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

Yong-Gao Chen

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arXiv · 2312.04120

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

Yong-Gao Chen

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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.

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Yong-Gao Chen. 2023-12-07. A conjecture of Erd\H{o}s on $p+2^k$. https://arxiv.org/abs/2312.04120

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