Abstract
Let $B$ be a subset of positive integers, and $\mathcal{P}$ the set of all positive primes. For a subset $A$ of positive integers, $A(x)$ denotes the number of integers in $A$ not exceeding $x$. Let $\mathcal{S}$ denote the set of integers of the form $p+b$ with $p\in \mathcal{P}$ and $b\in B$. In this paper, we prove that if $B(x)\gg \log x/\log\log x$ and $B(cx)\gg B(x)$ for some positive constant $c\lt 1$, then $\mathcal{S}(x)\gg x/\log \log x$. This result is best possible in a sense: For any positive integer $m$, we construct an explicit subset $B$ of positive integers with $B(x)\gg (\log x)^m$ and $B(cx)\gg B(x)$ for any positive constant $c\lt 1$ such that $\mathcal{S}(x)\ll x/\log\log x$. We also give an application to the integers of the form $p+2^{a^2}+2^{b^2}$, where $p\in \mathcal{P}$ and $a,b$ are integers. Two open problems are posed for further research.
Citation
Quan-Hui Yang. Yong-Gao Chen. "ON THE INTEGERS OF THE FORM $p+b$." Taiwanese J. Math. 18 (5) 1623 - 1631, 2014. https://doi.org/10.11650/tjm.18.2014.3155
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