I ask this question based on a comment of David Speyer in another question. What primes are of the form $$ \frac{p^2-1}{q^2-1} $$ where $p$ and $q$ are prime?
The first prime not apparently of this form is 17. The Diophantine equation $$ p^2-17q^2+16=0 $$ has solutions following a linear recurrence relation which has no primes in the first 1000 terms (only $(\pm1, 1)$ seeds may contain primes). But perhaps there is a better way to go about this?