Robert G. Wilson v, PhD ATP writes:
The below chart gives the general lower bounds for which the limit
of powers of two begins. As an example, 2^55 for the most part, is
the least exponent as a limit for all Mersenne numbers greater than
1,040,000. There are exceptions and this is the object of this
post.
I would like to see that there are no exceptions except for the
obviously untested Mersenne numbers. Once this is accomplished,
then I would like to see the Mersenne numbers at which a particular
exponent occurs also reduced.
Exponent, Mersenne Nbrs.
52 1,400
53 400,000
54 730,000
55 1,040,000
56 1,335,240
57 1,602,275
58 2,222,000
59 2,655,000
60 3,602,100
61 4,715,000
62 5,260,000
I'm not sure that I see what you're talking about, but I'll guess,
from the table you give, that you mean nearly all Mersenne numbers
with exponents above 1400 have been factored thru 2^52, nearly all
Mersenne numbers with exponents above 400,000 have been factored thru
2^53, and so on, and you would like someone to do trial factoring to
change those "nearly all"s to "all"s, increase those powers of 2, and
decrease the corresponding Mersenne exponents.
I believe that the vast majority of trial factoring presently being
done is the factoring step of GIMPS, prior to the LL test. For
example, nearly everyone that was doing trial factoring of small
exponents (already LL tested) is now doing ECM or P-1 factoring or has
gone back to LL testing; the only two people I know of that are still
running trial factoring of already LL tested exponents are
Jean-Charles Meyrignac and myself, and I'm just filling gaps in prior
trial factoring efforts, mostly between a known factor discovered by
GIMPS and a larger factor or the next power of two. Since I can't use
George's factorer for this "gap filling", I'm using mersfacgmp, which
is considerably slower, and I've only got one CPU, which usually
spends half its time doing cofactor pseudo-prime tests, looking for
complete factorizations.
So, unless other people assist, it's going to be a slow process, with
the exception of the GIMPS factoring. On the other hand, that means
that one person, even with only one computer, could make quite a
relative contribution, if they want to. For that matter, there still
aren't many people running ECM or P-1, which are typically faster than
trial factoring for the small exponents (up to, say, the 170000 limit
of Factor98) anyway. And the advantage of ECM and P-1 is best for the
smallest exponents, where their memory usage is less; the memory usage
of trial factoring is effectively unaffected by the exponent's value.
Other progress: all factors less than 2^32 for all exponents less than
25000 should be in lowM.txt (or factoredM.txt) now; some had been
missed by GMP-ECM because the starting bound is too large to get all
factors. All cofactors for exponents under about 14000 have been
pseudo-prime tested. 1742 composite Mersenne numbers are now known to
have either a prime or a pseudo-prime cofactor.
Will
