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Old 2012-02-23, 13:50   #23
R.D. Silverman
 
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Quote:
Originally Posted by aketilander View Post
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.
Both of you are just bandying words.
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Old 2012-02-23, 15:48   #24
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Originally Posted by R.D. Silverman View Post
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.
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.
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Old 2012-02-23, 16:22   #25
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Quote:
Originally Posted by R.D. Silverman View Post
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.
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!

Last fiddled with by aketilander on 2012-02-23 at 16:24
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Old 2012-02-23, 17:52   #26
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Originally Posted by xilman View Post
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.
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.
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Old 2012-02-23, 18:08   #27
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Quote:
Originally Posted by R.D. Silverman View Post
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.
Which brings up the point, we normalize 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.
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Old 2012-02-23, 21:01   #28
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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.
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Old 2012-02-23, 21:41   #29
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Quote:
Originally Posted by Fusion_power View Post
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.
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.
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Old 2012-02-23, 21:47   #30
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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.
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Old 2012-02-24, 02:27   #31
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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.
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
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Old 2012-02-24, 02:44   #32
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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.
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Old 2012-02-24, 09:54   #33
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Quote:
Originally Posted by R.D. Silverman View Post
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.
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 not commute must necessarily correspond to observables which can not be simultaneously measured to arbitrary precision. Examples of such pairs of operators include \{x, \ p_x} and \{E, \ t\} โ€” 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 \{x, \ (\hbar / i)\partial / \partial x\} and \{t, \ (\hbar / i)\partial / \partial t\}. 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.

Last fiddled with by xilman on 2012-02-24 at 09:57 Reason: Fix \TeX
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