Factorization attempt for a c214 regards Odd Perfect Numbers

[jchein1]

Hi every one,

Thanks to everyone who helped the factorization for a c163 I posted a couple weeks ago.

Before I finish my OPN (9 distinct factors) project, I need to get a complete factorization for a c214, which is a factor of 7^343 -1.

I am pretty sure c214 is the last roadblock for this project. The completed factorization is

7^343 - 1 = 7^(7^3) - 1 =

2 * 3 * 29 * 4733 * 3529 * 1074473 * 13473433 * 6106505825833677713 *
1373 * 8233 * 49393 * 734021 * 83517610741606021 *

8403170201386002840476080703299123251286617419186485806798706989812597310995759101855982323436346453490295844838291326800248443089057721410790484991729444190401836892041621145614063665607931920955659505589620515021 (c214)

I have now completed the 1115 th curve with limit B1=11000000; B2=1100000000 at the 30 digit level on a slow machine utilizing Dario Alpern’s ecm software without luck.

Please help and thank you in advance.


Regards

Joseph E.Z. Chein

[VJS]

What else have you done? P-1 to what bounds how about P+1 etc?

[akruppa]

I've started ECM with B1=11M.

The SNFS difficulty is 248 with a sextic - not easy, but feasible if a couple of people help sieving. So even if ECM does not get lucky again, we can definitely factor this one.

Alex

PS. : I've done P-1 with B1=10^8 and am currently doing 10^9.

[hlaiho]

http://www.loria.fr/%7Ezimmerma/records/c120-355 shows that
7^343-1 is already factored completely.

214 7, 343- 8403170201386002840476080703299123251286617419186485806798706989812597310995759101855982323436346453$
214 4902958448382913268002484430890577214107904849917294441904018368920416211456140636656079319209556595$
214 05589620515021
done 1575721793366942921257214180187566081951195327261 Shimoyama 11.08.04

Heikki

[philmoore]

http://www.cerias.purdue.edu/homes/ssw/cun/pmain505
also shows this number as completely factored. (Latest version as of 05 May 2005.) Also check http://www.cerias.purdue.edu/homes/ssw/cun/index.html
for the latest update.

[jchein1]

Dear Heikki, Philmoore, Alex and everyone,

Thank you very much for your valuable information. What a pleasure it is to here that c214 has been factored by Shimoyama on 11/08/2004 already. It’s saved me a great deal of time.

The completed factorization for c214 is

1575721793366942921257214180187566081951195327261 * p166.

Before I close this post, may I ask if you or anyone knows the factorization for

c427 = (19^361-1)/ (19^19-1)/ 84216527581

or any factorizations listing of p^n - 1, where p is a prime > 12 with high n’s? Please let me know.

I just completed the 346th curve for that c427 without luck. It’s time to call it quits. The c427 is much too big for any existing algorithm. Please don’t try. Many thanks.


Regards

Joseph E.Z. Chein

[akruppa]

Prof. Richard Brent's tables list factors of a^n+-1 for 12<a<100 and a^n < 10^255. Unfortunately the latter condition excludes 19^361-1.

Hisanori Mishima' tables list factors of cyclotomic numbers Phi_n(a) for a<=1000 and eulerphi(n)<=100. Again, the latter condition excludes 19^361-1.

I don't know of others that collect factorisation of such large a^n-1.

The c427 is far too large for SNFS, but ECM may have a good chance yet. I'll try a couple of curves.

Alex

[rogue]

Originally Posted by jchein1

I just completed the 346th curve for that c427 without luck. It’s time to call it quits. The c427 is much too big for any existing algorithm. Please don’t try. Many thanks.

346 curves with what B1?

[R.D. Silverman]

Originally Posted by jchein1

Dear Heikki, Philmoore, Alex and everyone,

Thank you very much for your valuable information. What a pleasure it is to here that c214 has been factored by Shimoyama on 11/08/2004 already. It’s saved me a great deal of time.

The completed factorization for c214 is

1575721793366942921257214180187566081951195327261 * p166.

Before I close this post, may I ask if you or anyone knows the factorization for

c427 = (19^361-1)/ (19^19-1)/ 84216527581

or any factorizations listing of p^n - 1, where p is a prime > 12 with high n’s? Please let me know.

I just completed the 346th curve for that c427 without luck. It’s time to call it quits. The c427 is much too big for any existing algorithm. Please don’t try. Many thanks.


Regards

Joseph E.Z. Chein

Hi,

How big is n?? Please note that if p = 1 mod 4, then p^(kp) - 1, k odd
has an Aurefeullian factorization that might help. Otherwise, Brent's
tables contain all that is publicly known.

[wblipp]

Originally Posted by akruppa

Prof. Richard Brent's tables list factors of a^n+-1 for 12<a<100 and a^n < 10^255. Unfortunately the latter condition excludes 19^361-1.

Richard Brent has a different file that lists factors for aⁿ±1 for a and n < 10⁴ and f>10⁹. This would contain any factors of this number known to Richard Brent. Unfortunately, it does not even show the 84216527581 factor that Joseph has already found.

It's also the best source I know for pⁿ-1 with p>12 and "large n" - although p>100 tends to have few factors with exponents above 100 and p>1000 tends to have few factors at all. OddPerfect.org has been contributing to these ranges as we prepare our first factors file.

[Keller]

[2005-06-12 18:09:58 GMT] Factor: probable factor returned by *CENSORED* (Athlon64_3400+)! Factor=4404877309587482066089969 Method=ECM B1=250000 Sigma=3960452341
[2005-06-12 18:09:58 GMT] Factor: Composite factor returned by *CENSORED*! Factor=581300074856809716701973568238586158736857742091321705834282135094473035267874767143193662215066561450964216295513378827083512261266014915528181747307393455020766994566080615418673082856139346577858493880082791410344804194670771991184755449759234316337033810904130156076844862334402121472790129337859325175391224529499684330384126802642899984651902174095890345457294139663461755277884279214872130395129 Method=ECM B1=250000 Sigma=3960452341

:P

Testing is being continued :)