Gordon Spence <[EMAIL PROTECTED]> writes,
in reply to Aaron Blosser ([EMAIL PROTECTED]):
> >Once this new one is verified, it will be interesting to see if there is a
> >prime either just below or just above it, to see if this elusive and highly
> >unverified "island" theory sticks in this case or not.
>
> I was wondering about this myself, might be worth getting a few exponents
> in the vicinity of the new number...
>
Well, _if_ (and it's still a big "if") the "island" theory holds, then we
would still expect any supposed "partner" of the new prime to have
an exponent a few hundred thousand different from the new
discovery. That's a lot of candidates, and lots of them are already
allocated.
Another approach (if you really believe in the "island" theory) would
be to use the new prime's exponent (once it's made public) to
refine the values of k & q in the predictor formula k*q^n and start
checking exponents in the 12/13/14 million range. Personally I feel
you've a better chance checking smaller untested exponents
allocated normally by PrimeNet.
BTW inspection of the tables of known primes of the forms k.2^n-1
and k.2^n+1 for odd k between 3 and 299 reveals some interesting
irregularities. Some values of k seem to generate far denser
patterns of primes than others; the density of Mersenne primes is
about average. Some values of k have distributions of primes which
are apparently much "clumpier" than Mersenne primes, however
other values of k have distributions of primes which appear to be
suspciously smooth. Without an underlying theoretical argument
as to why the "island" theory might be valid, I find it hard to explain
the distribution irregularities as anything other than statistical
artifacts based on inadequate observational data.
...................
There's been plenty of congratulations flying around - which I of
course endorse - but I feel we should also offer condolances to
Roland, Gordon and Joel for apparently being pushed down the
honours table!
With regard to the EFF prize - the new discovery (if verified) does
not represent anything radical in terms of hardware, algorithm or
exploitation of a new theorem, which is why I indicated in an earlier
message that I didn't think it would be contentious. After all, the
mathematical theory behind the Lucas-Lehmer test is "old hat", the
source code for several implementations is available for public
inspection, and several types of off-the-shelf consumer hardware
could be used to run the verification in a reasonable time.
The same might not be true for the larger EFF prize claims.
Regards
Brian Beesley
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