TITLE: Primes as quotients
QUESTION [14 upvotes]: 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?

REPLY [2 votes]: I think for the equation:
$$\frac{x^2-1}{y^2-1}=k$$
It is necessary to record decisions. We will use the solutions of the Pell equation.
$$p^2-ks^2=1$$
And then the solutions are of the form:
$$x=-p^2+2kps-ks^2$$
$$y=p^2-2ps+ks^2$$