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Old 2014-08-03, 04:06   #4
wblipp's Avatar
May 2003
New Haven

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Originally Posted by R.D. Silverman View Post
Wrong approach. What needs to be computed is the conditional probability that a large integer N has k prime factors given that it has no factors less than (say) N^1/a, for given a.
That's a good approach for the scenarios where you can calculate that distribution - you can start directly with the residual composite. The only way I know to calculate that distribution is to use the inclusion-exclusion and normalization approach described in your paper with Wagstaff. That works for cases where the total number of factors is a handful. But some cases of interest will have "a" of hundreds or thousands.

For these scenarios, I think you would be better off to start with the entire original number and the distibutions I described, then use Bayes to account for the known factoring efforts.

But it doesn't really matter what I think or you think - this is a question of comparing heuristics that can studied empirically. Perhaps we can get the OP to propose some "interesting" cases.
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