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2010-10-16, 18:34   #23
lorgix

Sep 2010
Scandinavia

61510 Posts

Quote:
 Originally Posted by Mini-Geek These "k=2" you are talking about are really k=1. Factors are of the form 2kp+1. In other words, they're mp+1, with m always even. With these factors, m=2 and k=1, since the factor is equal to 2*p+1. I'd bet that the factors of the k's break down, on average, like the factors of any natural number of about their size. And that the chance of any given k producing a factor is related to the equation given at http://www.mersenne.org/various/math.php: "(how_far_factored-1) / (exponent times Euler's constant (0.577...))". I don't know what that means precisely as far as how many k's will be smooth to such-and-such bounds, but the GIMPS Math page says "The chance of finding a factor and the factoring cost both vary with different B1 and B2 values. Dickman's function (see Knuth's Art of Computer Programming Vol. 2) is used to determine the probability of finding a factor, that is k is B1-smooth or B1-smooth with just one factor between B1 and B2. The program tries many values of B1 and if there is sufficient available memory several values of B2, selecting the B1 and B2 values that maximize the formula above."

I'll admit to this embarrassing mistake. 1=/=2

Next you are touching on some very interesting stuff. Some people might remember me and at least one other person on this forum asking about the precise algorithm for determining optimal bounds given certain conditions.

I have no reason to doubt that they break down essentially the same way, but it would be nice to see it, or have proof.

I should follow up on those references, and hope I know enough math to wrap my head around it. I love natural constants btw; memorized Pi to 1k decimal places... For Euler's I only know ~0.577215664901... which happens to be close to 3^-0.5.

 2010-10-19, 13:33 #24 lorgix     Sep 2010 Scandinavia 3·5·41 Posts Sweet, someone finished M937 less than 24hrs ago, by finding the factor: 46654722984595033623595915319018639089714063407438899506169 195bit monster.
 2010-10-19, 15:01 #25 alpertron     Aug 2002 Buenos Aires, Argentina 101110111002 Posts It appears that you did not read http://www.mersenne.org/report_facto...B1=Get+Factors . The 135-digit prime number that completes the factorization of M919 is not shown there.
2010-10-19, 15:39   #26
lorgix

Sep 2010
Scandinavia

26716 Posts

Quote:
 Originally Posted by alpertron It appears that you did not read http://www.mersenne.org/report_facto...B1=Get+Factors . The 135-digit prime number that completes the factorization of M919 is not shown there.
I don't understand what you mean...

The second largest prime factor of M937 was found yesterday. The largest prime factor is a 154digit number.

M919 is also fully factored, I don't know when that happened. But that's nice too.

 2010-10-19, 16:35 #27 alpertron     Aug 2002 Buenos Aires, Argentina 22×3×53 Posts The factor of M937 was not found by Primenet. I added it manually after copying it from http://www.mersenneforum.org/showpos...5&postcount=74 . The huge prime factor of M919 was found by Batalov and Dodson using the SNFS algorithm while the factor of M937 was found by Bos, Kleinjung, Lenstra and Montgomery using the ECM algorithm. This information was taken from the Cunningham project.
2010-10-19, 16:46   #28
lorgix

Sep 2010
Scandinavia

3×5×41 Posts

Quote:
 Originally Posted by alpertron The factor of M937 was not found by Primenet. I added it manually after copying it from http://www.mersenneforum.org/showpos...5&postcount=74 . The huge prime factor of M919 was found by Batalov and Dodson using the SNFS algorithm while the factor of M937 was found by Bos, Kleinjung, Lenstra and Montgomery using the ECM algorithm. This information was taken from the Cunningham project.
Ok, so the time I saw on PrimeNet was the time you added the factor.

Congratulations to everyone who cares, and particularly the few who went the extra miles to find the factors!

 2010-10-19, 16:53 #29 firejuggler     "Vincent" Apr 2010 Over the rainbow 22×7×103 Posts found a 'second' factor, this time for M77224373, 45'192'357'913'710'021'991 Last fiddled with by firejuggler on 2010-10-19 at 16:55
 2010-10-19, 20:12 #30 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 2·5,477 Posts M332207539 has a factor: 1078409779123917134183 (found that one about 20 months ago.) M53256451 has a factor: 1369868342874346481711 (a P-1 catch from this weekend)
 2010-10-20, 15:40 #31 lorgix     Sep 2010 Scandinavia 11478 Posts M2223187 has a factor: 99607667506275209464609 77bit k=2*2*2*2*3*101*4759*26717*36343
 2010-11-07, 08:01 #32 lorgix     Sep 2010 Scandinavia 3·5·41 Posts M3087479 has a factor: 69499497293731448560038350569 96bit k=2*2*3*1783*5501*8273*47777*241931 P-1, 5hrs ago.
 2010-11-24, 20:02 #33 cbright   Nov 2010 2 Posts M23355961 has a factor: 25246628074910152142119 k = 3^3 × 29^2 × 449 × 53011433 That was found by trial factoring. I guess the k value is about 171 times greater than the default GIMPS limit (at least when that number was first trial factored; according to the current bounds it's 684 times). So is that some kind of a record k value for trial factoring!? I was testing out mprime's ability to trial factor past the default limits and wasn't expecting it to work! The thing I don't understand is that back in 2006 I ran P-1 on that number with B1=1e6 and B2=1e8. It should've found that factor, right? B1 is easily large enough and B2 almost twice necessary. The machine was reliable; it returned lots of verified LLs and no known bad ones.

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