New paper on odd perfect numbers

[joshuazelinsky]

This is a new paper I coauthored with Sean Bibby and Pieter Vyncke. We prove new things about large prime factors of odd perfect numbers. The paper is pretty readable if one has only a very small amount of number theory background.

Recall that a number is perfect if the sum of its divisors less than the number is equal to the number. For example, 6 is perfect since the relevant divisors are 1, 2 and 3, and 1+2+3=6. One of the oldest unsolved problems in all of math is whether there are any odd numbers which are perfect.
In the last few years, there's been a bit of work on bounds about the largest prime factors of an odd perfect number N. In particular, assume that the the distinct prime factors of N are p1, p2, p3 ... pk with p1 < p2 < p3 ... < pk .
Acquaah and Konyagin proved that one must have pk < (3N)^(1/3) . Subsequently, using essentially the same techniques as them proved that p(k-1) < (2N)^(1/5), and that p(k-1)pk < 3N^(1/2). In a similar vein Luca and Pomerance using a similar technique showed that p1p2p3... pk < 2N^(17/26). In this paper , we prove that p(k-2) < (2N)^(1/6), and that p(k-2)p(k-1)pk < (2N)^(3/5). Note that 17/26 > 3/5, so this is an actual improvement over Luca and Pomerance's bound, although at the cost of only applying to the product three largest distinct prime factors rather than the product of all of them.

 

Comments

[sniffnoy]

Nice! I see you did not invert the numbering, then? :P

(Also, why is the text in this entry so large...?)

[joshuazelinsky]

This is the first paper the one which was about to be published. It is a possible sequel to this where the notation from this one leads to a conflict. Although in retrospect, if I had realized how much time went between recognizing that issue and this actually published, I would have suggested we spend the time to just go through and make the notation in this paper consistent.

As for why it is so big; not sure. Dreamwidth does weird things when copying and pasting rich text. My guess is there's something in the html doing that, but Dreamwidth also produces really icky html, so I'm not sure at a glance where it is there.

[sniffnoy]

This is the first paper the one which was about to be published. It is a possible sequel to this where the notation from this one leads to a conflict.

Ahh, right. I'd forgotten that.

As for why it is so big; not sure. Dreamwidth does weird things when copying and pasting rich text. My guess is there's something in the html doing that, but Dreamwidth also produces really icky html, so I'm not sure at a glance where it is there.

Oh, you're using the rich text editor. :P

[sniffnoy]

Hm, it looks like MathSciNet thinks you solo-authored this paper...? Then again, it looks like Bibby and Vyncke aren't in their system yet, so that's probably why, and presumably they'll update it once they are.

[joshuazelinsky]

Huh. I don't think that should make a difference. I'll send them a note just to make sure that gets corrected.

[joshuazelinsky]

They didn't reply to my email, but the paper is no longer visible on Mathscinet. So I'm guessing they are in the process of correcting it?

[sniffnoy]

I mean, it's probably just an automatic thing, I doubt it's necessary to contact them...