If $p\equiv3\pmod{4}$ and $q=2p+1$ is a prime then $q|(2^p-1)$ if $2^p-1$ is composite.
Also, prove that there are infinitely many primes $p$ for which $2^p-1$ is composite.
If $p\equiv3\pmod{4}$ and $q=2p+1$ is a prime then $q|(2^p-1)$ if $2^p-1$ is composite.
Also, prove that there are infinitely many primes $p$ for which $2^p-1$ is composite.