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Old 2009-07-03, 22:10   #1
Uncwilly
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Default 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....
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Old 2009-07-03, 22:58   #2
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Quote:
Originally Posted by Uncwilly View Post
what is the largest known k for a mersenne factor?
I bet two cents on 12108866376764397550161541865427518343038164488614374465439786407548571064766916617789179175969.
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Old 2009-07-03, 23:07   #3
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That number doesn't look too random.

Last fiddled with by Uncwilly on 2009-07-03 at 23:15
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Old 2009-07-04, 01:14   #4
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M487 has a factor with
Code:
k=84223018442647198252247712842154124067432692050617704182692120114149910097123865414637942529610347859467874274040269282312690175933098693564
That's 140 digits compared to ckdo's 95. I demand my two cents, ckdo!
Note that it has a smaller factor at k=5, so if you're restricting the query to smallest or non-largest factors, this one is excluded. (along with the rest I found here)
M1361 has a factor with:
Code:
k=12076886490053771065629428923805062632505373341069237464845439461102852813188005941197361329525867047950220904577332491798763952666073775918116482314948046414184092789065935655867091414347270600229052391558611823954583022546003127918295376515927661762878917550660356035433720414456776764202732155585641946789342054509601215
323 digits.
M1997 has a factor with:
Code:
k=8887636612434642293413323952745179210379088814858192078819218716982274188303607897062087246718438289092978102449686726574383849158784163402923419829024588632528041377425525647725618473812621891901558870255540306254031338646151565970697132291383891498121785270546179831619019008023864851082561554086447410669885269313331861241158271857813469282132517413683074988716723782579678291895559180729267173256201468158777356814638053196201039561137552822182661637471861841672368924954219327391866489230209981093772018895075769406080253230891706099
538 digits.
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 smallest-factor-only front: M727 has a 98-digit 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^x-1, 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 Mini-Geek on 2009-07-04 at 01:18
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Old 2009-07-04, 02:03   #5
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And lest my question was unclear, I meant the largest k for the smallest factor of a mersenne number.
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Old 2009-07-04, 02:51   #6
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Quote:
Originally Posted by Uncwilly View Post
And lest my question was unclear, I meant the largest k for the smallest factor of a mersenne number.
Oh, ok. Then ckdo's 95 digit k for M727 is the record. The smallest candidate for beating M727's k is M1061 with 320 digits. There are no known factors, so if SNFS is run and the smaller factor is at least about 99 digits, it will beat M727.
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).
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Old 2009-07-04, 04:14   #7
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Thanks!! That puts my "big" one in a harsh light, it is only ~10807438937185900. My best since v5 kicked in.
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Old 2009-07-04, 12:35   #8
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Quote:
Originally Posted by Uncwilly View Post
Thanks!! That puts my "big" one in a harsh light, it is only ~10807438937185900. My best since v5 kicked in.
Well, it's really no reason to poo-poo factors found to eliminate Mersenne candidates. A whole other category would be the largest k found by TF or P-1 (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.
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Old 2009-07-04, 16:35   #9
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Quote:
Originally Posted by Mini-Geek View Post
Well, it's really no reason to poo-poo factors found to eliminate Mersenne candidates.
No, a kill is a kill. 'Another one bites the dust.'

Mine was a P-1 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.
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Old 2009-07-04, 17:29   #10
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P-1 regularly turns up PRIME factors of more than 110 bits.

For instance : 60652538432058303954927083683377329 = 2 * 1631415054187968535232345168 * 37177871 + 1 is 116 bits, found by P-1 of course. k is 28 digits.

10807438937185900 is "only" 17 digits.

Jacob

Last fiddled with by S485122 on 2009-07-04 at 17:29 Reason: forgot to sign
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Old 2009-07-04, 22:42   #11
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I queried the database for large factors and found the following P-1 factorisation :

426315489966437174530195419710289226952407399 is a 45 digits, 149 bits factor of M17504141. k is 12177561011603973440633145599955154239, 38 digits. (P-1 : 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 2009-05-08 at 06:56 gives a k of 10820150346403195304476054699560, 32 digits.

Jacob

Last fiddled with by S485122 on 2009-07-04 at 22:42
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