> Also one must take into account the "pseudo-conjecture" that Curt Noll (I
> believe -- if I've misattributed, please forgive) has made about
> "Mersenne
> Islands". If you look at the distribution, they tend to clump, like
> galaxy clusters, with large voids between the islands.
On a whim, I figured I'd do a rough calculation of the %deviation from each
exponent to it's "center", based on the numbers you gave and see if they fit
any sort of pattern. Then, if you find one, you might be able to narrow
down where a potential "Gilligan" (Island Buddy) would be (I figure the
first one found would be "Skipper"). :-) I hereby release any claim on my
made up terms for island partners :-)
> Starting at about M13, you see that there are indeed islands:
>
> 13 521 \______ island centered at 564
> 14 607 /
~ 8.25% from center
> 16 2203 \______ island centered at 2242
> 17 2281 /
~ 1.7%
> 19 4253 \______ island centered at 4338
> 20 4423 /
~ 2.0%
> 21 9689 \______ island centered at 9815 (11213 MAY be part of this)
> 22 9941 /
~ 1.3%
> 24 19937 \
> 25 21701 >----- island centered at 21615
> 26 23209 /
~ 7.8%/7.4% (for smallest and largest exponents...a bit iffy)
> 27 44497 \______ POSSIBLE island at 65370
> 28 86243 /
~ 31.9% (this one is iffy also)
> 29 110503 \______ island centered at 121276
> 30 132049 /
~ 8.9%
> 32 756839 \______ island centered at 808136
> 33 859433 /
~ 6.3%
> 34 1257787 \______ island centered at 1328028
> 35 1398269 /
~ 5.3%
> ?? 2976221 \______ island centered at 2998799
> ?? 3021377 /
~ 0.8%
I'd say that from looking at the deviations between pairs in the "islands",
I'm not so sure we can call them pairs. The "groupings" we see between
them, especially given the "standalones" is more likely to be purely
coincidental and perhaps a bit of a projection on our part. When analyzed
strictly, I don't see any pattern there. Some of the definitions of pairs
are a bit fast and loose also. Perhaps we should only term two Mersenne
primes a pair if the exponents are less than 2% apart from their
average...just as a general guideline? Meaning that a good number of those
"pairs" above are nothing near each other, especially M27 and M28.
> Given this fit, I will estimate that Curt is correct, and that the
> potential M38 is in the region of the next island (where heavy testing
> recently occured) which should be around 6,300,000. If the
> pattern holds,
Perhaps the "center" points of the pair examples you gave appear to have a
loose linear correlation, but I think we may be playing a bit fast and loose
with the definition of a pair here...
> M39 may be relatively close to M38, too, so we ought not be too long in a
> fifth success, assuming M38 is valid, too, and not someone's
> cruel idea of
> a hoax...
It'd be silly for anyone to try and cheat the system, knowing there would be
double and triple checks on any claim. But then again, some people are
rather silly.
Aaron
________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
