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:
[LIST][*]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[/LIST]
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:[LIST][*]0 (XXX) 2[*]3 (XXX) 4[*]4 (XXX) ?[/LIST]
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?

[QUOTE=emily;290186]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?[/QUOTE]

Sigh...

Given fifty throws of a fair coin which shows up heads, what are the odds of the next throw being tails?

[QUOTE=emily;290186]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:
[LIST][*]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[/LIST]
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:[LIST][*]0 (XXX) 2[*]3 (XXX) 4[*]4 (XXX) ?[/LIST]
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?[/QUOTE]

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
[b]is[/b]. 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.

@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
[url]http://primes.utm.edu/notes/faq/NextMersenne.html[/url] and
[url]http://primes.utm.edu/mersenne/heuristic.html[/url]
. 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., [url=http://www.mersenneforum.org/showthread.php?t=15653]this[/url]. "Of this vicinity" means "active posters on this forum".)

[QUOTE=Dubslow;290196]@OP: RD Silverman is quick to jump on new guys.
.....
Many, many hours by many people have been put into studying the stats.[/QUOTE]:goodposting: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

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.

[QUOTE=Dubslow;290196]@OP: RD Silverman is quick to jump on new guys.
[/QUOTE]

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.

[QUOTE=R.D. Silverman;290194]your post
consists of jumbled numerology, nonsense, and nothing else.[/QUOTE]


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...!

[QUOTE=emily;290260]I said it's non-scientific, didn't try to present it as anything other than some quick non-scientific observations. ...![/QUOTE]

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.

[QUOTE=R.D. Silverman;290258]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.[/QUOTE]

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.

[QUOTE=R.D. Silverman;290263] FUTURE EVENTS ARE NOT DEPENDENT ON PAST EVENTS[/QUOTE]

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.

[QUOTE=R.D. Silverman;290263]It doesn't matter that the observations are not scientific. What matters is that they were made in the first place.[/QUOTE]

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 [B]simplest[/B] 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!

[QUOTE=philmoore;290493] 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.
[/QUOTE]

Rule out? We [b]know[/b] 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 [b]are[/b] 'truly random'. The next gap is unpredictable.

[QUOTE]
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.
[/QUOTE]

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

[QUOTE=R.D. Silverman;290494]She did not give "the wrong answer". What she gave was the equivalent of presenting [b]astrology[/b] at a physics convention. Her prose "was not even wrong".[/QUOTE]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, [B][U]if and [COLOR="DarkRed"]only if[/COLOR] they [COLOR="darkred"]choose[/COLOR] to speak up[/U][/B], 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.

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

:smile:

[QUOTE=R.D. Silverman;290494]If the gaps between Mersenne primes can be proven to conform to a Poisson process, then under any reasonable definition the
gaps [b]are[/b] 'truly random'.[/QUOTE]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.

Nature of Poisson Distributions?
 

[QUOTE=xilman;290532]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.[/QUOTE]

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.

[QUOTE=xilman;290532]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.[/QUOTE]

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.

[QUOTE=aketilander;290540]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.
[/QUOTE]


Contrawise:
And if the distribution of primes was [b]not[/b] 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 [b]in either case[/b]
your statement basically says nothing.

[QUOTE]
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?
[/QUOTE]

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.

[QUOTE=aketilander;290540]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.[/QUOTE]

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 [I]extremely[/I] similar.

[QUOTE=Mini-Geek;290550]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 [I]extremely[/I] similar.[/QUOTE]

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.

[QUOTE=aketilander;290552]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.[/QUOTE]

Both of you are just bandying words.

[QUOTE=R.D. Silverman;290545]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.[/QUOTE]

Allow me to add: Even assuming that the current conjectures are correct,
the density function is not ergodic. When we say the gaps are Poisson
distributed, we mean that they are LOCALLY Poisson (i.e. in the
neighborhood of any finite index). However, as p-->oo the gaps
become longer and longer. The number of expected M_p in [p, 2p]
is exp(gamma). Note that the interval [p, 2p] grows in length as
p grows, yet the number of expected M_p in the interval is constant.

[QUOTE=R.D. Silverman;290572]Allow me to add: Even assuming that the current conjectures are correct,
the density function is not ergodic. When we say the gaps are Poisson
distributed, we mean that they are LOCALLY Poisson (i.e. in the
neighborhood of any finite index). However, as p-->oo the gaps
become longer and longer. The number of expected M_p in [p, 2p]
is exp(gamma). Note that the interval [p, 2p] grows in length as
p grows, yet the number of expected M_p in the interval is constant.[/QUOTE]

I think you answered my question now, thank you! It is of course something quite different to try to prove that the distribution of primes are "LOCALLY Poisson" then to try to prove that the whole distribution from 2 to oo is a Poisson distribution. So I am very grateful, Thank you!

[QUOTE=xilman;290532]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.[/QUOTE]

How about the statistical question of whether the existing data is consistent with a Poisson process? The graph gives a pretty convincing "Chi-by-Eye," but it would be fun to have some "statistical confidence" statements. I've not seen anybody attempt this.

[QUOTE=R.D. Silverman;290572]Allow me to add: Even assuming that the current conjectures are correct, the density function is not ergodic. When we say the gaps are Poisson distributed, we mean that they are LOCALLY Poisson (i.e. in the neighborhood of any finite index). However, as p-->oo the gaps become longer and longer. The number of expected M_p in [p, 2p] is exp(gamma). Note that the interval [p, 2p] grows in length as p grows, yet the number of expected M_p in the interval is constant.[/QUOTE]

Which brings up the point, we [B]normalize[/B] the values before trying to fit them to a Poisson distribution, i.e., divide each value n by ln n. But as William points out, the existing data seem to fit a Poissonian pretty well, as judged by nearest neighbor spacings and other correlation measures.

The human brain is a marvelous thing. It seeks patterns, even when patterns do not exist. For example, if you watch the waves on the seashore long enough, you would think you could discern the pattern of large vs small waves. The waves represent a pseudo-random physical process which is not the same thing as a mathematically random process (which as RDS notes is not truly defined either).

So to add to the cnofusion :), I'll stipulate that in an effort to perceive a pattern, your mind throws up possibilities which then have to be waded through to determine that there is no discernible pattern.

Here is a different way to see this and it will illustrate the lack of pattern. Instead of expressing the known Mersenne Primes in base 10, try converting them to binary and then look at the distribution. If you are really good, you might see something amazing.

DarJones

p.s. RDS, would you care to have a discussion about whether or not there are any truly random mathematical processes? Note that I can prove that there are random physical processes, a la Heisenberg Uncertainty Principle.

[QUOTE=Fusion_power;290609]p.s. RDS, would you care to have a discussion about whether or not there are any truly random mathematical processes? Note that I can prove that there are random physical processes, a la Heisenberg Uncertainty Principle.[/QUOTE]

Did you know that HUP can be proven from basic mathematical principles?
Take a time sample from a signal. Assume the measurments have some
small error. Now, take the Fourier Transform of the signal. The errors
transform as well.

One can prove that the product of the errors in the original signal
times the errors in the transform are bounded from below.

This is what happens when one tries to measure position and velocity
vectors at the same time. One can be viewed as the Fourier Transform
of the other and the product of the errors is bounded from below.

More generally/abstractly, all waves and wave-like things have some form of uncertainty principle attached to them. For quantum particles, it just happens to be position and momentum. The only reason this became such a big deal is because of the classical counterintuitiveness of the result, which is exactly the same as saying that wave-particle duality is classically counterintuitive.

[QUOTE]Did you know that HUP can be proven from basic mathematical principles?
Take a time sample from a signal. Assume the measurments have some
small error. Now, take the Fourier Transform of the signal. The errors
transform as well. [/QUOTE]

But the question is whether or not there are any truly random mathematical processes. It is relatively easy to show that wave like processes are governed by HUP. Granted that HUP can be mathematically proven, can you then apply a HUP like constraint on a mathematical process.

Lets postulate that it does exist. We will call it the MUP (Mathematical Uncertainty Principle). Now an inherent property of MUP would be that some Mathematical Process would be truly random, therefore if you can identify a truly random mathematical process, you have inherently proven MUP.

The reason I am even opening up for this is: if the distribution of Mersenne Primes is governed by MUP, then they are truly random and if not, then they inherently can't be random and therefore there is a pattern to them.

And yes, I'm just having fun on a late Thursday evening.

DarJones

Define "mathematical process". I have no problem saying that the propogation of a wave is a mathematical process determined by the wave equation. Also, just because you have a random process does not mean that there is a "MUP". There might be something other than a "MUP" that implies randomness of some process.

[QUOTE=R.D. Silverman;290615]Did you know that HUP can be proven from basic mathematical principles?
Take a time sample from a signal. Assume the measurments have some
small error. Now, take the Fourier Transform of the signal. The errors
transform as well.

One can prove that the product of the errors in the original signal
times the errors in the transform are bounded from below.

This is what happens when one tries to measure position and velocity
vectors at the same time. One can be viewed as the Fourier Transform
of the other and the product of the errors is bounded from below.[/QUOTE]An alternative approach, and the one I first learned, doesn't appeal to time samples and Fourier transforms at all. It starts with the operator treatment of QM.

A physically observable quantity must be represented by a Hermitian operator, because only such operators have real eigenvalues. Two observables may have simultaneously precisely defined values if and only if the corresponding operators commute. Proof of necessity is simple. Proof of sufficiency is a bit trickier because of the possibility of degenerate eigenfunctions but not particularly difficult.

Two operators which do [b]not[/b] commute must necessarily correspond to observables which can not be simultaneously measured to arbitrary precision. Examples of such pairs of operators include [TEX]\{x, \ p_x}[/TEX] and [TEX]\{E, \ t\}[/TEX] — these are {position on a Cartesian axis, linear momentum along that axis} and {energy, time} respectively. The corresponding operators in the classical non-relativistic picture are [TEX]\{x, \ (\hbar / i)\partial / \partial x\}[/TEX] and [TEX]\{t, \ (\hbar / i)\partial / \partial t\}[/TEX]. It is easy to show that neither of these pairs of operators commute.

Incidentally, those two show that space and time are already on an equal footing in this picture, something that was noted very early on in the development of QM.

Bertrand's postulate
 

[QUOTE=aketilander;290540]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.[/QUOTE]

I think there must be something left out of that first sentence, for brevity perhaps, such as for integer x > some minimum value. (Because otherwise the disproof by example would be so trivial even I could do it; let x=1; the interval 2x > y > x contains no prime integers, and also no integers.) Ok, a bit of searching yields [URL]https://en.wikipedia.org/wiki/Bertrand%27s_postulate[/URL]

civility
 

[QUOTE=Uncwilly;290501]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, [B][U]if and [COLOR=DarkRed]only if[/COLOR] they [COLOR=darkred]choose[/COLOR] to speak up[/U][/B], 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.[/QUOTE]

Amen. I understand it can sometimes be irritating to cover again the same introductory ground with yet another novice lacking background, yet I think it sad that RDS appears to have driven emily off in well under a day elapsed time. How much better to be welcoming and pleasant to share what one loves with others. How did those who taught RDS act? How did any of math, science, or technology get developed, except by people paying attention, wondering, employing curiosity, coming up with hypotheses, trying things, talking with each other, and eventually teaching each other? Numbers can be used and enjoyed and puzzled over by people without the deep background in that specialty held by a rare few, just as music can be enjoyed without being a member of the London Philharmonic. I'd bet there are also few if any of us that could design and build an entire car or computer or house, much less invent a new better way of doing any of those, but enjoy their ready availability. RDS may have had as much to learn from emily or Dale Carnegie, about being civil and positive to people regardless of circumstances, as emily may have had to learn from RDS or Euler, about math, if measured by life impact.

One of the things that has impressed me and clearly others about some of the authors of current world class software in number theory is the decency, tact and courtesy with which they respond on forums or in email to anyone no matter what. (A skill I'm working on.) Carnegie teaches among other things that tone is more important than content, since it can help content be welcomed, or work against content being heard or accepted, or motivate active resistance to the content (to the possible detriment of both the recipient and sender). Carnegie teaches in effect to wrap the message like a present. Excelling in any field has perhaps more to do with people skill than technical skill.