Wikipedia:Reference desk/Archives/Mathematics/2020 January 10

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January 10

Odd perfect numbers

(This was brought to my attention when I found out today that the oddperfect.org domain is dead.)

Is there any evidence that suggests that there are odd perfect numbers?? Georgia guy (talk) 12:30, 10 January 2020 (UTC)

There's some discussion at Perfect number#Odd perfect numbers, giving some constraints that would apply to an odd perfect number. I'm not sure what would constitute evidence that they exist, other than some actual examples. AndrewWTaylor (talk) 13:38, 10 January 2020 (UTC)

You could have a heuristic argument, something like how the fact that the integral $\int _{2}^{\infty }{\frac {dx}{\log(2^{2^{x}}+1)}}$ converges suggests that there are only finitely many Fermat primes. (And the analogous integral for Mersenne primes diverges, suggesting that there are infinitely many of those.) But the article on odd perfect numbers seems to say that heuristic arguments suggest that there aren't any. Double sharp (talk) 15:44, 10 January 2020 (UTC)

But is there anything wrong that the heuristic arguments that suggest there are no odd perfect numbers imply?? Georgia guy (talk) 15:46, 10 January 2020 (UTC)

@Georgia guy: please edit this comment -- it is incomprehensible. --JBL (talk) 22:29, 10 January 2020 (UTC)

What I mean is: I want to know why the arguments that suggest that there are no odd perfect numbers do not prove there are no odd perfect numbers. Georgia guy (talk) 23:08, 10 January 2020 (UTC)

If you read (for example) the reference [1] that is attached to the statement about Pomerance's heuristic in the article Perfect number, you will find a description of the heuristic as well as a discussion of why it is just a heuristic, not a proof. (Roughly, it boils down to this: sometimes we can develop correct intuition about the behavior of the primes by assuming that they appear randomly in some sense; but they're aren't actually random and so you can't rigorously prove anything if you use such an assumption.) --JBL (talk) 02:54, 11 January 2020 (UTC)