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

Artyom Radomskii

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 $f(n)=n$.

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

Abstract

We show that for any polynomial with positive leading coefficient and irreducible over , if is large enough then there are two strings of consecutive positive integers and , such that , , , and is composite for any . This extends the result in [5] which showed the same result but with .

Paper Structure

This paper contains 7 sections, 4 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Setup
  4. Main Lemmas
  5. Deduction of Theorem \ref{['T1']} from Lemmas \ref{['L1']} and \ref{['L2']}
  6. Proof of Lemma \ref{['L1']}
  7. Acknowledgements

Key Result

Theorem 1.1

Let $f: \mathbb{Z}\to \mathbb{Z}$ be a polynomial of degree $B\geq 1$ with positive leading coefficient and irreducible over $\mathbb{Q}$. Let $0< \delta < C(1/2)$. Then for sufficiently large positive integer $N$ there are two strings of consecutive positive integers $I_{1}=\{n_1-m,\ldots, n_1+m\}$

Theorems & Definitions (6)

  • Theorem 1.1
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 6.1
  • proof