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Old 2012-02-21, 01:11   #1
emily
 
Feb 2012
Athens, Greece

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Default Distribution of Mersenne primes before and after couples of primes found

I made some non-scientific quick observations re the grand-scale distribution of found primes amongst all the tested exponents, paying particular attention to two M primes being found in consecutive million-exponents (e.g. 42M and 43M, M for million-exponents) and the gaps between the other primes before and after the prime couples: (when I say 'prime' I mean 'M-prime', M for Mersenne)

This list starts at the 0M range of exponents and goes up to 43M:
  • X
  • X (O:0)
  • XXX
  • XXX
  • O:2
  • X
  • O:6
  • X
  • O:6
  • X
  • O:3
  • XXX
  • XXX
  • O:4
  • X
  • O:1
  • X
  • O:4
  • XXX
  • XXX

In the 0M range there're 33 Mersenne primes, and 2 in the 1M range. I regard these as 'lone primes' (X). Then there is one prime in 2M and another one in 3M, I regard these as a 'couple' (XXX). For the next two million-ranges (4M and 5M) there is no M-prime, so I regard these as a 'gap' (O), which in this case it is a double-gap (O:2).

So here's the list of gaps before and after each prime couple:
  • 0 (XXX) 2
  • 3 (XXX) 4
  • 4 (XXX) ?

So the 'before-gap' prior to the second 'couple' is 3 (21M, 22M and 23M because the first prime of the couple is in 24M) and the 'after-gap' is 4. The 'before-gaps' so far have a sequence of (0, 3, 4) while the 'after-gaps' so far have a sequence of (2,4).

The question is what would we expect the 'after-gap' after the third and last (so far) 'couple' to be? Could we expect 6? (so the next 'lone M-prime' would be in 50M), 8? (next 'lone' in 52M), or something else?

Last fiddled with by emily on 2012-02-21 at 01:12
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Old 2012-02-21, 01:37   #2
chalsall
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"Chris Halsall"
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Quote:
Originally Posted by emily View Post
The question is what would we expect the 'after-gap' after the third and last (so far) 'couple' to be? Could we expect 6? (so the next 'lone M-prime' would be in 50M), 8? (next 'lone' in 52M), or something else?
Sigh...

Given fifty throws of a fair coin which shows up heads, what are the odds of the next throw being tails?
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Old 2012-02-21, 02:11   #3
R.D. Silverman
 
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Nov 2003

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Quote:
Originally Posted by emily View Post
I made some non-scientific quick observations re the grand-scale distribution of found primes amongst all the tested exponents, paying particular attention to two M primes being found in consecutive million-exponents (e.g. 42M and 43M, M for million-exponents) and the gaps between the other primes before and after the prime couples: (when I say 'prime' I mean 'M-prime', M for Mersenne)

This list starts at the 0M range of exponents and goes up to 43M:
  • X
  • X (O:0)
  • XXX
  • XXX
  • O:2
  • X
  • O:6
  • X
  • O:6
  • X
  • O:3
  • XXX
  • XXX
  • O:4
  • X
  • O:1
  • X
  • O:4
  • XXX
  • XXX

In the 0M range there're 33 Mersenne primes, and 2 in the 1M range. I regard these as 'lone primes' (X). Then there is one prime in 2M and another one in 3M, I regard these as a 'couple' (XXX). For the next two million-ranges (4M and 5M) there is no M-prime, so I regard these as a 'gap' (O), which in this case it is a double-gap (O:2).

So here's the list of gaps before and after each prime couple:
  • 0 (XXX) 2
  • 3 (XXX) 4
  • 4 (XXX) ?

So the 'before-gap' prior to the second 'couple' is 3 (21M, 22M and 23M because the first prime of the couple is in 24M) and the 'after-gap' is 4. The 'before-gaps' so far have a sequence of (0, 3, 4) while the 'after-gaps' so far have a sequence of (2,4).

The question is what would we expect the 'after-gap' after the third and last (so far) 'couple' to be? Could we expect 6? (so the next 'lone M-prime' would be in 50M), 8? (next 'lone' in 52M), or something else?
Look up "Poisson Distribution". Study it. Learn what it means to draw a
sample from a given density function. Indeed, learn what a density function
is. Go learn some statistics and probability theory. Learn what
conditional probability is. Study Bayes' Theorem. Come back here after you
have done so.

I don't know who you are or what your background is. But your post
consists of jumbled numerology, nonsense, and nothing else.
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Old 2012-02-21, 02:35   #4
Dubslow
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@OP: RD Silverman is quick to jump on new guys. OTOH, if you show a willingness and competence (he is very big on competence) to learn what he's talking about, then you should get along just fine.

Many, many hours by many people have been put into studying the stats. As more of a "light-reading" suggestion, try
http://primes.utm.edu/notes/faq/NextMersenne.html and
http://primes.utm.edu/mersenne/heuristic.html
. These two pages mention many of the things that RDS is talking about, and if you're not put off by it, then take his suggestion and read up (the "heavier reading") on those things. If you are put off by it (as I was, I hate statistics) then it's better to leave the thread alone, because the Math forum has literally the most knowledgeable mathematicians in the world lurking around. (I mean, e.g., this. "Of this vicinity" means "active posters on this forum".)

Last fiddled with by Dubslow on 2012-02-21 at 02:36
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Old 2012-02-21, 03:01   #5
Uncwilly
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Quote:
Originally Posted by Dubslow View Post
@OP: RD Silverman is quick to jump on new guys.
.....
Many, many hours by many people have been put into studying the stats.
I agree completely. Listen to Bob's (RDS) information and ignore his tone.

You also might try some light reading over at the MersenneWiki.org
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Old 2012-02-21, 06:55   #6
aketilander
 
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"Åke Tilander"
Apr 2011
Sandviken, Sweden

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Smile Poisson Distribution

Well we might add that there are strong arguments in favor of the distribution of Mersenne Primes is a Poisson Distribution, but there is no proofs as yet.
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Old 2012-02-21, 11:04   #7
R.D. Silverman
 
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Quote:
Originally Posted by Dubslow View Post
@OP: RD Silverman is quick to jump on new guys.
Learn to read. My post made it clear that I was NOT jumping on the
poster. I was jumping on the content of the post. Learn the difference.
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Old 2012-02-21, 11:22   #8
emily
 
Feb 2012
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Quote:
Originally Posted by R.D. Silverman View Post
your post
consists of jumbled numerology, nonsense, and nothing else.

I said it's non-scientific, didn't try to present it as anything other than some quick non-scientific observations. But I was stupid to post it on this forum where the discussion is apparently on a higher level. Thanks for the suggestions.

Oh and yeah, ofc I know this is a problem that very knowledgeable people are trying to solve for a very long time and still nobody knows for sure how the primes are distributed... Sigh...!
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Old 2012-02-21, 12:28   #9
R.D. Silverman
 
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Quote:
Originally Posted by emily View Post
I said it's non-scientific, didn't try to present it as anything other than some quick non-scientific observations. ...!
I don't think that you understand the problem.

It doesn't matter that the observations are not scientific. What matters
is that they were made in the first place. Making these 'observations'
violates some very basic fundamental principles and theorems in
probability and statistics. In particular: FUTURE EVENTS ARE NOT DEPENDENT ON PAST EVENTS in this domain. Learn what "independence"
means.

What you did was like walking into a medical convention and announcing
"I don't know anything about cellular chemistry, but I have some non-scientific observations about how to cure cancer".

You violated principles that are taught in the very first day of any class
on statistics and probability.
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Old 2012-02-21, 18:42   #10
Dubslow
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Quote:
Originally Posted by R.D. Silverman View Post
Learn to read. My post made it clear that I was NOT jumping on the
poster. I was jumping on the content of the post. Learn the difference.
Unfortunately, it is a subtle difference that most people don't make on first thought. Many of the residents here have learned the difference, and now hopefully OP has learned the difference.
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Old 2012-02-21, 20:39   #11
emily
 
Feb 2012
Athens, Greece

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Quote:
Originally Posted by R.D. Silverman View Post
FUTURE EVENTS ARE NOT DEPENDENT ON PAST EVENTS
My Caps-Lock key is mapped to the X-Windows keyboard language layout switch and I avoid using all-caps because most people equate it with yelling which tends to make them feel upset.

But there's no need to continue discussing something that isn't of interest to the Maths forum.

Last fiddled with by emily on 2012-02-21 at 20:50
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