Here is one of my favourite elementary number theory results.
If are positive integers such that , then is composite.
I like it because it is not immediately obvious that the relationship should be related to the primality of the sum of these four numbers. As an example and .
To prove this, let . Then we may write and where and have no common factors. In other words, extracts the common factors from and . This leads to or . Since and have no common factors, this means all the non-unit factors of divide (and all the non-unit factors of divide ). Then we may write where is a positive integer so that .
which is composite.
Note that we can similarly prove that is composite where is any positive integer.