# Talk:Perfect number

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## An improvement of Ochem and Rao's bound

A recent paper has an improvement of Ochem and Rao's bound on the number of total prime factors for an odd perfect number in terms of the number of distinct prime factors. However, I'm the author of the paper, so it seems like a potential COI issue for me to add it to the article. Can someone else a) decide if the result is significant enough that we should change the article and b) if so, make the relevant edit? JoshuaZ (talk) 18:27, 23 May 2018 (UTC)

My opinion is that this type of bounds is significant enough to be included in the article. And since Ochem and Rao's bound is now superseded by your result, which is published in a peer-reviewed journal indexed by the Maths Reviews of the AMS, it should not stay in the page as the best known of its kind. I think you should just go ahead and replace it by your bound. Sapphorain (talk) 19:58, 23 May 2018 (UTC)
This also sounds right to me. [I am aware the edit has already been made, just weighing in in case there is any reason for doubt.] --JBL (talk) 22:31, 23 May 2018 (UTC)

## Ochem's and Rao's paper about Odd perfect numbers

Apparently, you reverted my edit on perfect numberon grounds that the ref was a preprint. Im pretty sure that http://ftp.math.utah.edu/pub/tex/bib/toc/fibquart.html (type CTRL+F to find and then type "Volume 52, Number 3, August, 2014") backs up the assertion that this paper, this "remark" was indeed published in Fibonacci Quarterly. Indeed, a search on Rao's personal page at ens-lyon.fr (https://perso.ens-lyon.fr/michael.rao/publi.php?lang=en), we get the exact same information about publication data. Same volume, same number, same month, same page, same everything. It sounds, to me, that ens-lyon.fr and Fibonacci Quarterly are trustable references to put in a wikipedia page, and i therefore think we should come back to my version of perfect number.
BeKowz (talk) 17:32, 6 June 2018 (UTC)

There was no reference given for this paper with your contribution. Also, you mention it three times to source three different results, which don't seem to correspond to what actually is contained in the paper. Sapphorain (talk) 18:35, 6 June 2018 (UTC)
Oh. Ok. How about we revert to my edit, but this time with the correct <ref> (something like this : <ref>Pascal Ochem, Michaël Rao, Another remark on the radical of an odd perfect number, The Fibonacci Quarterly Vol 52(3) (2014), pp. 215-217.<ref> ?) And, well, what are the differences between what I wrote and what is in the paper ?? BeKowz (talk) 05:58, 9 June 2018 (UTC)
This is a very conditional result, that is with yet a further condition on the radical of an already hypothetical odd perfect number. I don’t think this deserves more than a very concise mention (if any), like « If N is a perfect number whose radical exceeds the square root of N, then its special prime exceeds 10^60 » (and with of course one single detailed link to the paper). I am not saying this type of research is uninteresting, but that it should not occupy a disproportionate space in the article. Sapphorain (talk) 18:53, 9 June 2018 (UTC)
Bekows, your comment said unconditionally that N has no prime factor less than 10^6, but the paper doesn't say that. It lists two conditions, at least one of which must hold, and one of those conditions is having no prime factors less than 10^6. 2604:2D80:4000:859F:D125:8C02:CE47:6E03 (talk) 12:46, 10 June 2018 (UTC)

## Reorganizing the section on restrictions about odd perfect numbers

If we add to the section on restrictions about odd perfect numbers the assumption ${\displaystyle p_{1} then we can directly use the same framework to talk about the the results on the largest prime factor, second largest and third largest by referring to bounds on ${\displaystyle p_{k}}$, ${\displaystyle p_{k-1}}$, and ${\displaystyle p_{k-2}}$. Would anyone object to that tweak? JoshuaZ (talk) 15:48, 22 October 2018 (UTC)

Actually, raising an objection to my own suggestion. This notation doesn't help that much since it doesn't naturally handle the Euler prime, so we cannot just talk about bounding ${\displaystyle p_{k}}$, ${\displaystyle p_{k-1}}$, and ${\displaystyle p_{k-2}}$. without losing some strength. JoshuaZ (talk) 16:40, 29 October 2018 (UTC)

## Pomerance and Luca's result on the radical of an odd perfect number

Luca and Pomerance have shown that if ${\displaystyle N}$ is an odd perfect number with ${\displaystyle N=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{k}^{a_{k}}}$ that one must have ${\displaystyle p_{1}p_{2}p_{3}\cdots p_{k}<2N^{\frac {17}{26}}}$. See https://www.math.dartmouth.edu/~carlp/LucaPomeranceNYJMstyle.pdf . Should this be added to the section on restrictions on an odd perfect number? My inclination is yes, but I worry about the section becoming overly long. JoshuaZ (talk) 04:08, 9 November 2018 (UTC)

## Klurman's result

I'm undoing my earlier edit since I've now gone and read Klurman's paper. Although some places say that he proved the result in the edit, in the actual paper he doesn't have an explicit constant but rather just a bound of the form ${\displaystyle CN^{\frac {9}{14}}}$ for some constant C and for N sufficiently large. I strongly suspect that this constant can be made explicit with a little work but this would be original research and shouldn't be in the article. Therefore, until someone publishes a version of Klurman's argument with an explicit constant, Pomerance and Luca's bound should be the one that stays listed in the article. My apologies for including the Klurman article without having actually read the paper carefully. JoshuaZ (talk) 19:12, 28 May 2019 (UTC)

I made an edit simplifying the initial paragraph of the lead and moving examples into the lead in this edit to make it accessible to younger readers or those with less formal mathematical education. I was reverted by Anita5192 with the reasoning "This not only is no clearer, it isn't even correct. 6 is divisible by -1, -2, and -3. The sum of its divisors is 0."

The assertion that what I wrote was "incorrect" is simply false. The first sentence read: In number theory, a perfect number is a positive integer that is equal to the sum of its divisors, excluding the number itself. By positive integers, we're therefore talking about natural numbers only, in which "-1" doesn't exist. I understand that in contemporary academia, number theory deals with ${\displaystyle \mathbb {Z} }$ and then the set of units is ${\displaystyle \{\pm 1\}}$ rather than just ${\displaystyle 1}$, but the point is that the lead should be understandable to the broadest possible group of people. In high school or primary school, when divisors and multiples are taught, only positive integers are considered (at least conventionally), and so the clarification that divisors must be positive doesn't add anything.

As for the reasoning behind me merging the example section into the lead, it seems fairly uncontroversial to me that it's easier for someone to learn a property about the natural numbers if they see an example or two. The current version of the lead now gives no examples of perfect numbers.

I would imagine that my eleven-year-old self would have struggled to parse the sentence In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). The sentence that follows it is even worse. But eleven-year-old me did understand what perfect numbers are without much difficulty. So something needs to be done so that an eleven-year-old can look at the lead of the article and learn what a perfect number is. Bilorv (he/him) (talk) 00:04, 30 June 2019 (UTC)

## Proposed change to odd perfect number section, upper bound on second largest prime factor, new reference, COI

We currently note that the second largest prime factor of an odd perfect number must be be greater than 104. I'd like to add that the second largest prime factor is at most ${\displaystyle (2N)^{1/5}}$. The citation for this is Zelinsky, Joshua (July 2019). "Upper bounds on the second largest prime factor of an odd perfect number". International Journal of Number Theory. 15 (6): 1183–1189. doi:10.1142/S1793042119500659. Retrieved 2 July 2019.. Since I'm the author, I have a COI for this, so I'd like to know if other people agree with adding this to the article. Note that the article also proves a few other results (in particular, an upper bound on the product of the largest two prime factors) but the other results are technical enough that they seem like they should not be cited in this article. Thoughts? JoshuaZ (talk) 12:36, 2 July 2019 (UTC)

I'm going to wait until Monday, and if no one objects, add in a citation to the paper. JoshuaZ (talk) 12:52, 5 July 2019 (UTC)

## The Next Perfect Number is 28 = 1 + 2 + 4 + 7 + 14

I added... The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next paragraph is confusing with explaining how 56 = 28 x 2. 2601:580:107:4A57:6D32:859F:D592:8ABB (talk) 11:24, 6 July 2019 (UTC)

The explanation given in the next paragraph of the lede is perfectly clear (provided of course that one takes the pain of reading the whole sentence...) Sapphorain (talk) 11:32, 6 July 2019 (UTC)
Hi there. The next paragraph is intended to illustrate a condition for perfect numbers in terms of the divisor function. The first paragraph is intended to be as accessible as possible to readers, but the divisor function is an important concept in number theory so it is helpful to mention it in the next paragraph. Bilorv (he/him) (talk) 11:38, 6 July 2019 (UTC)