Taiwanese Journal of Mathematics


Quan-Hui Yang and Yong-Gao Chen

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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.

Article information

Taiwanese J. Math., Volume 18, Number 5 (2014), 1623-1631.

First available in Project Euclid: 10 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes 11A41: Primes

Romanoff theorem Polignac conjecture prime set


Yang, Quan-Hui; Chen, Yong-Gao. ON THE INTEGERS OF THE FORM $p+b$. Taiwanese J. Math. 18 (2014), no. 5, 1623--1631. doi:10.11650/tjm.18.2014.3155. https://projecteuclid.org/euclid.twjm/1499706529

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