On 2 Jun 99, at 6:03, Aaron Blosser wrote:
> 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 thought "Noll's Island conjecture" related to there being particular
zones (exponents around k*q^n for some value k and for a value q
close to 2) where Mersenne primes were more likely to be found.
There might be 1, 2 or more Mersenne primes in a particular
"island", or an expected island might be missing (e.g. the gap
between 127 and 521), and there might be Mersenne primes in
unexpected places, but nevertheless some clustering is "predicted".
At the moment we don't know enough. If the purported new
Mersenne prime has an exponent close to 6 million, it fits the
conjecture rather well; if it's closer to 7 million, the fit is less good,
but it still adds weight. Conversely, the discovery of a Mersenne
prime with an exponent around 4.5 million, or around 9 million,
would do considerable damage to the statistical evidence for the
conjecture.
In fact, although the "smooth distribution" model is hard to
disprove, it would be actually be surprising if there weren't
"clusters" of some sort in the distribution - even if we can't explain
the underlying reasons. Although it doesn't seem to apply to
Mersenne numbers in particular, there is an interesting treatment of
"irregularities" in the distribution of prime numbers in chapter 3 of
"Prime Numbers and Computer Methods for Factorization" by Hans
Riesel (2nd ed) (Birkhauser, 1994) (ISBN 0-8176-3743-5, also
3-7643-3643-5), e.g. the numbers 21,817,283,854,511,250 + p for p
= 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61 are, very
surprisingly, all prime!
>
> There are still plenty of exponents to check below 6M, so there could very
> possibly be another undiscovered prime *less* than the one being examined
> right now, although the prize is awarded to the *first* megadigit prime
> *found*, not the first one above 1M digits... :-) For that matter, there
> are still a few double-checks left to prove that M3021377 is the 37th
> Mersenne Prime, or that M2976221 is the 36th. For all we know, there's
> another one lurking somewhere in the 2M range...
True enough. Though double-checking is approaching the 3 million
range quite fast now.
>
> While the prize money would be nice, I for one would think it cool enough
> just to find one, one reason I don't mind having even some fast machines
> doing double-check work. I hope people aren't going to be discouraged just
> because there's no big juicy carrot on a stick in front of them after
> this...
Not me. Though maybe I'll switch one PII (of four) from primary
tests to double checking, and maybe start a second P100 running
Yves Gallot's proth program instead of double checks. (Actually I've
been running proth on one P100 for about 3 months now, and have
"discovered" two new prime numbers with >20,000 digits)
The point here is that there is no particular reason to suspect that
the underlying distribution of Mersenne primes should be any
different from those of numbers of the form k*2^n+1. But there are a
lot more Proth numbers in the "quickly testable" range than there
are Mersenne numbers, so we may be able to get the statistical
data neccessary to (dis)prove the theory more quickly from testing
Proth numbers than by sticking to Mersenne numbers.
Regards
Brian Beesley
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