We know that $q=p^a$ and $q'=p'^b$ are odd prime powers and $p$ and $p'$ are of the form $4k+1$. Also $p\mid (q'^2+1)$ and $p'\mid (q^2+1)$. If $\pi(n)$ denotes the set of prime divisors of $n$, is it possible that the following relations are hold:
(1) $\pi(q^2-1)=\pi(q'^2-1)$
and
(2) $\pi(q^2+1)\cup \{p\}=\pi(q'^2+1)\cup \{p'\}$
We suspect that this is impossible but we can not prove it.