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Old 2004-06-03, 08:24   #4
geoff's Avatar
Mar 2003
New Zealand

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Originally Posted by dave_dm
Has anyone looked into working out such an algorithm?
I tried to work this out, I am not an expert so there may be some point I have missed:

Say you have done the expected number of curves to bound B1=m (and saved the stage one residues), and the next higher digit size d has optimal bound B1=n.

Let p,q be the probability of one curve finding a factor of size d with B1=m,B1=n respectively. Let r,s,t be the time to complete stage one to B1=m, stage one to B1=n, and stage two from B1=n respectively.

Then it is worth continuing from the saved residue if (q-p)/(s-r+t) > q/(s+t), or equivalently if the ratio (q-p)(s+t) / q(s-r+t) is greater than 1.

As an example I tried curves on 2^971-1 with a P4 2.66GHz and Prime95 (B2=100B1) using n=11e6, m=43e6. I got the probabilities p=1/79353, q=1/18461 by using Richard McIntosh's Maple program <> with factor size 50.5, and the resulting times were r=151, s=588, t=289 seconds. The ratio came to 0.927, so in this case it is better to start a fresh curve than continuing from the saved one.
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