> Anyway, still waiting to hear if ECM will,
> eventually, find all factors or if it leaves "factors" in the range...
The best way of thinking about this is that each curve at a given bound has
a small but non-zero probability of finding a factor of a certain length,
assuming that one exists. There are a very large number of curves
available, and so the probability of missing the factor on *all* the curves
can be made extremely low --- but non-zero. This handwaving approach is
far from rigorous but it is an extremely good approximation of the truth.
So yes, in principle it can leave factors undiscovered. In practice, it
will find them within a reasonable time --- but depending on your luck that
time could be short or it could be lengthy. A few examples from personal
experience might be illustrative. The largest factor I've found with ECM
was a 45-digit factor of 423*2^423+1 and was found in phase 1 with the
curve bound set at 3 million. I estimate I had a roughly one in twenty
thousand chance of finding that factor in the number of curves I ran. At
the other extreme, I recently used MPQS to finish off a factorization of an
89-digit integer, only to find that one of the factors had 30 digits. I had
already found a 33-digit factor of the number for which the C89 was the
cofactor, but missed the P30. I estimate that I had a less than 1% chance
of missing the smaller factor given the amount of ECM work I had done. In
the first case I was lucky, and the second I was unlucky. That's the nature
of ECM.
Paul
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