Long strings of composite values of polynomials and a basis of order 2

Abstract

We show that for any polynomial $f: \mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient and irreducible over $\mathbb{Q}$, if $N$ is large enough then there are two strings of consecutive positive integers $I_{1}=\{n_1-m,\ldots, n_1+m\}$ and $I_{2}=\{n_2-m, \ldots, n_2+m\}$, such that $m = [(\log N) (\log \log N)^{1/325565}]$, $I_{1}\cup I_{2} \subset [1, N]$, $N = n_1 + n_2$, and $f(n)$ is composite for any $n\in I_{1}\cup I_{2}$. This extends the result in [5] which showed the same result but with $f(n)=n$.