Proposition 36 of book IX in the elements

[MathematicalPhysicist]

i'm referring to this:http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX36.html

does someone know if there's on the web another proof of this proposition, perhaps in a more readerable way. (i don't like to read also the geometrical approach and sliding through the links to the other propositions and not understanding it, [the guide doesn't help either])?


thanks in advance.

 

[Nimz]

It's saying that any mersenne prime, P=2^q-1, multiplied by 2^q-1 will give a perfect number. First, consider the sum, S=1+2+4+8+...+2^k. Multiply by two to get 2S=2+4+8+16+...+2^k+1. Take the difference, and you get that S=2^k+1-1.

Any factor of the proposed perfect number will either be of the form 2^k or 2^kP. Using the above, it shouldn't be too hard to prove that the number said to be perfect is indeed perfect. That is, to show that:

1+...+2^q-1 + P+...+2^q-2P = 2^q-1P

 

[robert Ihnot]

The above is certainly correct, and I want to add that,

What is the sum of the divisors of M=p^a*q^b where p and q are prime?

The answer is Sum =(1+p+p^2+++p^a)(1+q+q^2+++q^b)
(This way of doing it also considers M itself to be a divisor of itself.)

So when you put this together with what has been said above, remembering that P=2^q-1 must be prime, since then its only divisors are 2^q-1 and 1, where the sum = 2^q, we are on the way!