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Let $s_2(m)$ be the sum of digits of $m$ in binary form.

I would like to ask the following question:

Is it true that for every $n\in \mathbb{N}$ there is at least one prime $p$ which has $s_2(p)=n$?

After some search I found some papers which study the sum of digits of all the primes below $N$ but not an answer to my question.
Any references (or an answer) would be appreciated!

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  • $\begingroup$ Is there actually an obvious counterexample to the statement that for any odd prime $p$ there is a $2^k>p$ with $p+2^k$ prime too? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 25, 2016 at 11:18
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    $\begingroup$ Erdos showed that there are arithmetical progressions of odd numbers $n$ such that there is no $k$ with $n+2^k$ prime. These arithmetical progressions satisfy the conditions of Dirichlet's Theorem, so there are prime numbers $p$ such that $p+2^k$ is never prime. See, e.g., math.dartmouth.edu/~carlp/PDF/covertalkunder.pdf $\endgroup$
    – Gerry Myerson
    Commented Jan 25, 2016 at 11:44
  • $\begingroup$ @GerryMyerson Thanks, very interesting! The counterexample at your link is quite big, actually I was unsuccessfully looking for $k$ with $2131+2^k$ prime, I wonder what happens with it... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 25, 2016 at 12:32
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    $\begingroup$ See en.wikipedia.org/wiki/… $\endgroup$
    – Gerry Myerson
    Commented Jan 25, 2016 at 12:42
  • $\begingroup$ @GerryMyerson wow thanks, I would never reach $2^{4583176}$ on my pc :D $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 25, 2016 at 14:18

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