Finding primes $p$ and $q$ such that $p$ divides $q^2 -4$ and $q$ divides $p^2-1$ Finding all sets of primes $p$ and $q$ such that $p$ divides $q^2 -4$ and $q$ divides $p^2-1$.
Hint :

$p$ divides $q+2$ or $q-2$ and $q$ divides $p+1$ or $p-1$

Consider one by one case : 

$p=l(q+2),q=m(p+1)\Rightarrow p=l(m(p+1)+2)=lmp+lm+2l\Rightarrow p= ??$

try other cases...


$p=l(q+2),q=m(p-1)$ 
$p=l(q-2),q=m(p-1)$ 
$p=l(q-2),q=m(p+1)$
Four cases would be formed and only two would be left .

$p+1 \equiv 0 \pmod q$

$q+2 \equiv 0 \pmod p$

After some estimation it would be found that only $p=5$ and $q=3$ can be the solution.