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

TL;DR

This paper proves that for large N, there exist two long strings of consecutive integers within [1, N] where a polynomial with positive leading coefficient and irreducibility over Q yields composite values, extending previous results for n.

Contribution

It generalizes earlier work by showing such composite strings exist for a broad class of polynomials, not just linear functions.

Findings

Existence of long composite strings for polynomial values

Extension of previous results from linear to polynomial functions

Quantitative bounds on the length of composite strings

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 $m = [(\log N) (\log \log N)^{1/325525}]$, $I_{1}\cup I_{2} \subset [1, N]$, $N = n_1 + n_2$, 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 $f(n)=n$.