How do I prove the following Argument:
No Mersenne Number is a Strong Pseudoprime to base $3$.
Can anyone build on from this or re-prove my argument:
Start by eliminating composite exponents $p$ of $2^p-1$, so then only prime exponents are left. If $2^p-1$ is composite with $p$ prime, then every prime divisor $q$ of $2^p-1$ has the form $2kp+1$. If the divisor $q$ of $2^p-1$ does not divide $3^p-1$, then $2^p-1$ is not a strong pseudoprime base $3$. Do this for all divisors $q$ of $2^p-1$. What next?