20200730, 09:23  #1 
Jul 2020
14_{8} Posts 
Are ECM curves with B1=50000 useful at all?
I recently did some ECM factoring (worktype 5, first factors of Mersenne numbers) on my laptop, mainly because those assignment do need less memory and finish fast (laptop does not run 24/7). Also, I like factors more than a LLresidue :)
However, I'm doubtful that those B1=50000 assignments will be useful for numbers like M2655361:

20200730, 10:05  #2 
"Oliver"
Sep 2017
Porta Westfalica, DE
2^{2}×3^{2}×13 Posts 
The interesting and useful part is it's stage 2 with 5,000,000.
In two days (e.g.), I found three factors to the following numbers in the 2M range with B1=50,000 (and B2=5,000,000): 2526553 2399933 2560849 That was lucky, of course. But I've been doing that for a long time now and found a lot of factors in that time. Last fiddled with by kruoli on 20200730 at 10:05 Reason: Semantic error. 
20200730, 10:14  #3  
Jun 2003
1335_{16} Posts 
Quote:
Quote:
250 curves at 50K is equivalent to 50 curves at 250K (in terms of effort). Even if we jump directly to 250K, you're saving 50/600 or about 8%. Hardly worth the complications in adding more rules to the server. Also, it is not like 50k ECM will magically stop at 25 digits. There is a small chance that it can find bigger factors (maybe as big as mid 30 digits). 

20200730, 10:15  #4  
"Oliver"
Sep 2017
Porta Westfalica, DE
1D4_{16} Posts 
Furthermore, welcome to the forum!
Quote:
Unforunately, that's the Gamblerâ€™s Fallacy. I hate that, too, but you cannot ignore it. 

20200730, 10:16  #5 
"Oliver"
Sep 2017
Porta Westfalica, DE
2^{2}×3^{2}×13 Posts 

20200730, 10:49  #6  
Jul 2020
2^{2}·3 Posts 
Thanks for your feedback. My assumption was that a single ECM curve is equally likely to find a factor like a P1 run, but for P1 computations are easier and you additionally gain extra "B1" because you can multiply the p of 2*k*p+1 Mersenne factors. Therefore, if we already did a P1 test with B1=2000000, the chance that a new ECM curve with B1=50000 will find something is probably very unlikely. And that's what I have experienced, never found a factor with this work type so far (with P1, quite a few). But at least it seems possible, given kruoli's examples!
Quote:
I know, but this is unlikely and we would find the factor with a larger curve, too. It's quite hard to optimize the ECM bounds given P1/trial factorization bounds, even if you have a M.Sc. in math :) Last fiddled with by keisentraut on 20200730 at 10:56 

20200730, 15:03  #7 
"Curtis"
Feb 2005
Riverside, CA
3×1,579 Posts 
You're welcome to manually set your ECM runs to 250k in that range; the server will convert them to a sortofequivalent number of 50k curves for the running tally.
While it won't change your chances of success much, it will improve your chances of finding a larger (say, 28+ digit) factor by a bit. You might do, say, 50 curves at 50k, and then move up to 250k. 
20200730, 15:10  #8 
"Curtis"
Feb 2005
Riverside, CA
3×1,579 Posts 
Consider the information you have before starting ECM versus the information you have after 1000 curves at B1=50k.
Before ECM, you have whatever TF data there is to tell you no factors below e..g 70 bits. There is the usual probability of a factor between 70 bits and 25 digits. After 1000 curves of ECM, you have ruled out factors less than or equal to 25 digits with 1  1/e^4 probability or better (the 4th power comes from the curve count being 4 times the "expected" number of curves for 25 digit factors). So, after 1000 curves you are quite sure there isn't a missed factor below 25 digits. But it's not like the 1000th curve is what convinced you each curve you run without finding a factor decreases the chance there is a small factor to be found. It doesn't decrease the chances by much that's why we run hundreds or thousands of curves at each level. 
20200730, 15:22  #9 
"Oliver"
Sep 2017
Porta Westfalica, DE
111010100_{2} Posts 
Thanks for the clarification.

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