Hi,
from stuff I saw on the list earlier I put together this estimate:
The chance for Mp being prime if it has no factor below 2^q is
2q/p (if q is reasonably big, at i.e. 2^q much bigger than 2p+1).
If we test the Mersennes in the 10M-digits range for factors
to, say, 2^80, the chance for, say, M33219281, to be a prime
is 1/207620. So the average money you�ll earn for a test in this
range is a little less than 50c. George has estimated that a test
will take about 1 year with current hardware.
We should try to increase this probability before starting any
full LL test, perhaps by using better factoring algorithms.
Will v19 have P-1 code that works with numbers of this size?
But even with P-1, we might not be able to factor that much
further. The time consuming part of P-1 are multiplications
mod Mp just as in the LL test, so the crossover point between
raising the factoring limit and starting the LL test will be a function
of p (the number of interations in the LL test) and less than linear,
that is, we might not get very far.
Do we have estimates already what the optimal bound
values will be for trying P-1 on a given Mp, to minimize the
(time of P-1 test) + (time of LL test * chance of not finding any factor) ?
What would REALLY help here is a transform that allows multiplying
without loss of accuracy (mod Mp), even if you can�t add/sub in the
transformed data. P-1 only exponentiates, you if could exponentiate
every transformed element and then transform back... talk about
bound values! I don�t know of any such transform, if you do, please
say.
Ciao,
Alex.
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