On 16 Oct 99, at 7:15, Jukka Santala wrote:
> I'm going to play clueless here, and without so much as a scientific reasoning
> note that since "we all know" composite exponents can't yield a Mersenne prime...
> is it somehow possible to "factor" this into any estimate on where the next
> Mersenne prime is? The most obivious way to play with this would seem to be to
> deal with a set of only the prime Mersenne exponents for the statistical play.
> However, I'm going to leave that for somebody else to dissect and blow apart as
> to why it wouldn't work, or try out on their statistical software. Other thoughts
> are trying the statistical approach either on the exponent itself, or the length
> of the whole expanded number, both within the set of all Mersenne numbers and
> only Mersenne numbers with prime exponents. And see if any patterns turn up.
They will, and they're very likely to be artificial.
You can analyse results of tossing a coin all you like; eventually,
you will find some trend which is statistically significant - even if
the coin really is fair. Knowledge of this trend will not help you
one iota in predicting the outcome of the next toss of the same coin.
If you could find something like "it's not worth trying exponents
congruent to 21 mod 22" from analysis of existing known Mersenne
primes, chances are that you'd be wrong; we just haven't found any
yet, our sample size being painfully small. However, it might be
worth putting a little effort into a mathematical argument as to why
the conjecture might be valid.
Note - the conjecture above may well have at least one known
counterexample - it's just an example, I didn't bother checking it!
Regards
Brian Beesley
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