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Math Help - How do you do this proof of perfect integer?

  1. #1
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    How do you do this proof of perfect integer?

    I'm completely stuck on this proof. I know the idea of how to do it, find the factors, add them, and show its the same, but I have no idea what the factors should be. Here's the question:

    Show that 2^(p-1) * ((2^p) - 1) is a perfect number when (2^p) -1 is prime.

    Thanks for any help, it is really really appreciated.
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  2. #2
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    All even perfect numbers are a power of two times a Mersenne prime

    Take a look at this (I'm working on the same problem btw). I'm still making sense of the sigma notation but hopefully this will get you going in the right direction.
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  3. #3
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    It's look pretty good, and helps, but our professor hasn't taught us sigma, so I don't know how much I can use, it's definitely a good basis though, thank you!
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  4. #4
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    I'm actually in the same position and was trying to extrapolate from what was there... but now I'm realizing that I would have to prove the correctness of their sigma function which leaves me just as stuck. Hopefully somebody else has an answer for the both of us.
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  5. #5
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    Ya, I hope so, it's a really confusing proof and I have no other idea of how to even start it.
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  6. #6
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    I posted the question on another forum and got this response:



    "I think a sketch of a proof could go about something like this:

    The sum of all the positive divisors to 2^(p-1) is:
    Sum ( 2^k ), k = 0 to k = p-1 = 2^p - 1 (known power series formula)

    For each term 2^k in the above series, except the last, there's another positive divisor (2^p - 1)*2^k that should be included. The sum of all of those are (2^p - 1)(2^(p-1) - 1).

    Adding these to together gives us:
    (2^p - 1)(2^(p-1) - 1) + 2^p - 1 = 2^(2p-1) - 2^p - 2^(p-1) + 1 + 2^p - 1 = 2^(2p-1) - 2^(p-1) = 2^(p-1)*(2^p - 1)

    Note that if 2^p - 1 where not a prime there would have been more divisors."



    This should do it! Hope this helps mathgirl.
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  7. #7
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    It does!! Thank you so much!! exactly what I needed!
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