At 07:34 PM 7/1/99 +0100, you wrote:
>My reading of the "island" theory is that the centre of the next
>"island" should be closer to 6 million than 7 million.
How so? If indeed p=6972593 is one of a pair in an island, then it is not
the middle. It is either the higher or the lower of the two. Since the
"pairs" have tended to fall approximately within 0.88 of each other (and
sometimes much closer), that means that if M38 is the upper of a pair, p *
.88 = 6135881 as the potential "lower bound". If M38 is assumed the lower
of a pair in an island, p / .87 = 7923401 as the potential upper
bound. There is currently no reason to assume 1) That the Island theory
holds or 2) That if it does, that M38 is speifically the higher or the
lower of the "pair". I choose to believe that the theory holds, "just
because it suits", and that M38 is the LOWER of the pair that will result,
again, "just because that still leaves me in the running" for finding a
HIGHER number, since I have several machines testing potential numbers in
that range.
What a selfish reason to believe a mathematical theory, eh? Then again,
this is all for fun, so since the money's been won, I want to find a bigger
prime, as we all do. ;-)
>Therefore, if
>"M38" has a mate, the "island" theory predicts that "M38" is the
>higher of a pair.
How so? We don't know WHICH of the bookends M38 might be... The number
that GIMPS recently found MIGHT be M39, with M38 still lurking (or even M40
-- we haven't exhaustively searched yet). How can you state one way or
the other, even IF the island theory holds, which end of the island the new
discovery is?
> _If_ a pair exists. _If_ the "island" theory really
>does hold water. I'd very much like to see the "island" theory
>proved; however, to the best of my knowledge, there isn't even a half-
>formed heuristic argument as to why "islands" should exist - it's
>just an observation of a (statistically insignificant) pattern.
Might big ifs, to be sure. However, the larger the "gaps" between the
supposed islands become, the more apparent the theory will be -- if the
theory holds. If we do find M39 in the range of +/- 10% of M38, it is not
conclusive proof of the Island theory, but one more nail to secure the
foundation.... I surmise we'll need far more statistical data before we
can even talk with reasonable confidence about Island Theory, and even
then, we'll lack mathematical/heuristic proof.
>There are still plenty of untested exponents in the 6 millions, so
>there's plenty of opportunity to find another Mersenne prime
>somewhere amongst them. Also, we aren't sure that there are no
>Mersenne primes in the high 3 millions, the 4 millions or the low to
>mid 5 millions; finding one in the "gap" would do the "island theory"
>serious damage.
Indeed. I'm predicting there are none, "just because it suits". TO me,
this is a much better prediction than the now-disproven "There are exactly
37 Mersenne Primes, no more and no less". ;-) Fun. Just For Fun.
Are we having fun yet?
Are we there yet?
Can I have a drink of water?
>Irrespective of the "island" theory, I'll predict that exactly one
>more Mersenne prime will be found with an exponent less than 10
>million, but I wouldn't risk more than $10 on a bet at even money
>odds, or venture to hazard an estimate of the value of its exponent
>in the event that my prediction turns out to be true.
I think you are correct, and I think its exponent will be consistent with
island theory.
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