prime is quite easy. It only involves finding testing all the possible remainders of integers when divided by 5 (recall equivalence classes) and prove that one of is prime.
Reference: Wikipedia, Image Credit: Optimus Convoy, Devian Art