**(Partial) Question in short:** Let $p$ be a monic integer polynomial of degree $n$. Is there a natural number $k$ with $0 \leq k \leq n$ such that $p+k$ is irreducible over the integers?

Longer version:

Let $p$ be a monic polynomial over the integers. Define the irreducibility measure $d(p)$ of $p$ as the smallest integer $k \geq 0$ such that $p+k$ is irreducible over the integers. Define $M_n:=$ sup $\{ d(p) | deg(p)=n \}$ for $n \geq 2$. Here $deg(p)$ is the degree of $p$.

Question: Is it true that $M_n \leq n$? (Answer no, by Joachim König). Is there a good bound for $M_n$ ?

The question is based on some small computer experiments.

edit: Sorry I forgot the condition that the polynomials are monic (I did all computer experiments with that assumption. The answer by Joachim König gives a counterexample in the non-monic case).

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