Find all prime numbers $p$ and $q$, such that $7p+q$ and $pq+11$ are also prime numbers.
Based on the fact that all primes, besides 2, are odd, I found that either $p$ or $q$ must be $2$ in order for $pq+11$ to be a prime number. From here, I found several pairs of $p$ and $q$ that work, but I don't know how to find all $p$ and $q$. I tried letting
and then adding the equations and using SFFT to get:
but it doesn't really help.