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Euclid's formula/Perfect Numbers #2

  1. Jan 30, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that the sum of the reciprocals of all divisors of a perfect number is equal to 2.

    2. Relevant equations

    Euclid's formula for perfect #'s

    3. The attempt at a solution

    I am so lost.
     
  2. jcsd
  3. Jan 30, 2010 #2
    Let d1, d2,...., dk be a (ordered) list of divisors of perfect number n where d1 = 1, dk = n. It is clear that the sum of all these divisors is 2n.

    Observe that d1dk = n implies d1 = n/dk, which in turn implies that 1/d1 = dk/n.

    Do this same thing for d2dk-1 = n implies..... etc
    Now you have each reciprocal written as d/n where d is a divisor of n, so add them up.
     
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