P56 ECM Factor of 19^193-1 for OddPerfect.org
 

On September 29th Ryan Propper, a Junior Computer Science student at Stamford University found a P56 factor of 19[sup]193[/sup]-1. It was found using the OddPerfect.org "Most Wanted" ECM Server at oddperfect.no-ip.com:8201. It is one of the the last two large factorizations necessary to prove that any odd perfect number exceeds 500 digits - the other one is presently undergoing SNFS factorization.

I believe this factor will make Richard Brent's Top 10 list, and Paul Zimmermann's Top 10 and Top 50 lists.

[CODE]A factor was found for 19_193M.C246 using GMP-ECM using factor method
ECM
Candidate number:
35008417433686092399041204805685245680564349948958346738667815977134616415
26072542425280118599704350405354644328925240219452934154556918376437783370
84634912593797074033580919314876929547322085269835389589845747161852022688
223622340105025530084481
Factor: 12165097652687289569559614927886409030606179279459368857
Factor Type: probable
Co-Factor:
28777752906860288850081076428679309196776615878836171685195569163102268080
65781052205925418824450826389548490377009471730554288544246080120281830112
3811564858580612611128165419600589500814633
Co-Factor Type: Probable
B1: 110000000
Sigma: 1470586190[/CODE]

How lucky! Congrats to Ryan Propper!

Looks like I picked the right number for SNFS. However, this means that we won't get to do a volunteer SNFS factorisation, which is a bit of a shame...

About 5,349-: I have a bit more than 110M relations so far and I expect I'll need ~160M, so sieving is a little more than 2/3 done.

Alex

[QUOTE=wblipp]On September 29th Ryan Propper, a Junior Computer Science student at Stamford University found a P56 factor of 19[sup]193[/sup]-1. It was found using the OddPerfect.org "Most Wanted" ECM Server at oddperfect.no-ip.com:8201. It is one of the the last two large factorizations necessary to prove that any odd perfect number exceeds 500 digits - the other one is presently undergoing SNFS factorization.

I believe this factor will make Richard Brent's Top 10 list, and Paul Zimmermann's Top 10 and Top 50 lists.

[CODE]A factor was found for 19_193M.C246 using GMP-ECM using factor method
ECM
Candidate number:
35008417433686092399041204805685245680564349948958346738667815977134616415
26072542425280118599704350405354644328925240219452934154556918376437783370
84634912593797074033580919314876929547322085269835389589845747161852022688
223622340105025530084481
Factor: 12165097652687289569559614927886409030606179279459368857
Factor Type: probable
Co-Factor:
28777752906860288850081076428679309196776615878836171685195569163102268080
65781052205925418824450826389548490377009471730554288544246080120281830112
3811564858580612611128165419600589500814633
Co-Factor Type: Probable
B1: 110000000
Sigma: 1470586190[/CODE][/QUOTE]


It seems like a lot of work to raise the exponent to 'only' 500.

Allow me to ask: Were any new ideas involved, or did the work merely
extend known methods? Were there any improvements made to the theory?
What would be needed to raise the exponent to something *much* larger?

I was hoping that the factoring work would raise the exponent to 10^6,
so that cranks who want to waste time trying to contruct an OPN might
be discouraged.......

[QUOTE=R.D. Silverman]Allow me to ask: Were any new ideas involved, or did the work merely extend known methods?[/QUOTE]
This factorization was a simple leveraging of the creative work done by many mathematicians such as yourself and the creators of GMP and ECMNET. I don't think it's accurate to call this "extending known methods" because there is no extension - it is "merely applying known methods." I'm one of those people that think one of the values of creative work is to allow other people to use it. Halmos once told me that if he ever learned a problem he was working on had practical applications he would cease immediately - does your disdain of "merely extending known method" come from a similar esthetic?

[QUOTE=R.D. Silverman]Were there any improvements made to the theory?[/QUOTE]
It appears likely that finishing the proof to 500 digits will require application of what I call the "excess exponent extension" on the oddperfect.org web pages. This appears to be an unpublished idea, but in the language of patents, it would have to be judged something "obvious to one skilled in the art."

[QUOTE=R.D. Silverman]What would be needed to raise the exponent to something *much* larger?

I was hoping that the factoring work would raise the exponent to 10^6[/QUOTE]
A million is tough. I'm working on a idea that looks promising for extending the boundary much higher than twice the existing factorization threshold. My preliminary rough estimates are that if we used that method and applied it to all factorizations below 250 digits that are listable in Richard Brent's factor tables, we could, in principle, extend the bound to half a million digits. But it looks like combinatorial explosion would stop us long before actually reaching that level. Ten thousand digits might be doable. Two thousand digits should be easy.

If you are correct that there are no odd perfect numbers, I don't see any way that these approaches could be extended to prove that - somewhat like proving Fermat's Last Theorem applied to ever higher exponents was never a path for proving it true. I've read that Sylvester, a 19th century mathematician and poet whose "web of conditions" is often quoted even today, thought he could prove odd perfect numbers do not exist through similar methods - once incorrectly thought he had done so. Someday I'll research that further.

William

The first paper of Brent and Cohen which proved that any odd perfect numbers must be greater than 10^160 utilized the fact that if an odd perfect number contained a prime power component greater than B, than the OPN would necessarily be greater than B^2. In the later paper of Brent, Cohen and te Riele in which they proved that an OPN must be greater than 10^300, they proved that under the same assumptions and some additional conditions which could in general be fairly easily verified, that the OPN must be greater than B^2.5 and even greater than B^n where n could in some cases be pushed close to 3. So I would not be surprised if current methods and factorization data could show that any OPN must be greater than 10^700 or something close.

Congrats William,

The major question is are you considering extending the project 10^700 is really quite large. Beyond the current limits of factoring but in a few years who knows?

It's also possible some of these numbers may factor easily with P-1 who knows.

I've ran a couple ecm curves (60-digit level) on the small composites but never reported anything. Perhaps a message board etc might help finish these off if you need them.

Actually, I am suggesting that it may be possible to extend the proof to 10^650 or 10^700 only on the basis of current factorizations, according to the method in the Brent, Cohen, and te Riele (or BCR) paper.

By the way, Bob Silverman asked if any new ideas were involved, or if the work was merely an extension of known methods. The BCR paper, as far as I can tell, relied only on ECM and P-1 factorizations. (Please correct me if I am wrong here, but they make no mention of NFS factorizations in their paper.) So what William has organized here is an extension to a higher range based on a collective factorization effort utilizing NFS as well. So this is technically a new extension, but, as William says, one that should be "obvious to one skilled in the art".

William, I'm curious, does your new idea for extending the result to a higher range require complete factorizations? I am sure that there must be quite a few cyclotomic numbers only partially factored that would be utilized to reach current bounds.

[QUOTE=R.D. Silverman]It seems like a lot of work to raise the exponent to 'only' 500.

Allow me to ask: Were any new ideas involved, or did the work merely
extend known methods? Were there any improvements made to the theory?
What would be needed to raise the exponent to something *much* larger?

I was hoping that the factoring work would raise the exponent to 10^6,
so that cranks who want to waste time trying to contruct an OPN might
be discouraged.......[/QUOTE]
Do you actually intend to piss people off, or are you just a natural jerk who can't control himself?

Maybe this means nothing to you, but even if I had an IQ of 500 I wouldn't even come close to talking down to people the way you do. Your intelligence is a gift from God and abusing people simply because they don't agree with you is immature and not deserving of respect.

For myself, I find humility to be the trait I most value in people, I would rather hang out with a humble, on welfare, retarded person, than someone who's a millionaire, intelligent, and likes to brag.

You have an arrogant heart Mr. Silverman, and you will be struck down for your pride at some point, I guarantee it. I suggest you change your attitude before God changes it for you.

You have been warned.

[QUOTE=jasong]Do you actually intend to piss people off[/QUOTE]
OH GOD! Please, not another Silverman Flame Thread. We have plenty of these already.

Bob asked hard questions that are perfectly reasonable when you are trying to dissuade someone from a line of inquiry. Unlike your response, there were no personal attacks in his questions. They are questions designed to entice stubborn people to think more deeply about what they are doing.

Embedded in the questions is Bob's often expressed prejudice that everybody should be doing leading edge research rather than finding uses for previously completed research. Embedded in the comments is Bob's belief that odd perfect numbers do not exist, although he has shunned several previous requests for more information about why he believes that. But we all have these kinds of quirks, and Bob is entitled to his.

Your personal attack on Bob is unwarranted in this instance. If you think his questions are unreasonable, then you are free to attack his questions - I've already provided my response to his questions and it has lead to some discussion on the mathematics of odd perfect numbers. Let's keep this thread about ideas, not people.

William

[QUOTE=wblipp]OH GOD! Please, not another Silverman Flame Thread. We have plenty of these already.

Bob asked hard questions that are perfectly reasonable when you are trying to dissuade someone from a line of inquiry. Unlike your response, there were no personal attacks in his questions. They are questions designed to entice stubborn people to think more deeply about what they are doing.

Embedded in the questions is Bob's often expressed prejudice that everybody should be doing leading edge research rather than finding uses for previously completed research. Embedded in the comments is Bob's belief that odd perfect numbers do not exist, although he has shunned several previous requests for more information about why he believes that. But we all have these kinds of quirks, and Bob is entitled to his.

Your personal attack on Bob is unwarranted in this instance. If you think his questions are unreasonable, then you are free to attack his questions - I've already provided my response to his questions and it has lead to some discussion on the mathematics of odd perfect numbers. Let's keep this thread about ideas, not people.

William[/QUOTE]

Part of the job of professors and experts is to DISSUADE others from lines
of enquiry that are likely to be unproductive.

It is also the job of a professor to encourage people to be *creative*.

It is also the role of a referee to question the validity of a line of research.
By presenting research in public, even if not formally presented, it is opened
to such scrutiny.

Perhaps when jasong grows up, he will understand this.

I do NOT take the view that everything must be 'leading edge research'.
However, IMO, it should be *creative* or it is not worth doing. Most
journals today have the editorial policy that a result that (say) merely
improves a bound while using known techniques should not be published.
That a publishable paper should contain new IDEAS. All I asked is
where the new ideas were in the work under discussion. It takes zero
intelligence to simply use old ideas and someone else's factoring software
to raise the OPN bound. It is not *creative*

And I don't think that raising the bound from 10^300 to 10^500 has much
value to mathematicians *unless* some new ideas were involved.

I think I hit a raw nerve with jasong simply because my comments ring true.

I have at times, suggested some elementary computational projects that
have not been extensively researched and which should allow for novices
to show some creativity. These comments, of course, have been ignored.

One such is the open question of whether every integer is the difference
of two powers. Another is the convexity of pi(x). Both should allow
clever people to show some creativity in cutting down on the amount of
arithmetic needed to investigate the questions.

[QUOTE=R.D. Silverman]Part of the job of professors and experts is to DISSUADE others from lines
of enquiry that are likely to be unproductive.
[/QUOTE]

This is true if someone has asked you to be their teacher. Trying to instruct someone who hasn't asked for your help is, as the I Ching would put it, "supreme folly".