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Old 2007-05-28, 23:51   #1
jasong
 
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"Jason Goatcher"
Mar 2005

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Default 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.
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Old 2007-05-29, 11:32   #2
R.D. Silverman
 
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"Bob Silverman"
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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.
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Old 2007-05-29, 12:31   #3
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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.
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Old 2007-05-29, 14:17   #4
philmoore
 
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Quote:
Originally Posted by R.D. Silverman View Post
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.
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.
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Old 2007-05-29, 14:34   #5
R.D. Silverman
 
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Quote:
Originally Posted by philmoore View Post
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.
It is not a joke.

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.
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Old 2007-05-29, 15:45   #6
Zeta-Flux
 
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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.
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Old 2007-05-29, 17:50   #7
R.D. Silverman
 
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Quote:
Originally Posted by Zeta-Flux View Post

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.
Allow me to repeat something that I think was orginally said by Dick Lehmer
(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???
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Old 2007-05-29, 18:35   #8
Zeta-Flux
 
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Quote:
I ask: what value is there in just raising this bound to (say) 10^500???
It depends. If someone is going to pay you a million dollars to do so, and you plan on spending that money to help poor orphans, I suppose there is a lot of value. If you are sitting in front of your computer day-in and day-out waiting to see if an OPN pops up, then probably little value.

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.
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Old 2007-05-29, 20:29   #9
jasong
 
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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.
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Old 2007-05-29, 22:32   #10
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Quote:
Originally Posted by jasong View Post
I really don't understand why so many people are confident OPNs(or even one OPN) don't exist.
As far as I can tell, prior to 2005 there were no serious heuristics about the existence of odd perfect numbers. In 2005 Joshua Zelinski and Carl Pomerance both created heuristics that say there shouldn't be any large ones, and we already know there aren't any small ones. However, both of these heuristics also say there shouldn't be any large even perfect numbers, either. Every Mersenne prime corresponds to an even perfect number, and most people believe there are an infinite number of these.

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.
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Old 2007-05-29, 22:57   #11
philmoore
 
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Quote:
Originally Posted by R.D. Silverman View Post
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.
So since this project is contributing to such articles in these publications, it therefore must have some value to mathematics.

Check out:
Hare, K. "New Techniques for Bounds on the Total Number of Prime Factors of an Odd Perfect Number." Math. Comput. 74, 1003-1008, 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/0000-000-00/...90-4/home.html
(to subscribers)
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