On 7 Feb 00, at 22:40, Wojciech Florek wrote:
>
> Hi all!
> Due to some reasons I've considered numbers in a form
> 3*2^n (3,6,12,....)
> and I've found that almost in each interval 3*2^n..3*2^(n+1)
> there are one, two or three exponents of Mersenne prime.
Isn't this really just saying that Mersenne primes have a similar
distribution to the series k*2^n?
I thought we already had a hypothesis suggesting that there should be
about 1.4 Mersenne primes per octave - on average - which is actually
a slightly more informative version of the hypothesis based on the
observation reported here.
> The first two: 2,3 are below or equal 3*2^0.
> `Almost' means that there is a true gap for n=6:
> there are no exponents between 3*64=192 and 3*128=384.
So the hypothesis has a counter-example ...
Sorry, I'll try to be more constructive.
If you have a hypothesis that there are, on average, k Mersenne
primes per octave but that the distribution is random, if you sample
the number of Mersenne primes per octave (starting at _any_ point)
then you should get something like a Poisson distribution with mean
k. It might be, from the limited sample we have, starting the
sampling interval at 3*2^n (as opposed to q*2^n for some other q)
gives a better/smoother fit than others - I don't know, I haven't
tried - but, in any case, the sample size we have to go on is
pitifully small for testing the hypothesis.
> The other possible gap is for n=20 3*2^20=3145728..6291456,
> but this is a reminder of the v17 bug (???).
We haven't fully searched this interval yet, and double-checking is
nowhere near complete. I'd guess that the bad v17 results have been
redone long since.
> Among 36 considered exponents (without 2,3) 25 can be written
> as 3*2^n+p OR (sometmes AND) 3*2^n-p, where p is a prime.
> On the other hand,
> 11 exponents are expressed as 3*2^n +/- c, where c is a composite
> number. I've considered only differences with interval
> limits. The smallest is 2203=3*512+(5*149)=3*1024-(11*79).
> The others are: 2281,11213,44497,86243,110503,132049,216091,
> 756839,859433,1257787,2976221. Note that the two largest known
> exponents are
> 3021377=3*2^20-124351 [prime!] 6972593=3*2^21+681137 [prime!]
This is an interesting observation. Do we have a handle on how likely
it is that an arbitary number of a similar size can be represented
this way? I have a (probably incorrect, gut) feeling that the
"composites" are under-represented.
However, even if we found a relationship like this which is true for
all known Mersenne primes, we wouldn't be sensible to use it as a
criterion for eliminating exponents without a decent proof (not just
a hand-waving argument) as to why the relationship _must_ hold for
Mersenne prime exponents.
Regards
Brian Beesley
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