P-1 bound optimisation
From SeventeenOrBust
This page is meant to serve as knowledge base for the attempt to optimise the choice of the bounds B1 and B2 for P-1- factoring in our project. In this sence, it is not a real wikipedia-article, but the features of mediawiki seem to fit better the necessities of knoledge exchange than the forum.
Feel free to add what you know, a collection of useful links would be cool, too. The discussion page might be a good place for coordination.
Basic outline
Garo has written an thorough article about factoring, but it is a bit long, so here a condensation:
Suppose I would like to factor a k,n-pair with n = N. Then there are bounds 
(given a particular computer) which optimize the ratio factors per time.
So the factoring attemt takes the time tO, and I find on average f_O factors, which save 1 < c < 2 PRP tests. I have saved cfOtN − t0 time, wheretN designs the time for a PRP-test of this N. If this value is negative, P-1 is not worth and should not be done.
This was the factor-througput-optimized case.
Now, I can use a part of my saved time t1 < cfOtN − t0 for increasing the bounds to 
, so that I find f1 < fO factors. I save cf1tN − t1 time.
I can again use a part of my remaining saved time... etc.
Somewhen, this last value becomes negative, and I have reached the project-optimal bounds. The infinitesimal gain (?) of increasing the bounds is zero.
ToDo
In order to optimise the bounds, we need first to determine
-the time it takes to run a P-1 test of one N, say 11M, as a function of B1,B2, first for a well defined machine,
-the time of a PRP-test of that very N, and
-the quantiles of the factor distribution for a lately completely sieved range. (And the count of tested k,n-pairs at all).
The K for which we have found a prime should be wiped out of the data, perhaps.
A next step would be to separate the different remaining K, but this is no urgent matter.
