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

Artyom Radomskii

paper

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

arXiv:2506.15641

Abstract

We show that for any polynomial $f : Z \to Z$ with positive leading coefficient and irreducible over $Q$ , if $N$ is large enough then there are two strings of consecutive positive integers $I_{1} = {n_{1} - m, \dots, n_{1} + m}$ and $I_{2} = {n_{2} - m, \dots, n_{2} + m}$ , such that $m = [(lo g N) (lo g lo g N)^{1/325565}]$ , $I_{1} \cup I_{2} \subset [1, N]$ , $N = n_{1} + n_{2}$ , 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 $f (n) = n$ .