Does ECM benefit from the trial factoring limits?
 

I know nearly null about the math, but I've read that like P-1 method, the ECM method needs higher bounds in to find larger factors but takes longer time to run. Does knowing the trial factoring limits provide benefit to the ECM bounds and/or the number of curves to try, like the case in P-1 method?

If you know that no factor has been found and that there has been TF work done up to a certain size, yes. ECM can then be done looking for factors larger than the largest factor searched for via TF.

[QUOTE=Uncwilly;552938]If you know that no factor has been found and that there has been TF work done up to a certain size, yes. ECM can then be done looking for factors larger than the largest factor searched for via TF.[/QUOTE]

Yeah, my thought is that we can use this to optimize assignment rules. Take M35731 as an example: the server is still actively giving out assignments with 3e6 as B1 (I just did one, haha), despite TF has shown it has no factor under 2^65; while if [URL]http://www.wraithx.net/math/ecmprobs/ecmprobs.html[/URL] is correct, even if 4700 curves are performed under that B1 value, the chance of finding a factor larger than 2^65 is 9.75e-6 (You can drag to select a range on that graph).

At this point I'm starting to think it as simply a waste of GHz-hours. If we can start giving assignments with B1=8e8 instead (this is the largest B1 listed at [URL]https://www.mersenne.org/report_ecm/[/URL], although in this case we should obviously use even higher bounds), and assume the time needed for a curve is proportional to B1, then the same CPU power can be used to test 17 curves, giving a success rate of roughly 2e-4. Still very small, but that is already a 20x improvement.

Maybe James H can look at the server code that does the ECM assignments. It might be time to tweak it.

I think you are confusing digits and bits. The TF was performed to 65 bits, and the probability you quoted is for finding a 65 digits factor using B1 = 3e6.

2^65 for that number is in the range of 2,324,976,294,838,206,465 (19 digits)
A B1 of 2000 and 500 curves is around 19 digits

[QUOTE=lycorn;552948]I think you are confusing digits and bits. The TF was performed to 65 bits, and the probability you quoted is for finding a 65 digits factor using B1 = 3e6.[/QUOTE]


Oh you're right. I also misinterpreted the graph at [url]http://www.wraithx.net/math/ecmprobs/ecmprobs.html[/url]. The "Success Chance" should mean the probability of finding (i.e. not missing) a factor in an interval if there is a factor there, not the overall probability of finding a factor.

If you run 4700 ECM curves with B1=3e6, the chance of missing a 40-digit factor is ~ 1/e (if there is one...) . Same holds for the various Size/B1 value / number of curves combinations in the table.

Conclusion: Not of much use
 

:picard: So the conclusion is that TF limits are too small to help anything in tweaking ECM parameters. Even if we have limits to 2^80 (25 digits), it is not obvious how much this will change the optimal crossing point of 5e4 and 2.5e5.

[QUOTE=Ensigm;552970]:picard: So the conclusion is that TF limits are too small to help anything in tweaking ECM parameters. Even if we have limits to 2^80 (25 digits), it is not obvious how much this will change the optimal crossing point of 5e4 and 2.5e5.[/QUOTE]

False- if you know TF was done to 82 bits, you know to not bother with ECM at the 20-digit level, nor at the 25-digit level. Skip right to T30 (B1=25e4).

Similar conclusions can be made about TF to, say, 79 bits- that rules out factors under 24 digits, so I would run less of a T25 and jump to T30-sized curves sooner. In B1 terms, less than a full set of curves at 5e4 and jump to B1=25e4 sooner.

However, it's rather unlikely to have a candidate number that one would TF to 78+ bits *and* consider ECM on. For the size of number for which we ECM, trial-factoring limits are usually 74 bits or lower and starting at B1=5e4 makes sense. So, in practice for GIMPS-factoring, TF doesn't influence our ECM choices.

[QUOTE=Ensigm;552944]Take M35731 as an example:.[/QUOTE]

This small exponent should be done by a prime95/GMP-ECM combination