A thought on the existence of an odd perfect number

[MathematicalPhysicist]

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?

 

[fresh_42]

What is known of the number of divisors of an odd number?

Quiet a lot:
https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers

 

[MathematicalPhysicist]

Quiet a lot:
https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers

I always wondered why such a difficult problem doesn't have a prize for the one who proves it.

But anyhow, back to my boring work of grading...

 

[mfb]

The prize is immortality in the mathematical literature.
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.

 

[Vanadium 50]

This is the great thing about mathematics. An odd perfect number probably doesn't even exist, but we know a lot about it!

 

[fresh_42]

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.

 

[mfb]

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.