SOLUTION: prove that a perfact number can be written as a sum of (2^n)-1 consicutine numbers for some n.
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Question 475361: prove that a perfact number can be written as a sum of (2^n)-1 consicutine numbers for some n.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
First, we have to assume that all perfect numbers N can be written as (2^k - 1))
for positive integer k.
Let m, m+1, ..., (m+2^n)-2 be the (2^n) - 1 consecutive numbers, in which their sum is
(2m + 2^n - 2)}{2} = (2^n - 1)(m + 2^{n-1} - 1))
We want to show that all perfect numbers N can be expressed in this form. However, if we set m = 1 the solution becomes trivial.
Alternatively, we can let (2^k)}{2})
in which N becomes the sum of the first (2^k) - 1 integers.
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