20190730, 04:50  #1 
"Juan Tutors"
Mar 2004
2^{4}×5×7 Posts 
About LL test cycles
I assume we know that we can test for LL test cycles pretty easily. Instead of checking for repeated residue, we can store the units digit of the residue after each cycle in ram or on hard drive, and periodically check for repeating digits in this string in less than 1 second, and then verifying a true cycle exists by running a number of steps equal to the cycle length. Is the reason we don't do that that the probability is so low from what we can tell?

20190730, 05:17  #2 
Jun 2003
3^{4}·5·13 Posts 

20190730, 15:39  #3 
"Juan Tutors"
Mar 2004
2^{4}×5×7 Posts 

20190730, 23:12  #4  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
3·19·107 Posts 
Quote:
See https://www.mersenneforum.org/showpo...1&postcount=10 

20190731, 03:58  #5  
"Juan Tutors"
Mar 2004
2^{4}·5·7 Posts 
Quote:


20190731, 04:27  #6 
Romulan Interpreter
"name field"
Jun 2011
Thailand
2^{3}×3^{2}×137 Posts 
Do you mean like 330 times slower than the LL test itself is done?
(Think about the fact that each iteration takes, say, 3 milliseconds, in an average processor, for an average exponent.) Joking apart, and from mathematical point of view, the length of a cycle would be closely related to the znorder of the factors. And smallest factors are larger than the exponent, by at least a factor of 2 (for p=4k+3) or 6 (for p=4k+1). If m turns out prime, there are no cycles (beside of the trivial 22222). For composites, possible cycles are larger than the number of iterations we do. We'll never find one... (even the trivial cycle for a prime, happens AFTER the test is finished) Last fiddled with by LaurV on 20190731 at 04:36 
20190731, 04:39  #7 
"Juan Tutors"
Mar 2004
2^{4}×5×7 Posts 
True! Perhaps much faster than that. And looking for a cycle is probably significantly faster than looking for other things. Nearly free, except for the fact that the string has to be stored, so it does take SOME memory and circles per iteration. Really depends on how useful or interesting the question is. I would estimate that it would add perhaps a minute per LL test based on absolutely nothing.

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