binary digit frequency in primes

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• I’m a bit lazy at the moment to do a proper search on this topic, so if anyone already knows the answer, please chime in. the primes, written in binary, all
Message 1 of 4 , Aug 29, 2012
I’m a bit lazy at the moment to do a proper search on this topic, so if anyone already knows the answer, please chime in.

the primes, written in binary, all have the first bit set to one and the last bit set to one except for the first prime “2”. what about the bits in the middle? is there a listing anywhere of the frequency of 1’s and 0’s of the inner bits? with analysis such even numbered bits vs. odd ones, and/or for just the prime numbered bits? I’m just curious...

James

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• This does not answer question on library search. ... I suspect the non-terminal digits are statistically indistinguishable for large primes from random, and I
Message 2 of 4 , Aug 30, 2012
This does not answer question on library search.
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I suspect the non-terminal digits are statistically indistinguishable for large primes from random, and I suspect--and this may be an error--that there has been no published result because it is easy to do and will almost certainly only generate expected results.
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Total tabulation for all small primes actually gives a fairly large number of digits.  I can comment in one way related to what I have been doing.  I was doing the four residue classes modulo 5 but hit 'close' when trying to 'scroll', so I started over with seperate PARI/GP windows checking more than just what I was initially for each of modulo 5, modulo 8, modulo 10 (final digits in our base) and modulo 12.  The comment I have is that if I am remembering correctly the three-way ties modulo 8 all have a deficit at 1 modulo 8.  That is, the occurrences in the primes through around 10^10 where some three of the possibilities out of four are the same count has the regularity of all of them having 1 as the exception and always with that one smaller.  I was about to take down notes on these four problems and report, since it takes up time and should be done (if it has not already been) in a more efficient computing environment.  But I can say
that if memory serves correctly this does mean there are fewer cases of the penultimate and pre-penultimate digits in binary both being zero than the other three possibilities consistently to a reasonably large number.  It feels a little like this is non-coincidental; but I think there is no known explanation other than coincidence, and it certainly is known not to hold over the long-term that one case is less represented than the other three (This is Chebotarev's Density Theorem applied to a specific case).
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I should give a detailed note on this in a few hours or so.

JGM

--- On Wed, 8/29/12, James J Youlton Jr <youjaes@...> wrote:

From: James J Youlton Jr <youjaes@...>
Subject: [PrimeNumbers] binary digit frequency in primes
Date: Wednesday, August 29, 2012, 6:09 PM

I’m a bit lazy at the moment to do a proper search on this topic, so if anyone already knows the answer, please chime in.

the primes, written in binary, all have the first bit set to one and the last bit set to one except for the first prime “2”. what about the bits in the middle? is there a listing anywhere of the frequency of 1’s and 0’s of the inner bits? with analysis such even numbered bits vs. odd ones, and/or for just the prime numbered bits? I’m just curious...

James

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]
• ... Dirichlet proved that all non-trivial residues are equally represented in the primes. So there are the same number of 4n+1 and 4n+3 primes. Therefore the
Message 3 of 4 , Aug 30, 2012
--- On Thu, 8/30/12, James J Youlton Jr <youjaes@...> wrote:
> the primes, written in binary, all have the first bit set to
> one and the last bit set to one except for the first prime
> “2”.  what about the bits in the middle?  is
> there a listing anywhere of the frequency of 1’s and
> 0’s  of the inner bits? with analysis such even
> numbered bits vs. odd ones, and/or for just the prime
> numbered bits?  I’m just curious...

Dirichlet proved that all non-trivial residues are equally represented in the primes. So there are the same number of 4n+1 and 4n+3 primes.
Therefore the 2s bit is uniform.
Ditto 8n+1, 8n+3, 8n+5, and 8n+7, thus the 4s bit is uniform.
Ditto in aeternum ...

Phil
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• Maybe I am not remembering correctly, but I believe that the infinitude in each relatively prime congruence class to a given modulus is all that Dirichlet
Message 4 of 4 , Aug 30, 2012
Maybe I am not remembering correctly, but I believe that the infinitude in each relatively prime congruence class to a given modulus is all that Dirichlet proved.  Cursory internet search does not disconfirm this suspicion, but it may be wrong anyway.  In any case, it is true that each class is eventually arbitrarily proportionally close to the average, and the strongest form is Chebotarev's or some generalization few here could decipher (most likely, i think).  However, the distribution of digits in binary could still be biased out to infinity (though it seems unlikely).  The first bit could be biased only for so long, the second again only so long, etc.  One could ultimately have any kind of limiting behavior to all bits in primes and still have the Chebotarev Density Theorem (or Dirichlet's, if memory fails me; or whatever would be most appropriate to refer to for the natural numbers if Chebotarev's is too much but Dirichlet only proved as
little as I remember).

At any rate, I will give the results I had on specifics on Saturday.  I got caught up on something with my IQ=5*10 earlier today (YES, I REALLY HAD TWO PROGRAMS COUNTING THE FOUR ALLOWABLE CLASSES FOR BOTH OF MODULO 5 AND 10 (AND THEN I THOUGHT I HAD HAD MY NEW COMP HACKED BECAUSE THEY DID NOT GIVE THE SAME RESULTS (IMAGINE: 2 is not congruent to 7 modulo 10))).

JIM M

--- On Thu, 8/30/12, Phil Carmody <thefatphil@...> wrote:

From: Phil Carmody <thefatphil@...>
Subject: Re: [PrimeNumbers] binary digit frequency in primes
Date: Thursday, August 30, 2012, 11:25 AM

--- On Thu, 8/30/12, James J Youlton Jr <youjaes@...> wrote:
> the primes, written in binary, all have the first bit set to
> one and the last bit set to one except for the first prime
> “2”.  what about the bits in the middle?  is
> there a listing anywhere of the frequency of 1’s and
> 0’s  of the inner bits? with analysis such even
> numbered bits vs. odd ones, and/or for just the prime
> numbered bits?  I’m just curious...

Dirichlet proved that all non-trivial residues are equally represented in the primes. So there are the same number of 4n+1 and 4n+3 primes.
Therefore the 2s bit is uniform.
Ditto 8n+1, 8n+3, 8n+5, and 8n+7, thus the 4s bit is uniform.
Ditto in aeternum ...

Phil
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