"Consecutive" irreducible polynomials - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T07:19:32Z http://mathoverflow.net/feeds/question/59956 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59956/consecutive-irreducible-polynomials "Consecutive" irreducible polynomials Ewan Delanoy 2011-03-29T07:07:02Z 2011-03-29T12:02:23Z <p>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 $P-a$ is irreducible in ${\mathbb Z}[X]$ for any $a\in I'$. Is that property already known to be true, and are bounds known for $C(d,l)$ ?</p>