A proof of Wilson's Theorem, a basic result from elementary number theory. The theorem can be strengthened into an iff result, thereby giving a test for primality. (Though in practice there are far more efficient tests.)
Number Theory is the study of whole numbers - also called integers. Since math began with the study of shapes and whole numbers, this is one of the oldest subjects around. However, it's also one of the most challenging. Many problems in number theory are easy to understand, but require a lot of cleverness to solve. It also does not require any advanced math to learn! Number Theory is for everyone.