20090703, 22:10  #1 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
10101000110000_{2} Posts 
Largest known k?
Idle curiosity:
We all know that it is not the size of the factor that counts, rather the size of the k. So, I was looking at results data, for my account. And got woundering, what is the largest known k for a mersenne factor? (What known factor yields the highest k?) I googled, no luck. Again, just idle curiosity.... 
20090703, 22:58  #2 
Dec 2007
Cleves, Germany
2×5×53 Posts 

20090703, 23:07  #3 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
10800_{10} Posts 
That number doesn't look too random.
Last fiddled with by Uncwilly on 20090703 at 23:15 
20090704, 01:14  #4 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
2×3×23×31 Posts 
M487 has a factor with
Code:
k=84223018442647198252247712842154124067432692050617704182692120114149910097123865414637942529610347859467874274040269282312690175933098693564 Note that it has a smaller factor at k=5, so if you're restricting the query to smallest or nonlargest factors, this one is excluded. (along with the rest I found here) M1361 has a factor with: Code:
k=12076886490053771065629428923805062632505373341069237464845439461102852813188005941197361329525867047950220904577332491798763952666073775918116482314948046414184092789065935655867091414347270600229052391558611823954583022546003127918295376515927661762878917550660356035433720414456776764202732155585641946789342054509601215 M1997 has a factor with: Code:
k=8887636612434642293413323952745179210379088814858192078819218716982274188303607897062087246718438289092978102449686726574383849158784163402923419829024588632528041377425525647725618473812621891901558870255540306254031338646151565970697132291383891498121785270546179831619019008023864851082561554086447410669885269313331861241158271857813469282132517413683074988716723782579678291895559180729267173256201468158777356814638053196201039561137552822182661637471861841672368924954219327391866489230209981093772018895075769406080253230891706099 I'm not 100% sure, but I doubt that's the largest known one, but Syd's DB is not really working right at the moment, so I won't look for any larger ones for now. On the smallestfactoronly front: M727 has a 98digit smallest factor, and...hm...is that ckdo's k? It's about the right size, I'll check...Yep! That's the one. In a quick glance in Syd's DB for p<2000 (in 2^x1, x prime, ignoring Mersenne primes) I think that's the largest k for a smallest factor. But you didn't specify that it had to be the k for the smallest factor, only for a factor. Last fiddled with by MiniGeek on 20090704 at 01:18 
20090704, 02:03  #5 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{4}×3^{3}×5^{2} Posts 
And lest my question was unclear, I meant the largest k for the smallest factor of a mersenne number.

20090704, 02:51  #6  
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
2·3·23·31 Posts 
Quote:
The candidates with p<2000 are M1061 (320 digits) M1237 (373 digits) M1277 (385 digits) M1619 (488 digits) M1657 (499 digits) M1669 (503 digits) M1753 (528 digits) The current record for SNFS factorization is M1039 (313 digits), and it was not semiprime (a 23 bit factor was known but SNFS was run on M1039 instead of GNFS on the cofactor). 

20090704, 04:14  #7 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
25060_{8} Posts 
Thanks!! That puts my "big" one in a harsh light, it is only ~10807438937185900. My best since v5 kicked in.

20090704, 12:35  #8 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
1000010110110_{2} Posts 
Well, it's really no reason to poopoo factors found to eliminate Mersenne candidates. A whole other category would be the largest k found by TF or P1 (and not SNFS or GNFS). I don't know of any easy way to find this. Yours just might be a good position on this list! Also, there aren't any that are anywhere near as large as M727's, (I think about 64 digits is the runner up, I think I recall seeing a p66 as a smallest factor of one of 'em) even among the NFSd ones, because there are usually small factors.

20090704, 16:35  #9  
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{4}·3^{3}·5^{2} Posts 
Quote:
Mine was a P1 on a 40M number (79.4 bit equiv.). I have been getting raw factors about 2 digits shorter up in the 332M range. Our "competition" reports the k, as opposed to the factor, found. 

20090704, 17:29  #10 
"Jacob"
Sep 2006
Brussels, Belgium
3477_{8} Posts 
P1 regularly turns up PRIME factors of more than 110 bits.
For instance : 60652538432058303954927083683377329 = 2 * 1631415054187968535232345168 * 37177871 + 1 is 116 bits, found by P1 of course. k is 28 digits. 10807438937185900 is "only" 17 digits. Jacob Last fiddled with by S485122 on 20090704 at 17:29 Reason: forgot to sign 
20090704, 22:42  #11 
"Jacob"
Sep 2006
Brussels, Belgium
5·7·53 Posts 
I queried the database for large factors and found the following P1 factorisation :
426315489966437174530195419710289226952407399 is a 45 digits, 149 bits factor of M17504141. k is 12177561011603973440633145599955154239, 38 digits. (P1 : 2 x 3 x 191 x 307 x 593 x 839 x 3593 x 3989 x 4051 x 6691 x 152429 x 2349679 x 17504141) If you insist on factorisations since v5 went on line : 865479351028708028377260890582073995281 at 39 digits, 130 bits, reported on 20090508 at 06:56 gives a k of 10820150346403195304476054699560, 32 digits. Jacob Last fiddled with by S485122 on 20090704 at 22:42 
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