While we were talking about Perfect Numbers in lecture today, I began wondering what significance these numbers have in Mathematics, and what, if any, are their real-world applications? So I did a little research, I couldn't find anything, (besides their strong connection with Mersenne Primes).

So I was wondering if you guys had any answers for me? I figure since they share such an interesting property: that the sum of their proper digits adds up to the number itself, and that they have such a cool name, there must be some sort of use or significance for these numbers. Any ideas?

Also, Andy mentioned today that we know that if an odd perfect number exists, than it is greater than or equal to 10^(300). But 10^300 is such a huge number!! I don't think I really know the difference between 10^(300) and say, infinity. So my other question would be, what is the mathematical motivation for finding this odd perfect number, if it exists? What would we do with it? And why are so many people looking for it?