Prove that $p(x)=(x-1)(x-2) \cdots (x-n) + 1$ is irreducible over $\mathbb{Z}$ for all $n \geq 1$, $n \neq 4$.
Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$