> Of course, the sequence that still remains unknown is
>
> 2
> M(2)=3
> M(3)=7
> M(7)=127
> M(127)=170141183460469231731687303715884105727
> M(170141183460469231731687303715884105727)=???
Yes, this sequence is interesting, but if someone finds a way to prove/
disprove the primality of M(M(127)), I think that would be far more
significant than actually proving/disproving that specific number prime.
(Assuming you don't find a factor, that is).
> I checked Chris Caldwell's pages on this, and Curt Noll's trial-factored
> M(M(127)) to 5.10^50, surprisingly low considering the size of M(127)
> itself, I noticed many other M(M(p)) as listed in
> http://www.garlic.com/~wedgingt/MMPstats.txt have only been tested to very
> low limits indeed.
The reason is relativly clear: the work of checking *even one* factor of
M(M(p)) is greater than the work required for an LL test on that number.
This is because of the need for p squarings modulo some number greater
than M(p).
> I wondered why there wasn't more work done on these - though I understand
> it's very hard to motivate people when Guy's law of small numbers no doubt
> applies, but everything M(M(61)) and above is currently unknown. It would be
> nice to see a few more results there.
I'm guessing that if a more optimized program were created, then perhaps
there would be more interest. The Selfridge prize for these numbers could
help. ...However, we have no way of knowing wether or not these numbers are
prime, unless we find a factor. Interestingly enough, when we find the next
Mersenne prime, searching for a factor of M(M(p)) might allow us to find an
even bigger prime. If for example, 6*M(p)+1 divides M(M(p)), then it must
be prime!
Wait, that might just be the reason to search! Will only searched up to
k=4 for M(M(6972593)), but if 2*k*M(p)+1 divides M(M(p)), then you've just
beaten the world record! Non-Mersenne's might once again grace the top
10 list!
-Lucas
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