Why are perfect numbers important? What is the best way of introducing these numbers to a first course on number theory? I could not find any application apart from the relation to Mersenne primes. Are there any other applications of perfect numbers?
Weclome to PF; I'll answer by asking questions. 1. why are the different classifications of numbers important? 2. how do you normally introduce concepts into the course? 3. why do you need more - do you know what Mersenne primes get used for? http://math.stackexchange.com/questions/701037/applications-of-perfect-numbers
To be honest, I doubt perfect numbers are useful at all. Surely in a number theory course, your students can appreciate a theory that doesn't have an application but just intrinsic beauty? If I were to teach the course, I would lay a link with history and the Pythagoreans.