Long strings of consecutive composite values of polynomials

Abstract

We show that for any polynomial $f$ from the integers to the integers, with positive leading coefficient and irreducible over the rationals, if $x$ is large enough then there is a string of $(\log x)(\log \log x{)}^{1/835}$ consecutive integers $n\in [1,x]$ for which $f(n)$ is composite. This improves a result of the first author, Konyagin, Maynard, Pomerance and Tao, which states that there are such strings of length $(\log x)(\log \log x{)}^{c_f}$, where $c_f$ depends on $f$ and $c_f$ is exponentially small in the degree of $f$ for some polynomials.

22 pages