20070528, 23:51  #1 
"Jason Goatcher"
Mar 2005
3·7·167 Posts 
could oddperfect's ecm progress page be improved?
I'm not sure how much work it would take, so I'm just going to throw this out there. If accomplishing the task isn't hard, but simply tedious, then I'd like to volunteer.
Put simply, the following would be nice: (1) a link to the optimal number of curves until progressing to the next B1/B2. Alternately, or additionally, it would be nice to know how much progress needs to be made before a new method is attempted on each number. (2) A way to go directly to active numbers. Maybe a clickable listing for Most Wanted, More Wanted, and Tiny Factorizations. I'm now going to go look at the HTML for those sites. For those who have no idea what I'm talking about, here is a link to the Most Wanted page. I had to scroll down more than half a page to find an active number. Really, the simplest thing that could be helpful, would be to list at the top of the page this link, as well as a list of the roadblock numbers so that people can copy and paste them into their search function. 
20070529, 11:32  #2 
"Bob Silverman"
Nov 2003
North of Boston
2^{2}×1,889 Posts 
Certainly the web page may be improved.
Since chasing odd perfect numbers is almost certainly a futile effort, the web page could be improved by deleting it. Hopefully, people will then turn to more productive computations. 
20070529, 12:31  #3 
Jul 2005
2·193 Posts 
The web page is automatically generated by the ecmnet server.
To change the information on that page you'll need C++ programming skills and a copy of the ecmnet (just stick "ecmnet" in google and you'll be able to find it). I'm sure Mark R would be happy to accept a tested patch/modification if it improves ECMNet. 
20070529, 14:17  #4 
"Phil"
Sep 2002
Tracktown, U.S.A.
19·59 Posts 
Surely you are joking, Mr, Silverman! Given that the published literature on odd perfect numbers continues to appear year after year after year in such publications as "Mathematics of Computations" and given that work done through and summarized on this web page has been cited in at least three publications by different authors that I know of, it appears to me that it is considered a valuable resource by professional mathematicians working in this area. At the very least, the impetus to factor cyclotomic integers seems to create more interest in the areas of factoring in which you yourself are working.

20070529, 14:34  #5  
"Bob Silverman"
Nov 2003
North of Boston
2^{2}×1,889 Posts 
Quote:
These publications are presenting conditions that restrict the form that may be taken by odd perfect numbers. They are furthering the art in proving that odd perfect numbers do not exist. They also contribute new mathematics. Simply applying a known algorithm to increase a numerical bound presents no new mathematics and contributes nothing toward a proof/disproof of existence. You might want to read my paper in Notices of the AMS: A Perspective on Computational Number Theory. 

20070529, 15:45  #6 
May 2003
3013_{8} Posts 
I have to agree with R.D. Silverman that is it (very probably) hopeless that one would ever find an odd perfect number through straightforward searching. So, from the point of view of "doing professional mathematics" such a search is not on top tier. There are likely other projects which are more beneficial to humanity, and/or mathematics. Or considered more "mainstream".
On the other hand, I can see some reasons that such a search is not pointless. 1) By doing these straightforward searches, one becomes more aware of the problem at hand, and may gain insights into how to attack the problem. It was only after a lot of "playing around" that I came up with the ideas for my recent paper. 2) Many of those professional papers on odd perfect numbers involve new ideas which greatly restrict the cases available. However, there are almost always a few special cases which need to be dealt with by straightforward (but tedious) computations. William's website becomes a clearinghouse for factorizations which help finish up these proofs. 3) Even if there are no odd perfect numbers, people are interested in this problem. It seems there is little one can do on the problem but computations. At least William's website consolidates the efforts of those workin on this problem. 
20070529, 17:50  #7  
"Bob Silverman"
Nov 2003
North of Boston
2^{2}·1,889 Posts 
Quote:
(but I am not sure of the source) "The purpose of computing is insight, not numbers". However, as far as I can tell, the OPN project is just computing for the numbers, and not insight. I see no point in chasing factorizations just so the lower bound on the size of OPN''s can be increased. When Brent established the bound of 10^300, his paper contained some new ideas. I ask: what value is there in just raising this bound to (say) 10^500??? 

20070529, 18:35  #8  
May 2003
3013_{8} Posts 
Quote:
For a more concrete example, I started participating in the project because I wanted to try my hand at an SNFS factorization *before* I delve into the actual mechanics of that factorization technique. I also am interested in a lot of questions about cyclotomic polynomials (and their factorizations under specialization), and find this project to be a good place to spend some computing time. 

20070529, 20:29  #9 
"Jason Goatcher"
Mar 2005
110110110011_{2} Posts 
I really don't understand why so many people are confident OPNs(or even one OPN) don't exist. I don't even pretend to understand the math behind the problem, but as someone who gets most of his news from the Internet, I am very cautious about what I believe or trust, and my intution, while it may be considered worthless to some people, tells me that Mr. Silverman has an unwarranted confidence about the problem.
As more and more factorizations are done, and as computers become more and more advanced, people will look at those factorizations and discover more and more possibilities to restrict where an OPN could possibly be. Maybe then it will be proven that an OPN doesn't exist. Until then, we can only guess. 
20070529, 22:32  #10  
"William"
May 2003
Near Grandkid
5^{3}×19 Posts 
Quote:
Bob used to urge people to ask Richard Brent. I'd had brief exchanges with Brent in the process of sending him thousands of factors for his tables, but I hadn't actually asked him before. Brent says that although he doubts that any exist, he wouldn't be particularly surprised if one were found, either. Bob continues to insist otherwise, though  apparently not really caring what Richard Brent says. My personal intuition is that if any OPNs exist, they will have lots of prime factors, almost all to low powers. In this region the problem starts to look like the linear algebra stage of QS  you have lots of "rows" that have to be combined so that they cancel just so. Low exponents are a way to limit the factor base. The analogy isn't exact of course  there are additional constraints that linear algebra cannot handle. The traditional searches have always run aground at the limits of factoring technology. But those limits are necessary to handle components with high exponents. If we restrict our attention to numbers with small exponents, huge searchable regions are accessible. And even if we don't find an OPN, we will add many more thousands of factors to collected tables. 

20070529, 22:57  #11  
"Phil"
Sep 2002
Tracktown, U.S.A.
10001100001_{2} Posts 
Quote:
Check out: Hare, K. "New Techniques for Bounds on the Total Number of Prime Factors of an Odd Perfect Number." Math. Comput. 74, 10031008, 2005. Nielsen, P. "Odd Perfect Numbers have at least nine Distinct Prime Factors", Math. Comput. posted May 9, 2007, available at http://www.ams.org/mcom/000000000/...904/home.html (to subscribers) 

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