[QUOTE=wblipp;381275]I'm not sure about the way I handled the small factor issue.[/QUOTE]

It looks like this approach is a bit optimistic on the number of small factors. I used factordb to look at about 200 exponents near 10[SUP]3[/SUP] and again near 10[SUP]4[/SUP]. Both ranges appear to have had thorough discovery of small exponents. Both ranges had within one standard deviation of the expected number of factors from 10[SUP]9[/SUP]to 10[SUP]13[/SUP], indicating the Poisson approximation is working well away from smallest factors. But they were 2.4 and 1.3 standard deviations smaller than expected for the number of factors up to 10[SUP]9[/SUP]. A better fit would be to model as no factors through another factor 10[sup]0.4[/sup] from the exponent.

[QUOTE=pdazzl;381276]The 10th prime of the 7M # was 18 digits. So the probability that this scenario was possible would be based off ln(18/7)? Then probability multiplied by the number of primes between 1M and 10M?[/QUOTE]

log[SUB]10[/SUB]7508981 = 6.88, so the expected number of factors through 10[SUP]18[/SUP] is ln(18/(6.88+0.4))=0.91. The probability of more than 9 such factors is 4.52*10[SUP]-8[/SUP].

This gives enough information to calculate the probability of success for most search strategies. To complete a comparison, you will need some model of the computational effort of various strategies. As mentioned earlier, I have determined that "select a bound and ECM to that bound until a number with 11 known factors is found" is both slow and stupid. A strategy of "ECM to some low bound, then extend promising candidates to higher bounds" will be much faster, but it remains to be determined the optimal number of tranches, the optimal tranche boundaries, and whether this is expected to be faster than working on 2[sup]7508981[/sup]

[QUOTE=wblipp;381301]



log[SUB]10[/SUB]7508981 = 6.88, so the expected number of factors through 10[SUP]18[/SUP] is ln(18/(6.88+0.4))=0.91. The probability of more than 9 such factors is 4.52*10[SUP]-8[/SUP].
[/QUOTE]


I calculate 586082 primes from 1M to 10M (1000003 is the 78499th prime number, 10000019 is the 664580th prime number).

That means the odds of any exponent even existing with more than 9 prime factors in the 1M-10M prime exponent range is about 2.6% ie: 4.52*10[SUP]-8[/SUP] multiplied by 586082 potential prime exponents from 1M-10M.

That sounds like this number is an outlier in terms of the odds.

[QUOTE=pdazzl;381306]That means the odds of any exponent even existing with more than 9 prime factors in the 1M-10M prime exponent range is about 2.6% ie: 4.52*10[SUP]-8[/SUP] multiplied by 586082 potential prime exponents from 1M-10M.[/QUOTE]

A. You mean with more than 9 prime factors less than 10[sup]18[/sup]
B. The odds change over the range - smaller exponents mean smaller primes are possible. At 10[sup]6[/sup] the expected number of primes increases to 1.032 and the probability of more than 9 such primes is 1.51*10[sup]-7[/sup].

[QUOTE=wblipp;381332]A. You mean with more than 9 prime factors less than 10[sup]18[/sup]
[/QUOTE]

Yes

[QUOTE=wblipp;381332]
B. The odds change over the range - smaller exponents mean smaller primes are possible. At 10[sup]6[/sup] the expected number of primes increases to 1.032 and the probability of more than 9 such primes is 1.51*10[sup]-7[/sup].
[/QUOTE]

I see what you mean and was starting to think about that after I posted. Not sure the proper way to average it out across the whole 1M-10M range. Though even taking that best case probability at the 1M mark and applying across the 1M-10M spectrum that is still around 8.8% probability that one even exists in there. That's why I'm thinking this one exponent may be a blue lobster that's worth throwing ECM at.

11th factor ECM'd! And only 25 digits. I'm running a PRP on the cofactor now.

[url]http://www.mersenne.ca/factor/1211907173840894224264391[/url]

Known prime factors (11 factors, 495.9 bits, 0.00660464% known):

At least 13 factors.....

[code]M7508981/known_factors is not prime. RES64: 2A94BAB0984542ED. We4: 2E4250B3,00000000
Known factors used for PRP test were: 1211907173840894224264391,18694135089678809,20333239254737,26356523311,281287549065522023,285341279,346309182073938289,367107436768162151,45053887,585700519,60071849[/code]

[offtopic]
Please use code sections or insert some spaces. Those long lines are impossible to read!
[/offtopic]


done.

[QUOTE=pdazzl;381347]11th factor ECM'd! And only 25 digits. I'm running a PRP on the cofactor now.

[url]http://www.mersenne.ca/factor/1211907173840894224264391[/url]

Known prime factors (11 factors, 495.9 bits, 0.00660464% known):[/QUOTE]

So the probability to find the prime factor is not very small as other people said before in this thread.

[QUOTE=alpertron;381356]So the probability to find the prime factor is not very small as other people said before in this thread.[/QUOTE]

Idiot.

If I were you, instead of making that response, I would investigate whether the probabilities are higher than you expected before or this is a stroke of good luck.

[QUOTE=alpertron;381358]If I were you, instead of making that response, I would investigate whether the probabilities are higher than you expected before or this is a stroke of good luck.[/QUOTE]

Idiot^2