A conjecture of Erdős on p+2^k

2023/12/07 by Yong-Gao Chen, Chen, Yong-Gao
Mathematics · #11A41 #11B13 #11B25 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2312.04120

openalex publication_date 2023/12/07 · openalex created_date 2023/12/09 · openalex updated_date 2026/07/28

Abstract

Let 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 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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