Long intervals of consecutive composite values of polynomials

Abstract

Let $f\in \mathbb{Q}[t]$ be an irreducible integer-valued polynomial of positive degree with positive leading coefficient. We show that there is a constant $c_f>0$ such that, for every sufficiently large real $X$, the interval $(X/2,X]$ contains a block of at least $c_f\log X\log \log X$ consecutive integers $n$ for which $f(n)$ is composite. The exponent of $\log \log X$ does not depend on the degree of $f$. This improves a result of Ford and Gabdullin, replacing a small fixed power of $\log \log X$ by $\log \log X$. The result includes the case $f(t)=t^2+1$.