Let $$f(x)=(x-1)^2(x-2)^2(x-3)^2\cdots(x-2013)^2+2014\tag{1}$$

Prove or disprove: $f(x)$ is reducible over $\mathbb Q$.

See :http://www-irma.u-strasbg.fr/~bugeaud/travaux/PolyaType.pdf

Question 2:

Let $$f(x)=(x-1)^2(x-2)^2(x-3)^2\cdots(x-2013)^2+d,d\in \mathbb N^{+}.\tag{1}$$

For which $d$ is the polynomial $f(x)$ is reducible over $\mathbb Q$?

I guess that $d=4k+3,d=4k+1,k\in \mathbb N^{+}$ and $d$ is prime numbers. Is is true? Do we have other cases?