Prove that $\frac{p^p-1}{p-1}$ is not prime if $p \equiv 1 \pmod 4$
Solution 1:
We may notice that $\frac{p^p-1}{p-1}=\Phi_p(p)$, where $\Phi_p$ is the $p$-th cyclotomic polynomial. If we assume that for some prime $q$ we have $\Phi_p(x)\equiv 0\pmod{q}$, then $x$ has order $p$ in $\mathbb{Z}/(q\mathbb{Z})^*$, hence $p\mid(q-1)$, or $q\equiv 1\pmod{p}$, by Lagrange's theorem. Additionally, the constraint $p\equiv 1\pmod{4}$ ensures that $\Phi_p(p)$ has a Aurifeuillean factorization.