On 26 Apr 2001, at 6:34, Hans Riesel wrote:
> Hi everybody,
>
> If 2^p-1 is known to be composite with no factor known, then so is
> 2^(2^p-1)-1.
Much as I hate to nitpick a far better mathematician than myself,
this is seems to be an overstatement.
It is certainly true that, for composite c, 2^c-1 is composite. It
does _not_ follow that no factors of 2^c-1 are known - even if c is
itself a Mersenne number.
There are certainly practical difficulties in finding such a factor.
If c = 2^p-1 is large enough that it is known to be composite but has
no known factors it's going to take some time - for example, using
trial factoring, the time to test a single factor is going to be
similar to the time taken to run a LL test on c. It would very likely
take longer to find a factor of 2^c-1 than it would to find a factor
of c. However, I suppose it's possible that someone might try, and
that (with luck and perseverance) they might even succeed, for one of
the smaller unfactored composite Mersenne numbers.
Question (deep) - if we did discover a factor of 2^(2^727-1)-1, would
that help us to find a factor of 2^727-1 ?
Regards
Brian Beesley
1775*2^332181+1 is prime! (100000 digits) Discovered 22-Apr-2001
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