On 14 Oct 99, at 9:57, [EMAIL PROTECTED] wrote:
> > #4: The Noll Island Theory is not valid. As more
> > Mersenne primes are collected, statistical effects
> > due to our small sample size will be lessened.
>
> Actually I think the "theory" is valid, at least in that we'd expect
> these sort of clusters - significance testing as you say is pretty
> much a lost cause with only 38 data.
If it sounds illogical that we have some sort of clustering
phenomenon, see Riesel, "Prime Numbers and Computer Methods for
Factorization" on super-dense clusters of prime numbers.
Sometimes you _can_ get significance from much less than 38 data
points. If you toss a coin only 8 times and it falls heads every
time, you can be 99% confident that the coin isn't fair. (You would
expect a fair coin to deliver a run of 8 matching results once in
every 128 trials)
> The problem though is it makes no difference where the next one is,
> you can always shape the conjecture to fit the existing data.
This method is widely used by astrophysicists ;-)
> Could anyone pick *one* new Mersenne prime that would either
> confirm or deny the island theory? I don't think you can. Just move
> the goalposts if you need to.
If the theory is that "Mersenne primes occur in pairs" and you define
a "pair" as being seperated from their mean by less than 5%, with a
gap of at least 50% between the higher of one pair and the lower of
the next, then it's already sunk.
If it turns out that M6972593 has no "partner" and we find one more
in the 5 million range, the statistical evidence (which is, at the
moment, tending towards significance) will be badly damaged.
Conversely, if the lowest currently undiscovered Mersenne prime turns
out to be a partner for M6972593, then the statistical evidence is
strengthened. But statistical evidence never _proves_ anything,
there's always a chance (even if vanishingly small) that the result
could arise accidentally. It's purely a question of whether you want
to risk being wrong once in 20 times (5% confidence level), once in
1000 times (0.1% confidence level), or some other figure.
But the _real_ problem is that statistics depends on sampling from a
population. The sample of Mersenne primes we have at the moment is
not only small, it's also _heavily_ biased in favour of the smaller
ones. This implies that statistical inferences from our sample _may_
be just as invalid as Gallup's use of telephone polling in the 1936
US Presidential election - in 1936, very few poorer people had
telephones.
> There's an older version of the Noll Island Conjecture that goes
> something like this. You can wait for a bus for hours and hours,
> then two come along at once. This was not called the Bus Island
> Conjecture - because it's common experience.
But there's a straightforward heuristic argument to explain the BIC,
which runs something like this. If you have a number of buses spaced
equally (in time) on a circular route, the first bus to stop to pick
up a passenger will be delayed. Passengers joining the queue "at
random" are now more likely to find the delayed bus arrives at their
stop, thus delaying it further. The delayed bus also has to stop more
frequently to let off passengers who have completed their journey.
This delays it more and more, allowing the following bus to catch up.
The following bus eventually becomes almost empty ...
In order to maintain an even schedule, it's neccessary to build in
"holding points" where a bus is artificially delayed to maintain
their spacing. If you don't do that, you might as well only have one
bus on the route. Apparently this argument is too subtle for urban
transport planners, who insist on inflicting us with "banana
specials", but that doesn't make it any less true!
If there's a similarly convincing heuristic argument as to why
Mersenne primes should occur in obvious bunches, I'm sure we'd all
like to hear about it.
Regards
Brian Beesley
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