A thought on the existence of an odd perfect number

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The discussion centers on the non-existence of odd perfect numbers, specifically through a proof by contradiction approach. It posits that if an odd perfect number exists, denoted as ##2n+1##, then the sum of its divisors, excluding 1 and itself, must equal ##2n##. The conversation highlights that this sum must contain an odd number of terms, leading to a contradiction. Additionally, it is established that an odd perfect number cannot be a square, reinforcing the argument against its existence.

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MathematicalPhysicist
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Well the most obvious approach to prove that such a number doesn't exist is by ad absurdum, or so I think.
Assume there exists an odd perfect number ##2n+1##, then by definition ##2n = \sum_{m\ne 1, 2n+1, m|(2n+1)}m##.

So, since m is odd (since 2n+1 is odd and it divides it), if you can prove that the sum has an odd number of terms then obviously we get a contradiction.
What is known of the number of divisors of an odd number?
 
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An odd perfect number cannot be a square. That alone is sufficient to make the sum of true divisors odd, so that's everything you get with that approach.
 
This is the great thing about mathematics. An odd perfect number probably doesn't even exist, but we know a lot about it!
 
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Vanadium 50 said:
This is the great thing about mathematics. An odd perfect number probably doesn't even exist, but we know a lot about it!
I prefer to phrase this in a positive way: Proofs of non-existence are much, much harder than proofs of existence. That's why lower bounds in complexity theory (NP=P), and those in number theory, e.g. FLT, are so incredibly complicated. Existence is often very easy: prove there is a group, ring, algebra, vector space: ##\{0\}##. Job done.
 
Vanadium 50 said:
This is the great thing about mathematics. An odd perfect number probably doesn't even exist, but we know a lot about it!
It sounds very familiar once you use physics jargon: Mathematicians set many exclusion limits.
 

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