Perfect numbers
- Perfect numbers, part I
- Perfect numbers, part II
- Perfect numbers, interlude (Challenge #11)
- Perfect numbers, part III
Recall that the first three perfect numbers are 6, 28, and 496. And if you tried computing 
as I suggested near the end of part I, you might have noticed that 8128 is perfect as well! Do you see any interesting patterns here? (Hint: try factoring…)
Post your discoveries as comments here (unless you already know the answer, in which case you should refrain from spoiling other people’s fun =).
So far, these perfect numbers are of the form 2^(n-1) * (2^n – 1), but only when (2^n – 1) is prime. Works for n=2, 3, 5, and 7, so maybe n itself should be prime?
I figured out the same thing as DB while I was on Thanksgiving break and away from the computer. Nice way to pass the time.
But it is not required that n be prime. I’m pretty sure that it works when n is 9.
DB: right. As it turns out, n does have to be prime, but that’s not a sufficient condition.
Steve: It is a nice way to pass the time, indeed. =) I don’t think 9 works, though. 2^8 * 511 = 130816, the sum of the proper divisors of which is 171696, not 130816. That’s because 511 is not prime; 2^(3*3) – 1 = (2^3 – 1)(2^6 + 2^3 + 1) = 7 * 73.
[...] Less Travelled does a 4-part post on Perfect numbers, part I (and part II and part III, with a challenge in between). It’s a pretty elementary subject, but one that people have been unable unlock [...]