20061013, 23:27  #1  
Sep 2006
Odenton, MD, USA
10100100_{2} Posts 
P95 Trial Factor speeds 40M vs 100M
I'm running Prime95 V24.14 on a XP SP2 P4 2.6GHz . I've noticed that it takes longer to trial factor a 40.5M exponent from 2^60 to 2^62 bits then doing a 100M exponent (approx 16mins vs approx 7mins). Just wondering why?
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20061014, 03:49  #2 
"Mark"
Feb 2003
Sydney
3×191 Posts 
Trialfactoring smaller exponents takes longer because there are more potential factors in a given size range eg 2^60 to 2^62. This is because any factor has to have the form q=2*k*P+1, where P is the prime exponent itself and k is an integer >=1. So for smaller exponents, potential factors are closer together.

20061014, 11:15  #3 
"Oliver"
Mar 2005
Germany
11×101 Posts 
Take a look here: http://mersenne.org/math.htm
as markr allready said: there are less candidates for factors in 100M mersennes from 60 to 62 bits (factor size) as in 40M mersennes. As a rule of thumb there are ~2.5 times more factor candidates for a 40M than for a 100M mersenne number in a specific range (factor size). On the other hand testing a single candidate for a 100M mersenne takes only one step longer than testing a single candidate for a 40M mersenne number. 26 steps for ~33M to ~67M 27 steps for ~67M to ~134M The number of steps for a single candidate is ceil(log2(exponent)). The duration of on step itself depends only on the size of the candidate. Since they are same size for your example each step takes the same ammount of time. 
20061015, 04:38  #4 
Sep 2006
Odenton, MD, USA
164_{10} Posts 
Thanks for the information. I've been reading the references and at first they didn't make a lot of sense to me. But, the more I look at the form q=2*k*P+1 and think about it, the more it is beginning to make sense.

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