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I know that if $p = 6k+1$ with $k$ integer, then $(2^p-p) \mod 6 \equiv 1$. I think that this means that $2^p-p$ could be prime. My question is: are there a finite or infinite number of primes which can be written like this?

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  • $\begingroup$ Note: All primes after 3 can be written in the form of $6k+1$ and $6k-1$ $\endgroup$ – user371530 Oct 23 '16 at 1:27
  • $\begingroup$ Also worth noting that non-prime may work: $2^9-9=503$ is prime $\endgroup$ – Joffan Oct 23 '16 at 1:51
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    $\begingroup$ oeis.org/A048744 gives the first $23$ values of $n$ for which $2^n-n$ is prime, including the prime values $n=2,3,13,19$, and $481801$. $\endgroup$ – Barry Cipra Nov 22 '16 at 23:05
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At this moment I do not have complete explanation of your question but with the use of my own properties of primes which I discovered last year I made a conclusion i.e.:

If (2p - 5) and (2p + 5) are primes than (2p - a) and (2p + a) are also primes only and only if "a" is a multiple of 5(which is rare but can come) or "a" is a prime number maybe "p" also.

7 also show similar property in (2p - 7) and (2p + 7)

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