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## Homework Statement

Prove that the polynomial $(x-1)(x-2)...(x-n) + 1$ is irreducible over Z for n\geq 1 and n \neq 4

## Homework Equations

N/A

## The Attempt at a Solution

Let $f(x) = (x-1)(x-2) \cdots (x-n) + 1$ and suppose $f(x) = h(x)g(x)$ for some $h,g \in \mathbb{Z}[x]$ where $\deg(h), \deg(g) < n$. Note that $f(x) = 1$ for $x \in (1,2,\ldots,n)$ and so $h(x)g(x) = 1$ for $x \in (1,2,\ldots,n)$. This implies that $h(x) = \pm 1$ and $g(x) = \pm 1$ for those $x$ values. Moreover, we must have $g(x) - h(x) = 0$. Since a polynomial of degree $m$ is determined by $m+1$ points we have that $h(x) = g(x)~\forall x$. This implies that $f(x) = g(x)^2$. Now consider $f(n+1) = g(n+1)^2$. We have that $n! + 1 = j^2$ where $j \in \mathbb{Z}$. I feel like I'm close but don't know where to go from here... Any help will be greatly appreciated!