Show that there are infinitely many primes which are $\pm 1 \mod 5$
Let $p$ be a prime dividing $N^{2} - 5$. Thus $N^{2} \equiv 5 \pmod{p}$, so by the very definition $5$ is a quadratic residue modulo $p$, so $p \equiv \pm 1 \pmod{5}$.
If $p = q_{i}$ for some $i$, then $p$ divides $N$, and since $p$ divides $N^2 - 5$, we have that $p$ divides $5$, a contradiction.