Re: [PrimeNumbers] binary digit frequency in primes
- 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))).
--- 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 ...
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