[ExI] mersenne primes again

[Mike Dougherty]

On Wed, Mar 31, 2010 at 5:03 PM, spike <spike66 at att.net> wrote:
> One last point.  The last 8 Mersennes form a kind of hockey stick, but what
> if the ratios of the last 8 primes been anomalously large instead of
> anomalously small?  We conjecture that there are infinitely many Mersenne
> primes, but had the hockey stick been pointed up instead of down, many of us
> would (most probably incorrectly) conclude that there is a finite number of
> Mersenne primes.
>
> More later.  Assignment please: think about everything you know regarding
> the definition of statistical significance, and what it means to you.
>

Statistical significance is significant only to those who consider
statistical signficance.  :)

Qualitatively, my first observation was that there appear to be
several "steps" along the progression even before the run-of-8 that
you've highlighted.  Draw 'best fit' lines between any two consecutive
primes and see how closely they approximate another.  ex: between
X-axis 20 & 26 it looks like three run-of-3, between 33 & 37, three
'steps' of 2.  Is there a pattern to the occurrence and length of
these steps?  Maybe the "obvious" answer is no.  Perhaps the
non-obvious answer is yes.  when did the latest run become
"statistically significant"?  4 in a line, 5?  What made the 8th
occurrence special?  Should we be able to predict (with measurable
degree of confidence) that the significant run will extend to a 9th
point along the highlighted curve, or will it shift up some amount as
we've seen around 10 times previously?

Have you looked a Penrose tiling?  from two basic shapes and some
matching rules a non-periodic pattern emerges that we recognize has
having an order (pattern) but without symmetry.  Maybe the 'order' in
which Mersenne primes manifest is another example of a non-periodic
pattern; one with too few examples for us to yet uncover the
underlying rules.  ...but that's why we keep searching for more
examples, right?