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Old 2015-06-23, 01:26   #1
mathgrad
 
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Default Prove 2^n cannot be a perfect number

Given n is an integer, prove that 2^n cannot be a perfect number.
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Old 2015-06-23, 02:02   #2
Batalov
 
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Ok, let's give it a try:

2^n is an even number.
Sum of its divisors (1 and many even numbers) is an odd number.
Hence, 2^n cannot be equal sum of its divisors. \qed

Last fiddled with by Batalov on 2015-06-23 at 03:28 Reason: its != it's
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Old 2015-06-23, 10:09   #3
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Nice job Batalov. The proof looks valid to me.

Regards,
Matt
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Old 2015-06-23, 10:20   #4
retina
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What about when n=0?

What about the more general k^n?

A much more interesting proof would be to show that no odd number can be a perfect number.
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Old 2015-06-23, 11:44   #5
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Quote:
Originally Posted by retina View Post
What about when n=0?

What about the more general k^n?

A much more interesting proof would be to show that no odd number can be a perfect number.
For n=0 k^n is odd but a list of proper divisors doesn't exist, the second statement can be partially worked out since odd +odd =even only odd numbers with an odd number of proper divisors can be perfect. Which also means that for k=odd only n=even need be considered, and yes I know this was likely all rhetorical

Last fiddled with by science_man_88 on 2015-06-23 at 11:47
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Old 2015-07-06, 19:45   #6
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Since the sum of the smaller factors of 2^n equals 1 + 2 + 4 + ... + 2^(n-1) = 2^n - 1
it is never equal to 2^n, hence 2^n is never perfect.

But I like Batalov's parity explanation better.
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Old 2016-03-18, 16:35   #7
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Prove b^n (n > 1), and b is prime:

Proof:

1 is a divisor of b^n for all natural numbers.the sum of all the divisors of b^n not counting 1 is a multiple of b. Adding one gives us a non multiple of b, which in order for b^n to be a perfect number, the divisors must add up to b^n (which should give us a multiple of b of course) and does not.
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Old 2016-03-19, 01:30   #8
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Quote:
Originally Posted by mathgrad View Post
Given n is an integer, prove that 2^n cannot be a perfect number.
Also see Wikipedia: almost perfect number. /JeppeSN
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