I used to obsess over prime number theory but it quickly becomes involved in highly advanced math. The basic concepts are really cool and easy to understand though. I don't know how much conversation this topic will get because it's pretty much accepted that this problem is currently impossible.

Composite numbers can be prime factorized easily and finding two numbers that multiply to it is as well. Prime numbers require brute force methods to check that they are in fact prime and use iterated algorithms that are time consuming for large primes. With a very large p and q, pq becomes so big that on top of figuring out a list of large primes there are so many possible combinations of all these to check for that equal a particular n, and there is only one distinct combination. The beauty of n=pq is that no matter how fast computers get the digits can just increase to stay ahead of their ability.

I think a breakthrough in prime number theory isn't going to happen for a while. They have been studied for such a long time and the basic problems of their nature don't seem to be something that can be solved with concise formulas. If decomposing primes were to become a real possibility the internet security business would be gone overnight.