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

2024/02/01 by Yuda Chen, Chen, Yuda, Xiangjun Dai +3
Mathematics · Physics and Astronomy · #11B25 #11Y16 #11Y55 #11Y60 #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #Primary 11P32 #Secondary 11B13

paper · pdf · doi:10.48550/arxiv.2402.06644

openalex publication_date 2024/02/01 · openalex created_date 2024/02/14 · openalex updated_date 2026/07/28

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 Erdős 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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