[QUOTE=MiniGeek;233559]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 [URL]http://www.mersenne.org/various/math.php[/URL]: "(how_far_factored1) / (exponent times [URL="http://www.utm.edu/research/primes/glossary/Gamma.html"]Euler's constant[/URL] (0.577...))". I don't know what that means precisely as far as how many k's will be smooth to suchandsuch 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 B1smooth or B1smooth 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."[/QUOTE] I'll admit to this embarrassing mistake. [B]1=/=2 [/B] 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. 
Sweet, someone finished M937 less than 24hrs ago, by finding the factor:
[B]46654722984595033623595915319018639089714063407438899506169 195bit monster. [/B] 
It appears that you did not read [url]http://www.mersenne.org/report_factors/?exp_lo=919&exp_hi=&exp_date=&fac_len=&dispdate=1&B1=Get+Factors[/url] . The 135digit prime number that completes the factorization of M919 is not shown there.

[QUOTE=alpertron;233871]It appears that you did not read [URL]http://www.mersenne.org/report_factors/?exp_lo=919&exp_hi=&exp_date=&fac_len=&dispdate=1&B1=Get+Factors[/URL] . The 135digit prime number that completes the factorization of M919 is not shown there.[/QUOTE]
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. :smile: 
The factor of M937 was not found by Primenet. I added it manually after copying it from [url]http://www.mersenneforum.org/showpost.php?p=233695&postcount=74[/url] . 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.

[QUOTE=alpertron;233886]The factor of M937 was not found by Primenet. I added it manually after copying it from [URL]http://www.mersenneforum.org/showpost.php?p=233695&postcount=74[/URL] . 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.[/QUOTE]
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! :smile: 
found a 'second' factor, this time for M77224373, [SIZE=2]45'192'357'913'710'021'991[/SIZE]

M332207539 has a factor: 1078409779123917134183 (found that one about 20 months ago.)
M53256451 has a factor: 1369868342874346481711 (a P1 catch from this weekend) 
M2223187 has a factor: 99607667506275209464609
77bit k=2*2*2*2*3*101*4759*26717*36343 
M3087479 has a factor: 69499497293731448560038350569
96bit k=2*2*3*1783*5501*8273*47777*241931 P1, 5hrs ago. 
M23355961 has a factor: 25246628074910152142119
k = 3^3 × 29^2 × 449 × 53011433 That was found by [I]trial factoring[/I]. 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 P1 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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