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-   -   Mersenne Primes and Benford's law (https://www.mersenneforum.org/showthread.php?t=25975)

kriesel 2020-09-17 18:39

Mersenne Primes and Benford's law
 
Someone mentioned Benford's law to me recently, and I decided to see how well the known Mersenne primes followed it. It turns out that the exponents follow it pretty well. The number of decimal digits in the known Mersenne primes do not though. Six is substantially more common a leading digit than expected. And that's the case whether the number of decimal digits is itself expressed in decimal, or in hexadecimal. Any ideas why?
See [URL]https://www.mersenneforum.org/showpost.php?p=557235&postcount=6[/URL] for details.

Uncwilly 2020-09-17 18:59

Why have you divide what should be 1 post into 2 parts in 2 threads?
I am tempted to fix this (absent a compelling reason).

Uncwilly 2020-09-17 19:01

The answer is that small data sets are small. And random distribution is random and smooth as one might expect. Flipping a coin 1000 times will yield runs of 6 heads or tails in a row, etc.

kriesel 2020-09-18 01:51

[QUOTE=Uncwilly;557247]Why have you divide what should be 1 post into 2 parts in 2 threads?[/QUOTE]Relax, nothing nefarious. There is logic to what I do, whether it is apparent or not.
A reference post in my blog I can edit, extend, and correct indefinitely. The purpose of the blog is to house my reference posts and threads.
This avoids bloat from duplication, and stale or wrong content lingering.

The post beginning this thread is brief and serves a different purpose, asking about the math, in the hope that a skilled mathematician can illuminate us. I can not update a post here after one hour. It seemed to me not at the level fit for math or number theory threads, so it's here.

I've tried a few different statistical tests on the observed distributions, and they are in considerable disagreement as to whether the variations could have been due to chance.
Pearson's, m and d.

Benford's law has scale invariance. Number size expressed in digit counts of different bases (2, 10, 16) seem like unit changes, which don't affect Benford's law. Over 50 samples or even 30 is workable in some statistics. [URL="https://en.wikipedia.org/wiki/Student%27s_t-distribution"]https://en.wikipedia.org/wiki/Student%27s_t-distribution[/URL]

pinhodecarlos 2020-09-18 06:25

You need to be careful about the sample size, there is a math reason to define it.

Dr Sardonicus 2020-09-18 12:12

About the small sample size, me three.

Benford's law is based on the theorem that, if α is a (positive) real irrational number, the fractional parts of α, 2*α, 3*α, etc are uniformly distributed in [0,1].

If N is "large," we would expect the fractional parts of k*α, k = 1 to N, to approximate a uniform distribution in [0,1] fairly well.

I don't think you can say much about uniformity if you take 51 values of k between 2 and 82589933.

xilman 2020-09-18 18:26

[QUOTE=kriesel;557263]Relax, nothing nefarious. There is logic to what I do, whether it is apparent or not.[/QUOTE]

Though this be method, yet there is madness in't

jwaltos 2020-09-19 01:08

[QUOTE=kriesel;557245] Benford's law..how well the known Mersenne primes followed it... Any ideas why?[/QUOTE]
I have looked at both aspects independently..and you may want to look at Finch's reference texts for some insight. Following some of the papers on Benford's law and its justification leads to some very intricate and general statistical results. Rather than look at the numerical properties of the Mersenne primes, I've looked at the algebraic roots and justification for designating a Mersenne prime as such. Both aspects of your question can be appropriately combined and generalized when you are able to compare and contrast the mathematics and logic comprising both. Rather than asking "why" understanding the "how" first provides the foundation for the former. The "to a man with a hammer every problem looks like a nail" adage is something to be aware of. Looking at your question topologically will give it a different spin and complicate things due to the other mathematical baggage that comes with it but you should see a few things specifically more clearly and you can crunch the numbers to bring these connections into view.
Sorry for the diatribe and I really appreciate your posts regarding gpu's..good stuff!

jwaltos 2020-09-20 09:31

kriesel, here's another "brick in the wall" for your question which again may provide some helpful insight:
[url]https://lance.fortnow.com/papers/files/alg.pdf[/url]

Quoting from the paper:
"3 External Structure
In this section we list papers that prove theorems using the external structure
of complexity theory. These theorems generally show that classes have certain
closure properties based on their external algebraic structure. We feel that
this study may prove more important as it may lead us to understand how to
separate complexity classes. If two complexity classes do not have the same
external algebraic structure then they cannot coincide."

This is my emphasis on understanding the general nature of your question then ensuring your analysis properly coordinates the appropriate concepts. As usual, the freedom to create conceptual bridges may allow you to invent/discover something new.

T.Rex 2021-09-15 03:06

I found this papier very interesting:
[url]https://faculty.math.illinois.edu/~jli135/paper/MersenneBenford.pdf[/url]
First, it seems to say that Mersenne primes are random.

Second, it means that looking at Mersenne exponents starting with 1,2,3, or 4 generates 3 times more primes than exponents starting with 5,6,7,8, or 9. Thus, should the GIMPS look first at these 1-4..... exponents?

[CODE]
1 13
2 10
3 7
4 5
5 2
6 3
7 2
8 3
9 2[/CODE]

slandrum 2021-09-15 03:37

[QUOTE=T.Rex;587893]I found this papier very interesting:
[url]https://faculty.math.illinois.edu/~jli135/paper/MersenneBenford.pdf[/url][URL="https://faculty.math.illinois.edu/~jli135/paper/MersenneBenford.pdf"]https://faculty.math.illinois.edu/~jli135/paper/MersenneBenford.pdf[/URL]

First, it seems to say that Mersenne primes are random.

Second, it means that looking at Mersenne exponents starting with 1,2,3, or 4 generates 3 times more primes than exponents starting with 5,6,7,8, or 9. Thus, should the GIMPS look first at these 1-4..... exponents?

[CODE]
1 13
2 10
3 7
4 5
5 2
6 3
7 2
8 3
9 2[/CODE][/QUOTE]

That's due to logarithmic bias. You take any sequence that increases exponentially (which the mersenne exponents do approximately) and you'll see a similar pattern in the leading digit. It actually gives no indication about where you can find the next mersenne prime, other than they get scarcer as the exponents get larger.

T.Rex 2021-09-15 04:23

[QUOTE=slandrum;587896]That's due to logarithmic bias. ... It get scarcer as the exponents get larger.[/QUOTE]

Hummmm Right.
My 2nd comment was stupid based on the first one.
First one seems to say that there is no rule enabling to find the next Mersenne prime based on the knowledge of previous known Mersenne primes exponents.

Dobri 2021-09-15 12:14

The Benford's law could indirectly be linked to a tendency toward maximum entropy.

The Gaussian distribution corresponds to maximum entropy on the entire axis from negative to positive infinity.

The uniform distribution has maximum entropy on a finite interval.

Then the exponential distribution has maximum entropy on the positive half line from zero to infinity.

The convolution of several exponential distributions results in a hypoexponential distribution on the positive half line.

The maximum entropy curve of the exponential distribution could potentially be what the decaying Benford's law curve tends to in terms of maximizing the entropy for some number theory cases.


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