# How many primes of the form $2^p-p$ with $p$ prime?

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?

• Note: All primes after 3 can be written in the form of $6k+1$ and $6k-1$ – user371530 Oct 23 '16 at 1:27
• Also worth noting that non-prime may work: $2^9-9=503$ is prime – Joffan Oct 23 '16 at 1:51
• 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 $481801$. – Barry Cipra Nov 22 '16 at 23:05