ECM Stage 2 RAM

[VBCurtis]

Originally Posted by storm5510

Could you define "level" in this context?

See https://www.mersenne.org/report_ecm/. Each column is a "level", a set of curves expected to find a factor of a specific size. It is customary to jump B1 to a higher level once the prescribed number of curves are complete at a lower level.

Citing ATH's B2 from GMP-ECM isn't very relevant for Prime95. They're two different programs using two different algorithms to do ECM. I will say that when I used P95 to do ECM, I used B2 = 150* B1, but I'm not sure it was faster at finding factors than the default B2 = 100* B1. I disagree that stage 2 time should be as long as stage 1 time; stage 2 taking anywhere from 40% to 70% of stage 1 time is perfectly normal for both GMP-ECM and Prime95.

If your goal is to maximize probability of finding a factor per unit of computation, you should not mess with the assignments as given. There are changes that might be a tiny bit better, say 10% more likely to find a factor per unit time, but there are lots of settings that will be quite a bit less efficient such as setting bounds suitable for a factor size much larger than you're likely to spend the time to run enough curves to find. If you're assigned curves with B1=50,000 and decide to use B1=500,000, your factors-found-per-curve will be much much better, but factors-found-per-time will be quite a bit worse (because you're running curves that take 10 times as long but not finding 10 times as many factors).

[storm5510]

Originally Posted by VBCurtis

See https://www.mersenne.org/report_ecm/. Each column is a "level", a set of curves expected to find a factor of a specific size. It is customary to jump B1 to a higher level once the prescribed number of curves are complete at a lower level...

This report is somewhat hard for me to follow. The first column B1 is 50,000 and the second is 250,000. What about the B1's in between? I would take this as meaning there are no bounds left to run below 50,000.

My HP is running assignments as given by the server. No changes. What I am doing with this Asus is experimentation. What I run is not assigned. I just comb through the existing records to find something with very little ECM work done.

[VBCurtis]

There is no particular reason to run B1's in between. You can if you like, but bookkeeping is easier if folks run t25 curves (B1=50k) or t30 curves (B1=250k). There's nothing wrong with picking arbitrary B1 values, but there is little to gain either. They're called t25 and t30 because those B1 sizes are most efficient for finding factors of those respective sizes. After the t25 column is "done", it is more time-efficient to target larger factors even though it's not impossible that a 25 digit factor is still out there.
Larger-bound curves will still find factors of a smaller size, and there is little harm in choosing B1 one level higher than the current column. Or, for that matter, any B1 between the current level's B1 and the next-higher B1.
Running curves smaller than recommended is a waste of time. Running curves larger increases the chance per curve to find a larger factor, at the cost of decreasing the chance per day of finding a relatively smaller factor.
The site I linked converts curves at odd B1 to the current wavefront linearly; that is, if your candidate has curves listed on the 50k column, and you run a curve at 100k, it'll count as two curves at 50k. It also takes twice as long as one curve at 50k.

[storm5510]

I found this a few nights ago when I was trying different things:

M250259 has a factor: 7756917629147385999280835759 (ECM curve 87, B1=16000, B2=2921135)

I was running a 100 curve test. I was not sending any of my experiments to the server, and was deleting the results files. I forgot the .spl file. So all of it went in. I should have been paying more attention.

I looked at the assignments on my HP. B1 is either 250,000 or 1,000,000. This goes along with the link to the report pages. I cannot say that I fully understand it, but enough to see how it is working.

[ATH]

Originally Posted by storm5510

This report is somewhat hard for me to follow. The first column B1 is 50,000 and the second is 250,000. What about the B1's in between? I would take this as meaning there are no bounds left to run below 50,000.

My HP is running assignments as given by the server. No changes. What I am doing with this Asus is experimentation. What I run is not assigned. I just comb through the existing records to find something with very little ECM work done.

https://www.mersenne.org/report_ecm/

Those "optimal" values was calculated >20 years ago. They are calculated so after ~280 curves at B1=50,000 there are "only" 1/e ~ 37 % risk/chance of missing a 25 digit factor and 63% chance of finding it, if there is one.

Then you go to B1=250,000 at the 30 digit level, but those curves will also find any missing 25 digit factor or even smaller factors. After ~640 curves there are again 63% chance of finding any 30 digit factor and now there are >99% chance of having found any 25 digit factor.

And so on at each digit level the calculated number of curves assures 63% chance of finding the factor at that level while assuring practically 100% chance of any factors at the earlier levels. Factors between the "levels" for example 28 digits and 32 digits will have difference odds of being found but will eventually be found at the next "level".

[storm5510]

Originally Posted by ATH

https://www.mersenne.org/report_ecm/

Those "optimal" values was calculated >20 years ago. They are calculated so after ~280 curves at B1=50,000 there are "only" 1/e ~ 37 % risk/chance of missing a 25 digit factor and 63% chance of finding it, if there is one.

Then you go to B1=250,000 at the 30 digit level, but those curves will also find any missing 25 digit factor or even smaller factors. After ~640 curves there are again 63% chance of finding any 30 digit factor and now there are >99% chance of having found any 25 digit factor.

And so on at each digit level the calculated number of curves assures 63% chance of finding the factor at that level while assuring practically 100% chance of any factors at the earlier levels. Factors between the "levels" for example 28 digits and 32 digits will have difference odds of being found but will eventually be found at the next "level".

Prime95 uses random numbers in the ECM process, as everyone knows Given that, I would guess that it would be nearly impossible to be 100% certain that every possible factor is found for any exponent. If a person were to keep increasing the bounds, perhaps a larger factor could be found.

I noticed something last night that I did not know. Someone had written that B1 was only used in Stage 1, Both B1 and B2 are used in Stage 2. Consider the sample below:

ECM2=<ID>,1,2,<Exponent>,-1,250000,25000000,50,<Knowns factors>

I saw that Stage 1 will begin with a very low "at prime" value and works its way up to something just short of the B1 value. Stage 2 starts just above the B1 value and proceeds to something near, or at, the B2 value. Basically, it is a continuous flow of primes, only breaking to grab more RAM in order to finish. A lot of this made more sense to me when I realized how it was working. Last week, I ran into an instance of curve 1 repeating because I had not allocated enough memory for the test.

As for the probability percentages, I am not an advanced mathematician. I know enough to do what I need. My area of expertise is hardware. I believe I can say that after 30 years of experience. If that doesn't make sense, consider the fact that I am 63 years old.

[ATH]

Originally Posted by storm5510

Prime95 uses random numbers in the ECM process, as everyone knows Given that, I would guess that it would be nearly impossible to be 100% certain that every possible factor is found for any exponent. If a person were to keep increasing the bounds, perhaps a larger factor could be found.

Only way to be 100% sure is if you find a factor and the remaining co-factor is a PRP or a Prime.

But if you run all the curves for 25digit and 30digit level you can be "nearly 100%" sure there are no factor at or below 25 digits, and if you then also run the curves for the 35 digit level, you can be "nearly 100%" sure there are no factors at or below 30 digits and so on.

[CRGreathouse]

Originally Posted by ATH

But if you run all the curves for 25digit and 30digit level you can be "nearly 100%" sure there are no factor at or below 25 digits

Right. I get about 99.9895% for GMP-ECM 7 at the default settings, using Alexander Kruppa's script.

Originally Posted by ATH

if you then also run the curves for the 35 digit level, you can be "nearly 100%" sure there are no factors at or below 30 digits

Yep, about 99.967%. My ugly code for both:

nr=1; \\ approximate as 1, not very important
1-(1-ecmprob (50000, 12746592, 5e24, nr, 2))^261*(1-ecmprob (250000, 128992510, 5e24, nr, -3))^513
1-(1-ecmprob (50000, 12746592, 5e29, nr, 2))^261*(1-ecmprob (250000, 128992510, 5e29, nr, -3))^513*(1-ecmprob (10^6, 1045563762, 5e29, nr, -6))^1071

[storm5510]

Everyone here will think I am nutty when I say that I keep a record of everything I run, except TF. So far this month, on this machine, I have ran 1,103 ECM's. 26 tests found a factor. That is 2.35% All of these had B1's of either 250,000 or 1,000,000. i didn't check the numbers on the HP. I suspect it would trend the same way.

Edit. Counting what the HP has done, drops the percentage to 1.7

[storm5510]

Originally Posted by axn

What are you trying to do? What is wrong with just completing the assignment you were given?

What was I trying to do? Learn something. I do not see a problem with that.

[PhilF]

My question is: Is there any difference in the probability of finding a factor when performing ECM on a small Mersenne number vs. a larger one (assuming plenty of memory is available)?