Distribution of Mersenne primes before and after couples of primes found

[philmoore]

Originally Posted by R.D. Silverman

It doesn't matter that the observations are not scientific. What matters is that they were made in the first place.

On the contrary, there is nothing wrong with making this sort of observation, people do it all the time. Casinos make a good living from people who think that they see a pattern in wins and losses that will enable them to predict when they are due for their next win. The problem here is that no one has ever proven that the distribution of Mersenne prime exponents is described by a Poisson distribution, that only appears to be the simplest explanation. However, because we know that the distribution of primes, or the distribution of Mersenne primes, is not truly random, we really cannot rule out the possibility that there is some underlying structure not yet discerned. The gaps and irregularities are certainly intriguing, but the central question is whether or not you would see similar sorts of gaps and irregularities in any randomly generated Poisson sequence. Have fun, Emily, and don't be too concerned by the comments of RDS; he's the forum's old-time schoolteacher who raps your knuckles with a ruler when you give the wrong answer. Of course, to him, he's not rapping your knuckles, he's rapping your answer!

[R.D. Silverman]

Originally Posted by philmoore

However, because we know that the distribution of primes, or the distribution of Mersenne primes, is not truly random, we really cannot rule out the possibility that there is some underlying structure not yet discerned.

Rule out? We know that there is a structure. Any sieve shows
the structure. But that is not the issue here.

The phrase "not truly random" is meaningless here because "truly random"
has not been defined. If the gaps between Mersenne primes can be proven to
conform to a Poisson process, then under any reasonable definition the
gaps are 'truly random'. The next gap is unpredictable.

The gaps and irregularities are certainly intriguing, but the central question is whether or not you would see similar sorts of gaps and irregularities in any randomly generated Poisson sequence. Have fun, Emily, and don't be too concerned by the comments of RDS; he's the forum's old-time schoolteacher who raps your knuckles with a ruler when you give the wrong answer.

She did not give "the wrong answer". What she gave was the equivalent
of presenting astrology at a physics convention. Her prose
"was not even wrong".

[Uncwilly]

Originally Posted by R.D. Silverman

She did not give "the wrong answer". What she gave was the equivalent of presenting astrology at a physics convention. Her prose "was not even wrong".

Firstly astrology is not a physics issue per se, it is an astronomy and medical/psychological issue.
Secondly, did not astrology (which should have been the proper term for astronomy) lead to astronomy?
Just be cause it seemed to you to be astrology, does not mean that it is of no value as a step to learning for the presenter. It is the duty of the more knowledgeable, if and only if they choose to speak up, to shepherd the presenter to a position of knowledge, knowledge sufficient so that they understand why astrology is wrong. Part of this self-assigned duty is to present the teaching in such a way that the newbie does not become so offended that they shut down to instruction.

I would ask Bob that he practice biding his time and let others the first couple of chances with newbies like Emily.

[Uncwilly]

RDS is the one on the right:
http://xkcd.com/386/

[Dubslow]

[xilman]

Originally Posted by R.D. Silverman

If the gaps between Mersenne primes can be proven to conform to a Poisson process, then under any reasonable definition the
gaps are 'truly random'.

Now there's an interesting question which perhaps you can answer for me. My mathematical skill and knowledge ar undoubtedly less than yours.

How would you go about proving that the gaps between Mersenne primes conform to a Poisson process?

Note: I'm not asking for a proof (though one woud be wonderful), only for a sketch of how such a proof may be attempted.

[aketilander]

Originally Posted by xilman

Now there's an interesting question which perhaps you can answer for me. My mathematical skill and knowledge ar undoubtedly less than yours.

How would you go about proving that the gaps between Mersenne primes conform to a Poisson process?

Note: I'm not asking for a proof (though one woud be wonderful), only for a sketch of how such a proof may be attempted.

Well I would like to expose my ignorance, but I have been wondering one thing:

There is a proof that there are at least 1 prime between X and 2X. If the distribution of primes were a true Poisson Distribution there would be a very, very, very small porobability that there were 0 primes between X and 2X.

So my question is: Is this not a proof that the distribution of primes is not a true Poisson Distribution only a distribution very similar to a Poisson Distribution?

I suppose to a matematician this is a dumb question, but even though I would like to pose it and hopefully learn someting from the answer.

[R.D. Silverman]

Originally Posted by xilman

Now there's an interesting question which perhaps you can answer for me. My mathematical skill and knowledge ar undoubtedly less than yours.

How would you go about proving that the gaps between Mersenne primes conform to a Poisson process?

Note: I'm not asking for a proof (though one woud be wonderful), only for a sketch of how such a proof may be attempted.

Noone knows. It may use a Tauberian approach or ergodic methods
similar to those uses by Tau & Greene to prove that there are arbitrarily
long AP's of primes. It might use analytic methods. Noone knows
how to even approach such a proof.

[R.D. Silverman]

Originally Posted by aketilander

Well I would like to expose my ignorance, but I have been wondering one thing:

There is a proof that there are at least 1 prime between X and 2X. If the distribution of primes were a true Poisson Distribution there would be a very, very, very small porobability that there were 0 primes between X and 2X.

Contrawise:
And if the distribution of primes was not a Poisson process then
there would be a very small probability that there were 0 primes between
X and 2X. Indeed. Since the probability is zero in either case
your statement basically says nothing.

So my question is: Is this not a proof that the distribution of primes is not a true Poisson Distribution only a distribution very similar to a Poisson Distribution?

You do not seem to know what a mathematical proof is. Your question
is ill-posed at best and nonsense at worst, especially since "similar
to a Poisson Distribution" is meaningless.

[TimSorbet]

Originally Posted by aketilander

There is a proof that there are at least 1 prime between X and 2X. If the distribution of primes were a true Poisson Distribution there would be a very, very, very small porobability that there were 0 primes between X and 2X.

Without already knowing that fact, how many primes would you suppose there are between 1000 and 2000? None or a great deal? It's pretty obvious there'd be a great deal. The proof that this extends infinitely means it at least might not be a true Poisson distribution, but doesn't preclude it being extremely similar.

[aketilander]

Originally Posted by Mini-Geek

Without already knowing that fact, how many primes would you suppose there are between 1000 and 2000? None or a great deal? It's pretty obvious there'd be a great deal. The proof that this extends infinitely means it at least might not be a true Poisson distribution, but doesn't preclude it being extremely similar.

Yes, you got my point. And if the distribution of Primes is not really a true Poisson distribution, only a distribution which is "extremely similar", it would of course be impossible to prove that it is a true Poisson distribution I suppose.