A note on Romanov's constant
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.
scientific article; zbMATH DE number 5082273