Question 3
Let $n\ge3$ be an integer and let $A_n$ be the set of positive integers less than $n$ and relatively prime to $n$. Define
$$P_n(x)=\sum_{k\in A_n}x^{k-1}.$$
(a) Prove that $P_n(x)$ is divisible by $x^r+1$ for some positive integer $r$.
(b) Find all such $n$ for which $P_n(x)$ is irreducible in $\mathbb Z[x]$.
$$P_n(x)=\sum_{k\in A_n}x^{k-1}.$$
(a) Prove that $P_n(x)$ is divisible by $x^r+1$ for some positive integer $r$.
(b) Find all such $n$ for which $P_n(x)$ is irreducible in $\mathbb Z[x]$.