Gratuitous factors thread

[Batalov]

Ben, you are not alone with the 3-split (on 11,226+).
(My near-repunit is only sweetened by a hidden "nice split".)

(25·10^223-1)/3 = 83333...3333 = 157 · C222
C222 = P55 · P55 · P114
P55  = 1306957603596747155756205207527556392595787608690473329
P55  = 3610883731712362153383889046706233893569388374490811951
P114 = 11247...19111

You're right, now we should all wait for the 4-splits!

[bsquared]

I knew I kept good company ;)

Nice work!

[Batalov]

(this one gave us some pain. Two (week-long) BLs in a row finished with trivial dependencies, but ...per aspera ad astra, or whatever.
The third one, after some tinkering and hopefully useful future tricks, finished happily.)

Factored (with maxal) two numbers:

2^743-3 = 5 . 919915458916081 . p53 . p76 . p80
p53 = 63529018304787469595760994544568916542616709131721179
p76 = 6161956699502951972272816735739214671420490437899755803072697139814876537011
p80 = 25696742367308592212803153065407421670388916498379493992349439722903096813916169

...and earlier, rather uneventfully,

2^745-3 = 23 . 53 . 59063 . 8186919763171 . 30496310582253756085729536479 . p67 . p109
p67=1574940336399388867361064992577820311163112464508298354353403078649

Now, I'll say, only two factors? That's a surprise, these days.

--Serge and maxal

[themaster]

148655129312786366154326965873005541392573294151300967
i just found this 54 digit factor using ecm with B1=2000
how lucky can u get

[bsquared]

Originally Posted by themaster

148655129312786366154326965873005541392573294151300967
i just found this 54 digit factor using ecm with B1=2000
how lucky can u get

Apparently pretty lucky, since gmp estimates that it would take an infinite number of curves to find that size factor with that B1. Even finding a 40 digit factor would take on average 3 Teracurves. What magical sigma was used, and what was the input number?

[jrk]

I call shenanigans.

[Andi47]

Originally Posted by themaster

148655129312786366154326965873005541392573294151300967
i just found this 54 digit factor using ecm with B1=2000
how lucky can u get

Can you please post the details of the lucky curve?

[10metreh]

If it was on the Alpertron applet, the factor could have been found by the Lehman method, not ECM.

[themaster]

Originally Posted by Andi47

Can you please post the details of the lucky curve?

unfortunately i cant i was using a program i had written to call gmp-ecm so i dont have the logs
it was an extremely lucky find
it was a RHP composite

[kenta]

My hobby: factoring n-1 for large primes n.

Consider the twentieth Mersenne prime M20 = 2^4423-1.

Then M20-1 = 2*(2^4422-1) =

2 * 3 * 3 * 7 * 23 * 67 * 67 * 89 * 683 * 1609 * 2011 * 4423 * 9649 * 13267 * 20857 * 22111 * 39799 * 283009 * 599479 * 6324667 * 7327657 * 193707721 * 12148690313 * 12371522263 * 361859649163 * 761838257287 * 6713103182899 * 224134035919267 * 3556355492892313 * 5157050159173695487 * 17153597302151518561 * 17904041241938148871927 * 59151549118532676874448563 * 1647072866431538116058878617811 * 87449423397425857942678833145441 * 1963672214729590922916323781834466879 * 49929707724752567469731915956762751258933207272739486748238351859309991348433 * 40393566547943595749562506243285884534929026356774912763863482259566537671583290150415083011252727505582091 * 245646981125691497673324668265536334044341262452177697864695233686173498977525877540362298849614068695233671 * 29792282327632127192280512714312339494458105715740509816040019161219528270861465666941470299423164525021764760664757557501816665197191248140710453823079834899917278481203481942074120698987141443607970695192539694488469929529584413885826254451155851081784465332583575562462448913571987013144129130422035667076921 * 81306434126435390369376308017426816467338589074376606953450887738659949122190481971292960301001128628269985908910250733571484380927682097166969483636698401864705393738719321415525908871375830643489767976984133538274257006197857712319103629206245907496601803359738478994241731519997695185318284051254656008033048015655006605203258597365919579712675545019366698923697429439095730189943

The one notable new factor comes from the primitive part of 2^2211+1, namely 1647072866431538116058878617811 (p31), which was found via ECM. I sent e-mail to Brent.

The factorization of the primitive part of 2^2211-1 turned out to be easy: 39799 * 12371522263 * P383


For nineteenth Mersenne prime M19 = 2^4253-1, the factorization of M19-1 is not complete because of (2^1063+1) C281 and 2126M C219.


For all smaller Mersenne primes, the factorization of M-1 may be accomplished from the Cunningham tables and other known sources. Of particular note are M18-1 (2^3217-2), which uses Arjen Bot's continuation of the Cunningham tables for 2^1608+1. M16-1 (2^2203-2) was made possible by this year's factorization by Silverman of 2^1101+1.

For the next Mersenne prime, 2^9689-2 looks out of reach. I found 8683987649357777 * C1227 for the primitive part of 2^4844+1 (assuming I calculated the primitive part correctly).

For the thirty-first Mersenne prime 2^216091-2 = 2*(2^216090-1), and 216090 = 2*3*3*5*7*7*7*7 which is 7-smooth, so 2^216090-1 might be an entertaining target for an ElevenSmooth-like project.

[Batalov]

(32·10^145+13)/9 = 355..557 = 37 · 3197004779 . C135
So, while cracking this C135...

> echo 300581646694173102754989957730357512537506582488896855340284419687744533384371572427030444849066320698471799488342573090961576245101059 | ~/bin/ecm -nn -B2scale 2 -sigma 1379705753 3e6
GMP-ECM 6.2.1 [powered by GMP 4.2.4] [ECM]
Input number is 300581646694173102754989957730357512537506582488896855340284419687744533384371572427030444849066320698471799488342573090961576245101059 (135 digits)
Using B1=3000000, B2=11414255590, polynomial Dickson(12), sigma=1379705753
Step 1 took 18333ms
Step 2 took 15497ms
********** Factor found in step 2: 1331962064897051431769453993617935390404440387816273704654346513
Found composite factor of 64 digits: 1331962064897051431769453993617935390404440387816273704654346513
Probable prime cofactor 225668323907862446214856345291670054173175166345927281111241519005977043 has 72 digits
Exit 10

Interesting, huh?

P.S. C64 = P30 . P34 (this is not important)