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Old 2012-08-21, 07:44   #3
Basketry That Evening!
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"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

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Originally Posted by fivemack View Post
Which second derivatives are you talking about? If you mean 'why are plots of log(B1) against number of digits concave', that is a corollary to the expected runtime of ECM being sub-exponential in the number of digits of the factor.

These figures from gmp-ecm come from the mathematics rather than from computer technology - you get slightly different ones if you optimise measured runtime rather than number of curves
No, he means this: Take, e.g., B1:

5e4 -- ratio to previous 4.5
25e4 -- ratio to previous 5.0
1e6 -- ratio to previous 4.0
3e6 -- ratio to previous 3.0
11e6 -- ratio to previous 3.67
43e6 -- ratio to previous 3.91

If you plot those ratios, the second derivative (which should be zero) swings from positive to negative and back, without every really settling at zero.

The same thing happens with necessary curves -- the ratio changes for each digit jump.

I've actually had this same question myself more than once (which is probably why I understood him).
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