20110226, 00:44  #1 
Bemusing Prompter
"Danny"
Dec 2002
California
2382_{10} Posts 
error rate and mitigation
As we know, each LL test has a small chance of returning an incorrect residue. From what I've seen, the error rate is in the neighborhood of 0.01 to 0.05. Suppose that the average rate is around 0.01 for a 35M exponent (per this thread).
Because the result of each iteration depends on that of the previous one, a single error will result in the entire test being incorrect. Thus, every single iteration will need to be correct in order for the final residue to be currect. Assuming a 0.99 chance of a good LL test, each iteration would have a p = 0.99999999971 chance of being correct. It would be reasonable to assume that each iteration has an equal chance of generating the correct residue. In terms of statistics, the 0.01 probability of a bad LL test is equal to: where n is the number of iterations, and q = 1  p. All else being equal, the error rate will increase as the exponent increases. In actuality, the error rate will probably be even higher because longer iterations will have a higher chance of going bad. For example, a 350M exponent would probably have less than a 0.9 chance of being correct. That having been said, what kind of factors can lead to bad LL tests? From what I gather, the most common causes are:
However, does anyone know if the following can affect the error rate?
As for ways to mitigate such errors, does anyone effective are Prime95's builtin errorchecking features? In other words, can I expect a much lower error rate if I enabled roundoff and/or SUM(INPUTS) errorchecking? Do both of these features add security, or are they redundant? That is, would it be a good idea to enable both? I've also heard that ECC memory can help reduce the risk of errors. Does anyone have any experience with them? 
20110226, 05:53  #2 
Bemusing Prompter
"Danny"
Dec 2002
California
2·3·397 Posts 
I forgot to mention the following factors:

20110228, 18:54  #3  
Sep 2002
Oeiras, Portugal
3×487 Posts 
Quote:
http://www.mersenneforum.org/showthr...t=13476&page=1 

20110411, 22:06  #4 
Bemusing Prompter
"Danny"
Dec 2002
California
2·3·397 Posts 
I still have a few unanswered question, though. For example, would frequent reboots have an impact on the error rate? Would a different algorithm be more or less likely to generate an incorrect result?

20110412, 02:14  #5  
Dec 2010
Monticello
5×359 Posts 
Quote:
The algorithm with the fastest time to run or smallest memory footprint, all else being equal, has the least likelihood of an incorrect result since it minimises the opportunity for error. Beyond that, failure of TF,ECM, or P1 to find a factor when one should be found results in the LL and LLD checks, and, as a result, has a negligible effect on the likelihood of failing to identify a mersenne prime. If these processes result in factors, those are checked quite quickly by the server when reported, I believe. 

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