At 08:22 AM 6/19/99 -0600, you wrote:
> >There is a conjecture that the nth Mersenne exponent resulting in a prime
> >is approximately (3/2)^n. linear relationship. The linear
> regression >parameters yield the relation M(n) = 1.4796^n + c, where c
> is a small >constant. 1.4796 is pretty close to 3/2.
>
>So, based on this conjecture, what would you guess M38 to be (roughly)?
>Should we start a pool to see who can guess the closest to the real
>exponent? And those of you who know for sure cannot participate! :-)
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.
Starting at about M13, you see that there are indeed islands:
13 521 \______ island centered at 564
14 607 /
15 1279 standalone
16 2203 \______ island centered at 2242
17 2281 /
18 3217 standalone
19 4253 \______ island centered at 4338
20 4423 /
21 9689 \______ island centered at 9815 (11213 MAY be part of this)
22 9941 /
23 11213 standalone
24 19937 \
25 21701 >----- island centered at 21615
26 23209 /
27 44497 \______ POSSIBLE island at 65370
28 86243 /
29 110503 \______ island centered at 121276
30 132049 /
31 216091 standalone
32 756839 \______ island centered at 808136
33 859433 /
34 1257787 \______ island centered at 1328028
35 1398269 /
?? 2976221 \______ island centered at 2998799
?? 3021377 /
Now, fit the center of the ISLANDS that have more than one member into a
linear progression:
564
2242
4338
9815
21615
65370
121276
216091
808136
1328028
2998799
Graph that you, and you get a VERY close linear fit. I haven't done it
mathematically, but at first glance a guestimate (first suggested by Noll,
IIRC) is that Mersenne Primes are distributed into islands ROUGHLY a linear
double from the last island, with the occasional empty "gap" (such as the
region around p=400000, where if island theory held, we should have
expected a couple primes).
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,
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...
IMO, the longer it takes us to hear the official word on M38, the more
likely it is NOT a valid prime, but some dimwit trying to submit a zero
residue for prize money. If the "confirming test" comes back NON-prime,
then several additional tests will have to be done to find out for sure,
and that will take longer.
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