a prime divisor of $1+q+q^2+q^3+q^4$ Suppose $q$ is a prime power. Can we say anything about a prime divisor of $1+q+q^2+q^3+q^4$? Is any prime divisor congruent to 1 modulo 10?

Let $p$ be a prime divisor of $1+q+q^2+q^3+q^4$, then
$$1+q+q^2+q^3+q^4 \equiv 0 \pmod{p} \implies q^5-1 \equiv 0 \pmod{p}$$
So $q$ has order $1$ or $5$ in the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. If it has order $1$, then $q\equiv 1 \pmod{p}$, hence $$5 \equiv 1+q+q^2+q^3+q^4 \equiv 0 \pmod{p} \implies p=5$$
If $q$ has order $5$, from $|(\mathbb{Z}/p\mathbb{Z})^{\times}| = p-1$, we obtain $5|(p-1)$.
Hence every prime divisor of $1+q+q^2+q^3+q^4$ is either $5$ or congruent to $1$ modulo $5$. Since the number $1+q+q^2+q^3+q^4$ is always odd, its prime divisor is either $5$ or congruent to $1$ modulo $10$.
