Is there an integer polynomial $ A \in {\mathbb Z} [ X ]$ of degree $d\geq 2$ such that for any integer $n\in {\mathbb Z}$ , $ A(n) $ is a squarefree integer?
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No. WLOG $A$ is irreducible. Pick a sufficiently large prime $p$ dividing $A(k)$ for some $k$ (there are infinitely many such primes, for example by the argument here). In particular pick $p$ large enough so that it is relatively prime to the coefficients of $A$ and to $d$, and so that it does not divide the discriminant of $A$. Then the congruence $A(x) \equiv 0 \bmod p^2$ has a solution by Hensel's lemma. 

