The Prime Detector

You are given a string of 1s. Match it only when its length is a composite number — that is, not prime and not 1.

This is the most famous regular expression ever written, and it does something regex is not supposed to be able to do: arithmetic. The idea is to ask whether the run can be divided into two or more equal groups. If it can, the length is composite. If no division works, it is prime. Invert it with a negative lookahead and you have a prime detector in twelve characters.