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24417Re: [PrimeNumbers] binary digit frequency in primes

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  • James Merickel
    Aug 30, 2012
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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 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. 
      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).
      I should give a detailed note on this in a few hours or so.

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

      From: James J Youlton Jr <youjaes@...>
      Subject: [PrimeNumbers] binary digit frequency in primes
      To: primenumbers@yahoogroups.com
      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...


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