It's not *that* arduous to factor 2^11 - 1. The possible prime factors of
the Mersenne primes 2^p - 1 are [Fermat] just (2p)k + 1, so 23 is the
smallest factor of 2^11 - 1 you have to check, and it works: 2047 = 23*89
(And even if one didn't use Fermat's result, 23 is only the fifth prime for
which any sort of calculator might be necessary.)
Incidentally, the possible factors of 2^23 - 1 are of the form 46k + 1, and
the first one you check (47) works: 2^23 - 1 = 8 388 607 = 47*178481. For
2^29 - 1, it's the second one you have to check (233), and for 2^37 - 1,
it's also the second (223). The first tough one is 2^41 - 1, which has a
smallest factor of 13367,which is 82(163) +1, the 163rd one you have to
check (well, not really; there are some obvious composites among 82k + 1,
like 165). It gets easy again for
2^43 - 1: the third one you have to check (431) works. It gets hard again
until you get to
2^73 - 1 (which has a smallest factor of 439) and 2^83 - 1 (which has 167)
For more info, see the Mersenne site, www.mersenne.org
mark snyder
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