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24418Re: [PrimeNumbers] binary digit frequency in primes

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  • Phil Carmody
    Aug 30, 2012
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      --- 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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