20200808, 16:33  #1 
Aug 2020
2·3·5 Posts 
Does ECM benefit from the trial factoring limits?
I know nearly null about the math, but I've read that like P1 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 P1 method?

20200808, 18:37  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{2}×7×307 Posts 
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.

20200808, 19:52  #3  
Aug 2020
2·3·5 Posts 
Quote:
At this point I'm starting to think it as simply a waste of GHzhours. If we can start giving assignments with B1=8e8 instead (this is the largest B1 listed at https://www.mersenne.org/report_ecm/, 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 2e4. Still very small, but that is already a 20x improvement. 

20200808, 20:09  #4 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2194_{16} Posts 
Maybe James H can look at the server code that does the ECM assignments. It might be time to tweak it.

20200808, 20:14  #5 
Sep 2002
Oeiras, Portugal
1,399 Posts 
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.

20200808, 20:26  #6 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2^{2}·7·307 Posts 
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 Last fiddled with by Uncwilly on 20200808 at 20:32 
20200808, 21:18  #7  
Aug 2020
2·3·5 Posts 
Quote:
Oh you're right. I also misinterpreted the graph at http://www.wraithx.net/math/ecmprobs/ecmprobs.html. 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. 

20200808, 22:19  #8 
Sep 2002
Oeiras, Portugal
1,399 Posts 
If you run 4700 ECM curves with B1=3e6, the chance of missing a 40digit factor is ~ 1/e (if there is one...) . Same holds for the various Size/B1 value / number of curves combinations in the table.

20200808, 22:50  #9 
Aug 2020
2×3×5 Posts 
Conclusion: Not of much use
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.
Last fiddled with by Ensigm on 20200808 at 23:16 
20200809, 03:51  #10  
"Curtis"
Feb 2005
Riverside, CA
19·227 Posts 
Quote:
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 T30sized 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, trialfactoring limits are usually 74 bits or lower and starting at B1=5e4 makes sense. So, in practice for GIMPSfactoring, TF doesn't influence our ECM choices. 

20200809, 04:04  #11 
P90 years forever!
Aug 2002
Yeehaw, FL
7,151 Posts 

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