So let's talk about an important topic that plays a very, very significant role in the development of number theory, which is the topic of perfect numbers. And this goes back to the Greek classification of numbers. And to understand that, we need to introduce one idea. The a la quote parts of a number are what we would now call the proper divisors, the numbers less than the number that are also divisors of the number. We have three important classifications of numbers based on the a la quote parts. The first are the deficient numbers, where the a la quote parts of the number add to less than the number itself. In some sense, what we have is a number that you cannot form by putting its parts together. So for example, ten is a deficient number, the a la quote parts, the proper divisors 1, 2, and 5, and if I put the proper divisors together, 1 plus 2 plus 5, what I get is less than 10. Well, if I can have deficient numbers, I can go the other direction and have abundant numbers, where the a la quote parts are more than the number. So for example, 12 is an abundant number, the proper divisors of 12, 1, 2, 3, 4, and 6, and if I add all of those together, what I get is more than 12. Well, if I have less than the number, more than the number, it seems reasonable that I could have exactly equal to the number, and I can have the perfect numbers, where the a la quote parts are equal to the number. Now, for those of you who have a classical background, which is to say if you know something about Greek and Latin, you'll probably realize that perfect is not actually a Greek term. I said this emerges from Greek number theory, but the term itself, perfect number, is a Latin term. Perfect comes from the Latin word meaning complete, and in this case the idea is that the parts are completely equal to the number. And we find that 6 is a perfect number, because if I look at the a la quote parts of 6, they are 1, 2, and 3, and if I add 1 plus 2 plus 3, I get 6. Now, perfect numbers test the limits of empiricism, test our ability to do anything useful in numbers theory by trial and error, and part of the problem is that, well, let's say that I found my perfect number 6, well, how many others can I find, and what can I say about these perfect numbers? So, well, let's see what happens. So I'm going to put down the numbers 1 through 20, and I'm going to try and find the perfect numbers. So 1 has no a la quote parts at all, so it's convenient to just ignore 1 whenever we talk about Greek number theory, or number theory in general. All primes are definitely deficient, because the only a la quote part a prime has is 1, so we'll strike the primes from the list. And we can find the a la quote parts of the remaining numbers. So 4 has a la quote parts 1 and 2. So those don't add to 4, they're less than 4, so 4 is deficient. 6, we already knew was perfect. 8, 1, 2, and 4 is the a la quote parts. Not adding to 8, again, deficient number. 9, deficient. 10, also deficient. 12, again, a la quote parts 1, 2, 3, 4, and 6, which is abundant. 14, not perfect. 15, not perfect. 16, not perfect. 18, not perfect. Here's something to note, it is an abundant number, the second abundant number that we've actually come across. And then 20, also an abundant number, third abundant number we've come across. And so in our numbers from 1 through 20, the only perfect number is 6. Now we can try to find other perfect numbers, and if we had some patience, we would find the next one, it's actually 28, and we say, oh great, let's keep going. And if we keep going, it takes us a little while to get to the next perfect number, 496. And maybe if we have lots and lots and lots of patience, we can find the next perfect number after that, 8,128. And hey, I have nothing to do for the next few hours, so let's see if we can find the next one. That takes us a while. So here's a different way of playing the game of mathematics. What we can do is we can begin with whatever assumptions we want, and derive whatever conclusions we can in the hopes that something we find is going to be useful. And how that works is, well suppose I begin with a set of assumptions, I'll call those statement A, and I go for a derive, and I figure out whatever I find, and eventually I find something interesting. I end up with my statement B. Well I started with my statement A, and assuming that my derivations were valid, that I used the proper rules of inference, I end up with my statement B, well that is a proof. And what that tells me is that if I start at A, I can end with statement B, and I will have proven the conditional statement if A, then B. And with any luck, this will actually be a useful conditional. While if you're unlucky, this will be something you want to prove. If you're lucky, you'll find something that's rather unexpected and potentially new. So how about those perfect numbers? Well, the simplest case to start with is N is a product of two and only two prime numbers. Now again, we can start with any assumption we want and see where our derivations take us. So let's start with, well let's suppose N is a perfect number. And that N is P times Q. Now that tells me that the proper divisors of N are going to be 1, P, and Q. And you may want to verify that that's actually true. And the sum of the proper divisors is going to be 1 plus P plus Q. And our assumption that N was perfect means that the sum of the proper divisors should be the number itself. And so if we go on our derive, I start with N as a product of two primes. That's also perfect. Then I know that the number itself is the sum of the proper divisors. And so that tells me this really neat relationship between P and Q. Product equals the sum of the individual piece. And let's do a little bit more algebra. P is Q plus 1 over Q minus 1 after all the dust settles. And again, this proves a conditional. What we started with is N equals P times Q, where N is perfect. What we ended with is that P has to be related to Q in a very specific fashion. So our starting premise, N is perfect.  And we have just proven this statement. Well, let's play the game a little farther. So what we did is we just proved that if I have P and Q prime and N as a product, if N is a perfect number, then P and Q have a very specific relationship to each other. And this is great because it tells us something about perfect numbers. Well, almost. The thing to remember here is that we are starting with the antecedent I have in my hands a perfect number. And then this tells me something about the primes that compose it. Well, that's not actually what I want. What I'd like to do is I'd like to end up with the perfect number. And what I need is the converse of this statement. If something, then N is perfect. The good news is this is a purely algebraic proof, which turns out to be completely and totally reversible. So I can actually start here and then there. Now, don't take my word for it. Don't ever believe that the converse is always true. We actually have to run through the proof. So again, our starting point, we have P and Q primes. N is a product of two primes. Again, you can make whatever assumptions that you want to. In this case, I'll make the assumption that P is Q plus one over Q minus one. Now, this is essentially just retracing our algebraic proof in the opposite direction. I'll do a little bit of algebra. I'll do a little bit of algebra. Well, PQ is N. N is the sum of its proper divisors. So if this is true, it follows that N is going to be a perfect number. And so I can actually join those two things together. N is perfect if and only if P has a particular relationship to Q. And now I'm ready to find perfect numbers. Or are we? We'll see.