20060907, 19:45  #1  
∂^{2}ω=0
Sep 2002
República de California
11755_{10} Posts 
Official Odd Perfect Number thread
First, I owe some folks an apology:
Earlier today I merged several older threads on the subject of OPNs, but at some point during my tidyingup I must've clicked the wrong link or button, because I inadvertently wound up harddeleting the whole result. Mike (a.k.a. Xyzzy) says it is theoretically possible to reconstruct the original threads from server backups of the SQL database, but as that would be a major headache, and since most of the meaningful content in the old threads was links to OPN references, unless someone has a massive objection, I'd like to just move on  lesson for me is in future to never harddelete anything, especially since the forum now has a nifty mergethreadswithtimedexpireofredirects feature for the moderators (you can currently see several redirects in the Math forum to threads I've moved which have this timedexpire feature). I apologize for any inconvenience this may cause, and to the folks who took the time to post to the deleted threads. The perils of housecleaning  sometimes good stuff inadvertently winds up going out ith the garbage. So, please feel free to repost links to any of your favorite OPN references and continue/reconstruct the deleted discussion, which can more or less be summarized as "discussion of the webofconstraints evolution w.r.to OPNs, the current proven lower limit on the number of factors an OPN must have, and the utility of numerical search for OPNs." I'll start with some related links: Odd Perfect Number  from Wolfram MathWorld oddperfect.org  Odd Perfect Number Search Odd Perfect Numbers  A Factoring Challenge ========================================= ...and a recent email exchange from the NumberTheory mailing list: Quote:
Quote:
Quote:
Last fiddled with by ewmayer on 20060913 at 22:22 Reason: Added URLs 

20060908, 00:49  #2 
"Phil"
Sep 2002
Tracktown, U.S.A.
2^{5}·5·7 Posts 
I see I must have had quite a few posts deleted in that process, but at least we probably won't miss all the heated exchanges about whether OPN research is even worthwhile...
On the other hand, I wouldn't be too quick to merge threads, as I remember discussions about several distinct topics that probably deserve separate threads, i.e., number of factors, size of an OPN, heuristics, etc. Fortunately, some of the more interesting discussions actually occurred in the factoring thread, but if anyone remembers anything interesting about what was deleted, feel free to ask about it and perhaps if I can't come up with references, Pace Nielsen or Joseph Chein or Kevin Hare, all contributors to this forum who have also published papers on odd perfect numbers, can help us out. William Lipp (oddperfect.org) is also a good resource. 
20061110, 08:41  #3 
2×101 Posts 
On some recent results re: OPNs
Hi all. I was able to show recently that, given an odd perfect number
N = (p^k)*(m^2) with special prime factor p, then it must be true that p^k < (2/3)*(m^2). I will post here the details of the proof as soon as I have the paper published in a local journal. Meanwhile, this result shows in particular that m^2 > (sqrt(6)/2)*(10^250) (since N > 10^500, according to http://www.oddperfect.org), which somehow strengthens Cohen's 1987 result that (p_sub{i})^(2*alpha_sub{i}) > 10^20 for some i. (Here (p_sub{i}) divides m; see http://en.wikipedia.org/wiki/Perfect_number.) Best regards, Jose 
20061110, 15:12  #4 
May 2003
7×13×17 Posts 
It is always good to see someone else interested in OPN's. Keep it up.
Not to discourage you, but your result doesn't seem that difficult. I think it was Pomerance who proved that for an odd perfect number, there is no decomposition N=AB with gcd(A,B)=1 and \sigma(A)=2B, \sigma(B)=A. In particular, \sigma(m^2)\neq p^k. But we do know p^k \sigma(m^2). Therefore, 3p^k <= \sigma(m^2) <= 2m^2; which recovers your result. (That is, if I didn't make a mistake.) 
20061110, 21:55  #5 
"Jason Goatcher"
Mar 2005
3×7×167 Posts 
I didn't understand a bit of the last two posts, but I still think OPNs are way cool if they exist.
/me plans on doing a few more curves on wblipp's servers, even though he doesn't understand one bit of the math involved. Hey, guys, if we find an OPN, maybe we could mail a hard copy of it and all it's factors to Mr. Silverman? Can we, PLEEEEEAAAAASSSSEEE? Huh, huh, can we? Huh? Last fiddled with by jasong on 20061110 at 21:58 
20061111, 07:46  #6  
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Quote:
Alex 

20061111, 19:13  #7 
"Jason Goatcher"
Mar 2005
6663_{8} Posts 

20061111, 19:20  #8 
"Jason Goatcher"
Mar 2005
3×7×167 Posts 
While we're on the subject of perfect numbers:
If we did find an OPN, wouldn't it be possible that there are even perfect numbers that don't correspond to Mersenne primes? (I apologize if this question isn't mathematically robust, I'm not sure how to change it) 
20061111, 19:27  #9 
"Nancy"
Aug 2002
Alexandria
4643_{8} Posts 
It's been proven by Euler that every even perfect number is of the form 2^{n1}*(2^{n}1), where 2^{n}1 must be prime. It's been known since Euclid's times that every Mersenne prime (although they didn't call them "Mersenne" as Father Mersenne's work was still two millenia off) leads to a perfect number, so even numbers are perfect if and only if they are of the stated form.
Alex Last fiddled with by akruppa on 20061111 at 19:29 
20061111, 23:55  #10 
"Nancy"
Aug 2002
Alexandria
100110100011_{2} Posts 
Heh... there is a new result on odd perfect numbers: "Odd perfect numbers have a prime factor exceeding 10^8" by Takeshi Goto and Yasuo Ohno. This research brought to you by the E. W. Dijkstra foundation...
Alex 
20061118, 23:59  #11  
"Jason Goatcher"
Mar 2005
3×7×167 Posts 
Quote:
Wblipp seems like a nice guy, I'm not interested in offending him, accidentally or otherwise. 

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