A note on Romanov's constant
type: Item
instance of: scholarly article
title: A note on Romanov's constant
zbMATH Open document ID: 1121.11068
published in: Acta Mathematica Hungarica
publication date: 2007-01-08T00:00:00Z
review text: It was shown by \textit{N. P. Romanov} [Math. Ann. 109, 668--678 (1934; Zbl 0009.00801)] that a positive proportion of integers can be written as a prime plus a power of 2. Define \[ R_1:=\liminf_{N\to \infty} N^{-1}\#\{n\leq N:n=p+2^m\}. \] Then \textit{Y.-G. Chen} and \textit{X.-G. Sun} [J. Number Theory 106, 275--284 (2004; Zbl 1049.11106)] showed that \(R_1\geq 0.0868\). In the present paper this is improved to \[ R_1\geq 0.09368. \] (A very slightly weaker bound has been obtained independently and simultaneously by \textit{L. Habsieger} and \textit{X. F. Roblot} [Acta Arith. 122, 45--50 (2006; Zbl 1131.11064)]. The paper shows that there is little room for improvement in the current line of analysis, and goes on to discuss the connection with the Goldbach-Linnik problem, in which one represents an even integer as a sum of two primes and a fixed number of powers of 2. In particular, various hypotheses under which it suffices to use two powers of 2 are highlighted.
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 5082273
zbMATH Keywords: Romanov's constant
zbMATH Keywords: generalized twin prime problem
zbMATH Keywords: Goldbach-Linnik problem
zbMATH Keywords: Romanov's theorem, numerical constant
zbMATH Keywords: Goldbach-Linnik problem, sieve method
zbMATH Keywords: power of 2
author: János Pintz
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
full work available at URL: s10474-006-0060-6
OpenAlex ID: W2005537236
DOI: 10.1007/S10474-006-0060-6
description: scientific article; zbMATH DE number 5082273