20200917, 18:39  #1 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2×3^{2}×7×47 Posts 
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 https://www.mersenneforum.org/showpo...35&postcount=6 for details. 
20200917, 18:59  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
5·7·17^{2} Posts 
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). 
20200917, 19:01  #3 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
5·7·17^{2} Posts 
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.

20200918, 01:51  #4  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2·3^{2}·7·47 Posts 
Quote:
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. https://en.wikipedia.org/wiki/Studen...tdistribution Last fiddled with by kriesel on 20200918 at 02:16 

20200918, 06:25  #5 
"Carlos Pinho"
Oct 2011
Milton Keynes, UK
3·1,663 Posts 
You need to be careful about the sample size, there is a math reason to define it.

20200918, 12:12  #6 
Feb 2017
Nowhere
23×223 Posts 
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. Last fiddled with by Dr Sardonicus on 20200918 at 12:18 Reason: Change symbol for positive real from z to α 
20200918, 18:26  #7 
Bamboozled!
"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across
2B10_{16} Posts 

20200919, 01:08  #8  
Apr 2012
Gracie on alert.
110010101_{2} Posts 
Quote:
Sorry for the diatribe and I really appreciate your posts regarding gpu's..good stuff! Last fiddled with by Uncwilly on 20200919 at 01:13 Reason: clarrity (uncwilly: fixed quote tag) 

20200920, 09:31  #9 
Apr 2012
Gracie on alert.
625_{8} Posts 
kriesel, here's another "brick in the wall" for your question which again may provide some helpful insight:
https://lance.fortnow.com/papers/files/alg.pdf 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. 
20210915, 03:06  #10 
Feb 2004
France
2·461 Posts 
I found this papier very interesting:
https://faculty.math.illinois.edu/~j...nneBenford.pdf 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 14..... exponents? Code:
1 13 2 10 3 7 4 5 5 2 6 3 7 2 8 3 9 2 Last fiddled with by axn on 20210915 at 10:35 Reason: duplicate url 
20210915, 03:37  #11  
Jan 2021
California
367_{8} Posts 
Quote:
Last fiddled with by slandrum on 20210915 at 03:40 

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