5- Table Discussion and OddPerfect.org

[akruppa]

Sorry, no... had an exam yesterday and will have another (the last one!) on the 21st. I'll play with sqrt again after that.

Alex

[akruppa]

Turns out all I had to do was change a #define...

Original number had 244 digits:
2180075438084173168593947502718622130302257527967068979673625647025361320317722693777938884468406733797422512728450828995171625828330838682415089213199694141322162661316080591217910460859207203227286693244038606742662977922009304165840148925781
Probable prime factor 1 has 108 digits:
465248728728895394653153888943818531375975270853944206732179708325578302943996260672261958710669770611963071
Probable prime factor 2 has 136 digits:
4685827823840272482505664867616517656515469121675968181975198207240785431020525292215919392072745559212525121242512397104031773498892011

Found right on the first dependency.

Well, that was it!

Alex

[Mystwalker]

Excellent work, Alex!

And clearly not an ECM miss.

[R.D. Silverman]

Originally Posted by akruppa

Turns out all I had to do was change a #define...

Original number had 244 digits:
2180075438084173168593947502718622130302257527967068979673625647025361320317722693777938884468406733797422512728450828995171625828330838682415089213199694141322162661316080591217910460859207203227286693244038606742662977922009304165840148925781
Probable prime factor 1 has 108 digits:
465248728728895394653153888943818531375975270853944206732179708325578302943996260672261958710669770611963071
Probable prime factor 2 has 136 digits:
4685827823840272482505664867616517656515469121675968181975198207240785431020525292215919392072745559212525121242512397104031773498892011

Found right on the first dependency.

Well, that was it!

Alex

Nice result......

[jchein1]

Dear Alex,


May I ask you, is 5,349- is OddPerfectSearch's only roadblock left to prove any opn > 10^500 ?

What a great pleasure it is to see your final result on the 5^349-1. Congratulations.


Cheers,


Joseph E.Z. Chein

[philmoore]

Congratulations, Alex, on the completion of an impressive factorization! I see that this is the second-largest penultimate factor found of a Cunningham number.

To answer Joseph Chein's question, William has said that he believes that this was the last remaining roadblock, but that actually putting all this together in a proof will be necessary to verify it. The methods of the first Brent and Cohen paper should now be sufficient to show that any OPN > 10^500, but the methods of the second paper (with te Riele) should be able to push this limit somewhat higher, perhaps 10^625, or even greater. I don't know even if anyone has yet established that 2⁵²⁰(2⁵²¹-1) and 2⁶⁰⁶(2⁶⁰⁷-1) at 314 and 366 decimal digits, respectively, are in fact the 13th and 14th perfect numbers in order of size. The next perfect number, 2¹²⁷⁸(2¹²⁷⁹-1) at 770 digits, presumably 15th, may actually be proven so given a few more roadblock SNFS factorizations. But the next one, 2²²⁰²(2²²⁰³-1) at 1327 digits, definitely seems out of reach without new methods, as it would require SNFS factorizations of numbers with over 400 digits.

[jchein1]

Dear Philmoore,

Thanks. I guess you mean that William and his company might use BCR’s “lifting” algorithms to pushing opn > 10^625 or higher ?.


Regards

Joseph E.Z. Chein

[akruppa]

Thanks, Dennis, Bob, Joseph and Phil!

Alex

[philmoore]

Originally Posted by jchein1

Dear Philmoore,

Thanks. I guess you mean that William and his company might use BCR’s “lifting” algorithms to pushing opn > 10^625 or higher ?.

Regards
Joseph E.Z. Chein

We should ask William about this. For those who are interested, the relevant paper is at:
http://wwwmaths.anu.edu.au/~brent/pub/pub116.html

[wblipp]

Originally Posted by philmoore

We should ask William about this. For those who are interested, the relevant paper is at:
http://wwwmaths.anu.edu.au/~brent/pub/pub116.html

Yes, the methods of the BCR paper can be used to stretch a boundary proof. But I'm under the impression that the work to stretch a boundary is essentially lost when another factorization is achieved, so we don't want to undertake that task until we are confident that we have reached a roadblock that is beyond the project's factoring resources. The next roadblock is still undergoing ECM at the 55 digit level (6453 curves have been run at B1=110M). Partway through the 60 digit level this number becomes a reasonably qualified candidate for SNFS, and we will see about interest. It would be a very large but not record breaking undertaking, and there will be less widespread interest than Alex's recent factorization because the base is not in the Cunningham range.

All of this premature because I continue to work very long hours so that we don't even have to the tools to complete an "unstretched" proof, so it's early to squabble about whether now is the time to stretch the proof.

[R.D. Silverman]

Originally Posted by wblipp

Yes, the methods of the BCR paper can be used to stretch a boundary proof. But I'm under the impression that the work to stretch a boundary is essentially lost when another factorization is achieved, so we don't want to undertake that task until we are confident that we have reached a roadblock that is beyond the project's factoring resources. The next roadblock is still undergoing ECM at the 55 digit level (6453 curves have been run at B1=110M). Partway through the 60 digit level this number becomes a reasonably qualified candidate for SNFS, and we will see about interest. It would be a very large but not record breaking undertaking, and there will be less widespread interest than Alex's recent factorization because the base is not in the Cunningham range.

All of this premature because I continue to work very long hours so that we don't even have to the tools to complete an "unstretched" proof, so it's early to squabble about whether now is the time to stretch the proof.

OK. Suppose those undertaking this project succeed in raising the bound.
I have no doubt that they will succeed.

Allow me to ask:

What insight is gained by this computation?

Allow me to quote Hamming:

The purpose of computing is insight, not numbers.

If this computation leads to any new mathematical insights about the
problem, I will applaud it heartily. Until then, it is just mindless computing.

On the other hand, if raising the bound helps convince people of the
unlikelihood that OPN's exist, and thereby dissuades such people from
trying to actually find an OPN, then I will also applaud the effort.