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A conjecture of Erdős on $p+2^k$

Yong-Gao Chen

Abstract

Let $\mathcal{U}$ be the set of positive odd integers that cannot be represented as the sum of a prime and a power of two. In this paper, we prove that $\mathcal{U}$ is not a union of finitely many infinite arithmetic progressions and a set of asymptotic density zero. This gives a negative answer to a conjecture of P. Erd\H os. We pose several problems and a conjecture for further research.

A conjecture of Erdős on $p+2^k$

Abstract

Let be the set of positive odd integers that cannot be represented as the sum of a prime and a power of two. In this paper, we prove that is not a union of finitely many infinite arithmetic progressions and a set of asymptotic density zero. This gives a negative answer to a conjecture of P. Erd\H os. We pose several problems and a conjecture for further research.
Paper Structure (5 sections, 18 theorems, 189 equations)

This paper contains 5 sections, 18 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1']}
  3. A negative answer to Conjecture A
  4. Proofs of Theorem \ref{['thm1b']}, Theorem \ref{['thm1c']} and their corollaries
  5. Proofs of Theorems \ref{['thm2']} and \ref{['thm3']}

Key Result

Theorem 1.1

For any set $S$ of asymptotic density zero, $\mathcal{U}\cup S$ is not a union of finitely many infinite arithmetic progressions and a set of asymptotic density zero.

Theorems & Definitions (33)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.9
  • Corollary 1.10
  • Conjecture 1.14
  • Theorem 1.15
  • ...and 23 more