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arXiv · 2402.06644

Some results on a conjecture of de Polignac about numbers of the form $p + 2^k$

Yuda Chen·Xiangjun Dai·Huixi Li
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Abstract

We have primarily obtained three results on numbers of the form $p + 2^k$. Firstly, we have constructed many arithmetic progressions, each of which does not contain numbers of the form $p + 2^k$, disproving a conjecture by Erd\H{o}s as Chen did recently. Secondly, we have verified a conjecture by Chen that any arithmetic progression that do not contain numbers of the from $p + 2^k$ must have a common difference which is at least 11184810. Thirdly, we have improved the existing upper bound estimate for the density of numbers that can be expressed in the form $p + 2^k$ to $0.490341088858244$.

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Yuda Chen, Xiangjun Dai, Huixi Li. 2024-02-01. Some results on a conjecture of de Polignac about numbers of the form $p + 2^k$. https://arxiv.org/abs/2402.06644

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