If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then it is easy to see that for any integer $m$, at least one of the polynomials $P(m+1),P(m+2),P(m+3),P(m+4)$ is irreducible in ${\mathbb Z}[X]$. I believe a much stronger property holds, namely that for any degree $d\geq 2$ and length $l\geq 1$, there is a constant $C(d,l)$ such that for any polynomial $P\in {\mathbb Z}[X]$ of degree $d$, in any interval $I=\lbrace m+1,m+2, \ldots ,m+C(d,l) \rbrace$ we may encounter a subinterval of length $l$, $I'=\lbrace m+j+1,m+j+2, \ldots ,m+j+l \rbrace$ with $j+l \leq C(d,l)$ such that $Pa$ is irreducible in ${\mathbb Z}[X]$ for any $a\in I'$. Is that property already known to be true, and are bounds known for $C(d,l)$ ?
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