This is the last in a series of posts about perfect numbers. A quick recap of the series so far: in part I, I defined perfect numbers as positive integers n for which , where denotes the sum of the divisors of n. I also revealed that if n is factored into prime powers as , then can be calculated as follows:
In part II, we saw where this formula actually comes from. Finally, in the challenge interlude, DB and Steve Gilberg (and you?) found that the first four perfect numbers seem to all be of the form ; DB additionally claimed that this only works when is prime.
Well, it’s true: is perfect whenever is prime. But you don’t have to take my word for it—let’s prove it! Since n is perfect if , we want to show that if is prime. Applying the formula for , this is just some straightforward algebra. (Note: by “straightforward” I don’t mean “you’re dumb if you don’t see it immediately”; I mean “if you’re patient and persistent, you should be able to work it out for yourself”. Mathematicians are fond of saying that things are “obvious” or “straightforward”, but this is what they actually mean.)
Voila! Just what we wanted to show. Note that the restriction that must be prime is very important: the formula for assumes that n is factored as a product of prime powers, so the computation above is invalid if can be factored further.
So, the question becomes, when is a number of the form prime? Well, first of all, m must be prime; if m can be factored as , then can also be factored, as . But is it enough for m to be prime?
Notice that the first four perfect numbers correspond to the first four primes, 2, 3, 5, and 7:
The next one, however, dashes our hopes: . Now, it is important to note that this doesn’t necessarily mean that isn’t perfect: we have only proven above that is perfect when is prime, which doesn’t necessarily say anything about what happens when it isn’t. However, it turns out that this is indeed true. In fact, more is true: all even perfect numbers must be of the form , with prime. There are no other kinds of even perfect numbers. So, we saw that 11 doesn’t give us a perfect number, but it turns out that the next three primes (13, 17, and 19) do:
are all perfect. But then a few more primes get skipped; the next perfect number corresponds to 31.
Primes of the form might ring a bell for long-time readers of this blog: these are the Mersenne primes! As of right now, we know of 44 Mersenne primes, and therefore we know about 44 perfect numbers. The largest known Mersenne prime has almost ten million digits, so the largest known perfect number has about twice that many!
Now, what about odd perfect numbers? Are there any odd numbers which equal the sum of their proper divisors?
…No one knows!!