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Proposition 36 of book IX in the elements

  1. Mar 31, 2006 #1
    i'm refering 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.
     
  2. jcsd
  3. Apr 3, 2006 #2
    It's saying that any mersenne prime, P=2q-1, multiplied by 2q-1 will give a perfect number. First, consider the sum, S=1+2+4+8+...+2k. Multiply by two to get 2S=2+4+8+16+...+2k+1. Take the difference, and you get that S=2k+1-1.

    Any factor of the proposed perfect number will either be of the form 2k or 2kP. 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+...+2q-1 + P+...+2q-2P = 2q-1P
     
    Last edited: Apr 3, 2006
  4. Apr 4, 2006 #3
    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!
     
    Last edited: Apr 4, 2006
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