About Odd Pefect Numbers

[Cxxo]

If all perfect numbers are the sum of consecutive powers of 2, why does anyone think that there could be an odd perfect number?

Can someone explain this please?

[Batalov]

Originally Posted by Cxxo

If all perfect numbers are the sum of consecutive powers of 2, why does anyone think that there could be an odd perfect number?

Can someone explain this please?

I am pretty sure that Mr.google can.

Did you read Mathworld and wiki (see chapters about even and odd perfect numbers) before asking?

[Cxxo]

Originally Posted by Batalov

I am pretty sure that Mr.google can.

Did you read Mathworld and wiki (see chapters about even and odd perfect numbers) before asking?

Yes, but nothing I've read addresses the fact that perfect numbers are the sums of consecutive powers of 2. I find it extremely unlikely that I'm the first to discover that, so I'm simply wondering why there is still speculation about the existence of odd perfect numbers.

[Batalov]

Oh. No, you didn't discover that.

What you are saying is that a sum of consecutive powers of 2 = (a power of 2 minus 1) times (a power of 2). That is obvious. After you will see this equality, continue reading about even perfect numbers. From the word "Euclid..."

Originally Posted by Cxxo

Yes, but nothing I've read addresses the fact that (all?) perfect numbers are the sums of consecutive powers of 2.

The even ones. And you didn't prove that this is true for all PNs. This statement does not imply anything about odd PNs.

[Cxxo]

Originally Posted by Batalov

Oh. No, you didn't discover that.

What you are saying is that a sum of consecutive powers of 2 = (a power of 2 minus 1) times (a power of 2). That is obvious. After you will see this equality, continue reading about even perfect numbers. From the word "Euclid..."


The even ones. And you didn't prove that this is true for all PNs. This statement does not imply anything about odd PNs.

I see. I'm not very good at math, so I don't understand 100%, but I will think about it a little more.

[Aramis Wyler]

I think Batalov is simply saying that just because the consecutive powers of 2 are PNs, that doesn't mean that all PNs are consecutive powers of 2. Only that the consecutive powers of 2 are currently the only known category of perfect numbers. Just because we know a way to form a perfect number doesn't prove that there are not other ways.

[Dubslow]

Yes, that is correct, but:

the main takeaway here is that we can prove that all PNs that are even by assumption can be written as sums of powers of twos (when and only when a certain Mersenne number is prime).

But since this proof requires evenness as an assumption, it says nothing about the existence or form of odd PNs. (This is very clearly a much harder subject, having withstood centuries of mathematicians' concerted effort on the matter.)

[LaurV]

Originally Posted by Aramis Wyler

Just because we know a way to form a perfect number doesn't prove that there are not other ways.

We can prove that there are no other "even" ways. All even perfect numbers are product of a prime mersenne and its inferior power of 2. There is no other "way".

Unfortunately this shades no light on the "odd" side, and generally, nothing from the "even" can be applied to the "odd" side, therefore we have no idea if there are any odd perfect numbers or not. Most probably there are none. But who knows? Big numbers are full of surprises.

edit: crosspost with Dubslow, sorry