User talk:Bopomkova

October 2019
Hello, I'm Deacon Vorbis. I noticed that you added or changed content in an article, Perfect number, but you didn't provide a reliable source. It's been removed and archived in the page history for now, but if you'd like to include a citation and re-add it, please do so. You can have a look at the tutorial on citing sources, or if you think I made a mistake, you can leave me a message on my talk page. Thank you. –Deacon Vorbis (carbon &bull; videos) 14:48, 22 October 2019 (UTC)

By the way, while proving that an odd perfect number must have a remainder of 1 when divided by 4 is an impressive result for a 13 year old, this result has been known for at least a few hundred years. This is included for example in the stronger statement in the article that " N is of the form N ≡ 1 (mod 12), N ≡ 117 (mod 468), or N ≡ 81 (mod 324)." which is already in the article and has a source. JoshuaZ (talk) 17:10, 22 October 2019 (UTC)

Follow up conversation with JoshuaZ (talk)
Is it known yet that N must have 4n+2 factors? Bopomkova

Yes. That's a easy corollary of Euler's form for an odd perfect number. JoshuaZ (talk) 19:18, 26 October 2019 (UTC)

Is it known yet that N does not exist? Bopomkova

No, if so, the page would be very different. We don't even have any method that looks likely to prove that no such number exists. JoshuaZ (talk) 02:17, 5 November 2019 (UTC)

Well, I guess that's something. I think I have part of a potential proof. I'm still working on it though. Bopomkova

Is it known yet that the sum of the factors of 2N is 6N? Bopomkova
 * Yes, that's known to follow from the fact that $$\sigma(n)$$ is a multiplicative function. JoshuaZ (talk) 02:49, 8 November 2019 (UTC)

Has a formula been created to find $$\sigma(\sigma(n)$$) for any perfect number n? Bopomkova 10:55, 12 November 2019 (EST)

Rephrasing this: Has it been proven that a Perfect Number with one factor of 2 must be divisible by 3, with two factors of 2 divisible by 7, three factors of 2 divisible by 15, etc.? Bopomkova 10:09, 19 November 2019 (EST)

Also, is 6 the only perfect number such that $$\sigma(\sigma(6)$$) is also a perfect number?

I don't know how much has been done about looking at $$f(n)=\sigma(\sigma(n))$$, but it shouldn't be that hard to rephrase standard results in terms of that. In particular, the classification of even perfect numbers gives a pretty straightforward description of what $$f(n)$$ should look like. As for the final question, yes 6 should be the only such number. If N is even and perfect with $$N = (2^p-1)(2^{p-1})$$ then $$f(n)=(2^{p+1}-1)2^p$$ which could only be perfect if $$p+1$$ is prime (by the Euclid-Euler theorem for even perfect numbers), and the only pair where $$p$$ and $$p+1$$ are both prime are 2 and 3. In the case of N odd, it isn't hard to show that $$f(n)$$ will always be abundant. In general, any question of this sort is going to have an already known easy answer or is going to be extremely difficult, with little middleground. JoshuaZ (talk) 20:22, 22 November 2019 (UTC)

Is it known yet that for any perfect number, n, where n = 2p−1(2p − 1) and k = 2p − 1 that n = $$\sum_{x=1}^{k} x$$ ? Bopomkova 10:38, 10 March 2020 (EST)
 * Yes. This has been known since the ancient Greeks. There may be some locations which are better to ask your math questions than here. math.stackexchange and reddit's /r/math will probably be better for that purpose. JoshuaZ (talk) 18:34, 17 March 2020 (UTC)