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In considering the binary expansion of prime numbers, I'm interesting in the skew of digits towards 0 or 1.

I searched through other questions and arrived at: Last digits of primes

I just want to confirm my suspicion that if I exclude the most significant binary digit and the 3 least significant binary digits (in avoidance of issues from 2 and 5), and if i consider each binary position independently, the distribution across primes should tend to a 50/50 balance of 0s and 1s?

Should this follow as well for distinct semiprimes?

Anything other impacts (e.g. 3, 7, ...)?

Wolf Larson
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  • What is the issue with $5$ that you are trying to avoid? What do the last $3$ bits have to do with it? Am I missing something obvious? – saulspatz Jun 17 '19 at 20:22
  • @saulpatz the li ked question excluded 2 and 5. –  Jun 17 '19 at 20:34
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    The linked question was about decimal digits. That's why $5$ is special. But that's irrelevant for binary digits. – Robert Israel Jun 17 '19 at 20:47
  • My intuition led me to believe that since 5 and its multiples were excluded from being represented within a binary expansion, it would have an impact in a similar but lesser way that 2 has an impact (that the least significant digit is never 0). What I'm gathering is that my intuition was wrong. What else is new? :-) I will look into understanding the references that have been provided. Thanks. – Wolf Larson Jun 17 '19 at 21:17
  • Would it be correct to say then that you're talking about primes in octal? If so, https://oeis.org/A004682 would be quite relevant. – Robert Soupe Jun 18 '19 at 02:54

1 Answers1

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The last binary digit is, of course, $1$ except for the prime $2$. Every other binary digit, including the second-last and third-last, should be (asymptotically) equally likely to be $0$ or $1$, because of (the strong form of) Dirichlet's theorem on primes in arithmetic progressions: asymptotically, for any fixed $m$, the primes are evenly distributed among the odd congruence classes mod $2^m$.

Robert Israel
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  • Thanks. I will take a look at your reference. In reviewing the source of my error, looks like I assumed that 5 and its multiples would have an impact in biasing the 3 least significant digits. In looking at the first 8 multiples of 5, all combinations of 0s and 1s are reflected in those 3 digits. I gather this is what I missed on my initial review. – Wolf Larson Jun 17 '19 at 21:30