A conjecture of Erd\H{o}s on $p+2^k$

Yong-Gao Chen

A conjecture of Erd\H{o}s on $p+2^k$

Number Theory 2024-02-20 v3 Combinatorics

Authors: 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.

Keywords

Erdős conjecture sum of prime and power of two arithmetic progressions asymptotic density additive number theory positive odd integers

Cite

@article{arxiv.2312.04120,
  title  = {A conjecture of Erd\H{o}s on $p+2^k$},
  author = {Yong-Gao Chen},
  journal= {arXiv preprint arXiv:2312.04120},
  year   = {2024}
}

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