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By Bertrand' postulate, there exists a prime between $2^n$ and $2^{n+1}$.

For every $n$, is there a prime $p < 2^n$ such that $2^n+p$ is a prime?

The smallest such primes are listed in OEIS A056206.

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    $\begingroup$ It seems like you’re hoping for an effective form of Dickson’s conjecture. $\endgroup$
    – Charles
    Jun 1, 2023 at 19:49
  • $\begingroup$ @Charles Similar. My question contains a constraint $p < 2^n$. The prime $p$ could be large according to the value $n$. $\endgroup$
    – P.-S. Park
    Jun 2, 2023 at 0:54
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    $\begingroup$ It's not even known whether every power of two is a difference of two primes, is it? $\endgroup$ Jun 2, 2023 at 2:45

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