<s>[INST] Let $x \in \mathbb{N}$ and let $p$ be a prime divisor of $x^4+x^3+x^2+x+1$, prove that $p = 5$ or $p \equiv 1 \mod{5}$ 
Let $x \in \mathbb{N}$ and let $p$ be a prime divisor of $x^4+x^3+x^2+x+1$, prove that $p = 5$ or $p \equiv 1 \mod{5}$

Note that $(x^4+x^3+x^2+x+1)(x-1) = x^5 - 1$.
I tried reducing the equation $\mod{5}$, which gave me some information on $x$, but I was not able to utilize this information.
 [/INST] If $r= ord _p(x)$ then by Fermat $r\mid p-1$ and by proposition $r\mid 5$. 
If $r=5$ then $p-1 =5k$ so $p\equiv_5 1$
If $r=1$ then $x=1$ so $p\mid 5$ and thus $p=5$.
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