Chris Nash <[EMAIL PROTECTED]> writes, in reply to my earlier message
> >Similarly I can find no statistically convincing evidence, even at
> >the 5% level, that the "Noll islands" really do exist.
>
> I wonder how many more "confirming instances" we'd have to find before we
> could even get a result as statistically weak as 5%? My statistical
> background is pretty awful but I know that these sort of analyses are order
> sqrt(n)... maybe we'll be having the same discussion in a "few years" time
> after M(148) is discovered and the sample size is four times larger... of
> course M(148) will probably have 10^24 digits...
It depends what we find. If we were to find a group (say 1, 2 or 3)
Mersenne primes around 6 million & another group around 12
million, with empty "oceans" between them, then we would
probably get a result significant at 5%. (Which means that there
would be less than 1 chance in 20 of the Noll island theory being
wrong). On the contrary, if we start finding Mersenne primes where
Noll expects oceans, and nothing where Noll expects islands, then
the statistical significance would actually fall with more data.
For prime numbers in general, there is a theorem (the Prime
Number Theorem) which predicts the statistical probability that a
randomly-selected integer will be prime; the value is proportional to
the inverse of the natural logarithm of the number selected.
I do not believe that there is any reason for believing that Mersenne
numbers are in any way non-randomly selected in this respect,
except for the obvious one that they are all odd, which simply
doubles the probability predicted by the PNT.
If we want more data to examine the Noll island theory, then we
should be able to get it from the distribution of large primes of other
forms e.g. k*2^n+1 - unless these can be ruled out for some
reason, for which I can think of no particular justification. They're all
odd, too, so the PNT would predict that the probability of 2^p-1 and
k*2^q+1 being prime should be the same, after scaling by their
natural logarithms.
There are a number of values of k for which k*2^n+1 has been
tested for primality over quite a large range of values of n - over
300,000 in the cases k=3, 5 and 7.
I leave the detailed computation, and the drawing of the neccessary
inferences, as an exercise.
Regards
Brian Beesley
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