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Old 2009-08-02, 16:21   #1
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Default prime distribution near mersenne primes

from which resource can we get prime distrubution near mersenne primes
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Old 2009-08-02, 17:09   #2
10metreh
 
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Quote:
Originally Posted by Unregistered View Post
from which resource can we get prime distrubution near mersenne primes


What on earth do you mean? Do you mean "are the primes still randomly distributed around Mersenne numbers"?

Mersenne primes (and similar primes like Riesel primes) are the only primes we can find at these very high levels, due to the Lucas-Lehmer test and similar algorithms. As far as we know, there is nothing unusual about the distribution of primes around Mersenne primes.

Let me ask: do you mean Mersenne primes lower down (eg M521) or ones like M43112609?

Last fiddled with by 10metreh on 2009-08-02 at 17:12
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Old 2009-08-02, 18:55   #3
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Here are some basic statistics from a program I just whipped up. It looks below and above a given Mersenne prime in a range large enough that you'd expect to see about 1000 primes, then counts the primes in those ranges. The expected rage is something like 970 to 1030. Results, in the format [below, above]:

2^61 - 1: [1049, 967]
2^89 - 1: [999, 1003]
2^107 - 1: [975, 976]
2^127 - 1: [986, 986]
2^521 - 1: [967, 999]
2^607 - 1: [985, 999]

The expected (Poisson) deviation for a sample this size is sqrt(12000) ≈ 110. The actual deviation was 109. So I'd say that's pretty usual.

Last fiddled with by CRGreathouse on 2009-08-02 at 19:00 Reason: ≈ lest some pedant complain
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Old 2009-08-02, 20:15   #4
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The next two are
2^1279 - 1: [978, 1054]
2^2203 - 1: [1056, 1024]
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Old 2009-08-02, 20:53   #5
R.D. Silverman
 
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Quote:
Originally Posted by Unregistered View Post
from which resource can we get prime distrubution near mersenne primes
The question is at best poorly posed.

What do you mean by "prime distribution"?

The actual distribution of prime numbers is not known. There are
some open questions, but there is insufficient evidence to even
characterize these questions as conjectures.

We do have some Tauberian type theorems. The Prime Number
Theorem is one such: The number of primes between 2 and N is
asymptotically N/log(N). [a better approximation is LI(N)].
However, their *distribution* over the interval [2,N] is NOT KNOWN.

Please do not make the mistake of confusing a counting function
(essentially a cumulative distribution function) with a distribution
(i.e. density) function.

And there is no reason AT ALL to believe that the distribution of primes
around Mersenne primes will be any different than the distribution of
primes in general. The question itself is "wrong headed".
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Old 2009-08-02, 21:02   #6
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Quote:
Originally Posted by CRGreathouse View Post
Here are some basic statistics from a program I just whipped up. It looks below and above a given Mersenne prime in a range large enough that you'd expect to see about 1000 primes, then counts the primes in those ranges.

Monsieur: I would have expected better from you. The original poster
asked about the *distribution* of primes. You have discussed the
prime counting function. The CDF is not the same as the PDF. The
OP asked for the latter. You have replied with the former.

Knowing the count of primes in an interval is NOT the same as knowing their
distribution. (as you know).
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Old 2009-08-02, 21:59   #7
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I was of course trying to give some sensible interpretation to the original question. I though it was likely that it was prompted by the gambler's fallacy or the inverse gambler's fallacy, that finding one prime made it less (resp., more) likely that its neighbors were prime.

I chose 1000 expected primes as a 'small' interval that was still large enough to analyze. Perhaps if I had chosen a smaller range and considered the exact placements in that range you would have been more satisfied. Of course with a small enough range there are certain patterns based on the value of a Mersenne prime mod small primes.

Frankly, I'm accustomed to interpreting questions that are phrased imprecisely loosely.
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Old 2009-08-02, 23:02   #8
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Quote:
Originally Posted by R.D. Silverman View Post
The original poster asked about the *distribution* of primes.
... but, based on context, may not have composed a completely accurate wording. Interpreting it too tightly would, in that case, be a mistake.

If CR's initial interpretation is wrong, that can be straightened out through continued interaction with the questioner.

Last fiddled with by cheesehead on 2009-08-02 at 23:02
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Old 2009-08-02, 23:22   #9
R.D. Silverman
 
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Quote:
Originally Posted by cheesehead View Post
... but, based on context, may not have composed a completely accurate wording. Interpreting it too tightly would, in that case, be a mistake.

.
I disagree. Part of learning mathematics is learning to ask the
right questions.

Part of learning mathematics is learning that words such as 'distribution'
have an exact meaning.

If I answer a question as asked, but it is not what was meant,
it [b]should[b] be an impetus for the OP to learn (or ask) WHY his/her question was wrong. And, IMO, if it is NOT such an impetus then the
OP should not have asked in the first place.

My assumption, upon reading a mathematical question using the word
'distribution' is to assume that it has its correct meaning. To do otherwise
would be an insult to the original poster, because interpreting it otherwise
would imply that the OP did not know what he/she was talking about.

Since I know nothing about the OP or his background, I prefer to
believe that the word 'distribution' was used correctly.

Last fiddled with by R.D. Silverman on 2009-08-02 at 23:23 Reason: typo
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Old 2009-08-02, 23:47   #10
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Quote:
Originally Posted by R.D. Silverman View Post
If I answer a question as asked, but it is not what was meant, it [b]should[b] be an impetus for the OP to learn (or ask) WHY his/her question was wrong.
Yes, but I was referring to CR's answer.

Quote:
And, IMO, if it is NOT such an impetus then the OP should not have asked in the first place.
... or else your answer's being not what was meant was not the ideal impetus for that particular questioner.
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Old 2009-08-03, 15:41   #11
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Quote:
Originally Posted by cheesehead View Post
Yes, but I was referring to CR's answer.

... or else your answer's being not what was meant was not the ideal impetus for that particular questioner.
When I was an undergrad (and even for one teacher in high school), a
frequent response to a question that was poorly posed would be:

"Go think about what you just asked. Get back to us when you have
reformulated your question more precisely".

I has some SUPERB teachers.

In forcing students to not only solve problems, but also to think about
the questions they asked, they were forcing students to learn about
mathematics. Most mathematics is not learned from lecture, but by first
learning the basics from lectures, then learning to apply it ON YOUR OWN.

One should NEVER ask questions (in any field of study), if one does not
understand the definitions of words used in the questions.

We learn by DOING, and not by sitting and having answers handed to us.


And whether respondents to my posts like it or not, there are some
pre-requisites for the study of number theory. Being told that one
lacks the pre-requisites is not (and should not be) an insult, but for some
bizarre reason it is often taken as such within this newsgroup.

I could have chosen to simply say to the OP "You do not understand
what you are asking. Go learn the meaning of the word 'distribution',
then get back to us". OR I could also have said
"From the lack of clarity in your question, I judge that you lack the pre-
requisites to understand a meaningful answer. May I suggest that you study
some elementary number theory and elementary mathematical statistics
and get back to us?"

There is no personal judgment in such a response. Yet, I suspect that the
OP would have responded with "you are an a**hole". I don't understand
why people respond this way, but they do. Telling someone that they
lack the background to discuss a technical subject is not a slur on their
character. But people take it as such. WHY????

Instead of such a response, I took the time to point out WHY the question
was poorly posed. The response from the OP: silence.
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