Brian Beesley wrote:
>Suppose we multiply 2^727-1 by 100!. The reason for doing this is
>that this just about guarantees that there is some combination of
>factors a, b such that (a-b) << sqrt(a). Therefore we should be able
>to use Fermat's method (or anything else which might be better) and
>get a factorization in a reasonable time.
But the guarantee is not there.
your a ~ sqrt(2^727 * 100!) ~ 10^188
sqrt(a) ~ 10^94
There are (only) 2^100 ~ 10^30 ways to choose the combinations of factors
of 100! , which are the degrees of freedom. But they need to be chosen so
that a-b << 10^94 for a (and b) near 10^188. For f1, a factor of 2^727-1,
near 10^100 it will be nearly impossible (not guaranteed) for (f1 * some
combination of the factors of 100!) to be within one part in 10^94 of a
certain number ~10^188. The combination of factors of 100! need to be
within 10^-6 of a certain number of size ~ 10^88. 10^30 chances at hitting
a number ~10^88 to within 10^-6 leaves you several dozen orders of
magnitude short of a guarantee.
Best regards,
Todd Sauke
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