Geoffrey Faivre-Malloy writes:
M16384 has a factor:
3178457030898592746194675570663374420833971377365687459461386297851584459031
8073180374859604847822828243686877928403667633015295
Further, if you try to divide this into M8192 (2^8192 - 1), you should
find that it factors that as well. That is, I checked all the factors
of the number above that my run of ecmfactor printed and they all
divided M8192 or an even smaller Mersenne (and therefore M16384,
M32768, etc).
The factoredM.txt and lowM.txt files in mersdata.tgz on my web pages
list, among other factors:
M( 256 )C: 59649589127497217
M( 512 )C: 1238926361552897
M( 1024 )C: 2424833
M( 1024 )C: 7455602825647884208337395736200454918783366342657
M( 2048 )C: 45592577
M( 2048 )C: 6487031809
M( 2048 )C: 4659775785220018543264560743076778192897
To here, the Mersenne numbers are completely factored (and are
therefore in factoredM.txt); from here on, the other (presumably
larger) factors are not known.
M( 4096 )C: 319489
M( 4096 )C: 974849
M( 8192 )C: 114689
M( 8192 )C: 26017793
M( 8192 )C: 63766529
M( 8192 )C: 190274191361
M( 8192 )C: 1256132134125569
M( 16384 )C: 2710954639361
M( 16384 )C: 2663848877152141313
M( 16384 )C: 3603109844542291969
M( 16384 )C: 319546020820551643220672513
My data contains no factors of M(32768) = M(2^15) or higher that are
not also factors of a smaller Mersenne number.
The ecm3 program of the mers package knows that composite exponent
Mersenne numbers have factors equal to smaller Mersenne numbers and
pulls all those factors out using repeated GCD's before trying to
factor the Mersenne number it is told to factor. E.g., it should
never print 319489 as a factor of M(8192) or any larger Mersenne,
since 319489 factors M(4096) but it will print 319489 as a factor of
M(4096) because it factors no smaller Mersenne number.
Will
http://www.garlic.com/~wedgingt/mersdata.tgz
http://www.garlic.com/~wedgingt/mers.tgz
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