## Taiwanese Journal of Mathematics

### ON THE INTEGERS OF THE FORM $p+b$

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 10 July 2017

https://projecteuclid.org/euclid.twjm/1499706529

Digital Object Identifier
doi:10.11650/tjm.18.2014.3155

Mathematical Reviews number (MathSciNet)
MR3265080

Zentralblatt MATH identifier
1357.11099

#### Citation

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

#### References

• Y.-G. Chen, On integers of the forms $k^r-2^n$ and $k^r2^n+1$, J. Number Theory, 98 (2003), 310-319.
• Y.-G. Chen, Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers, Math. Comp., 74 (2005), 1025-1031.
• Y.-G. Chen, On integers of the forms $k\pm 2^n$ and $k2^n\pm 1$, J. Number Theory, 125 (2007), 14-25.
• Y.-G. Chen, R. Feng and N. Templier, Fermat numbers and integers of the form $a^k+a^l+p^{\alpha}$, Acta Arith., 135 (2008), 51-61.
• Y.-G. Chen, Romanoff theorem in a sparse set, Sci. China Math., 53 (2010), 2195-2202.
• F. Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comp., 29 (1975), 79-81.
• J. G. van der Corput, On de Polignac's conjecture, Simon Stevin, 27 (1950), 99-105.
• R. Crocker, On the sum of a prime and two powers of two, Pacific J. Math., 36 (1971), 103-107.
• R. Crocker, On the sum of two squares and two powers of $k$, Colloq. Math., 112 (2008), 235-267.
• P. Erdős, On integers of the form $2^k+p$ and some related problems, Summa Brasil. Math., 2 (1950), 113-123.
• M. Filaseta, C. Finch and M. Kozek, On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture, J. Number Theory, 128 (2008), 1916-1940.
• H. Halberstam and H.-E. Richert, Sieve Methods, Academic Press, New York, 1974.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
• F. Luca and P. St\ptmrs ǎnic\ptmrs ǎ, Fibonacci numbers that are not sums of two prime powers, Proc. Amer. Math. Soc., 133 (2005), 1887-1890.
• M. B. Nathanson, Additive Number Theory: The Classical Bases, Grad. Texts Math., 164, Springer-Verlag, New York, 1996.
• H. Pan, On the integers not of the form $p+2^a+2^b$, Acta Arith., 148 (2011), 55-61.
• H. Pan and W. Zhang, On the integers of the form $p^2+b^2+2^n$ and $b_1^2+b_2^2+2^{n^2}$, Math. Comp., 80 (2011), 1849-1864.
• A. de Polignac, Six propositions arithmologiques d\ptmrs éduites du crible d'Eratosth\ptmrs éne, Nouv. Ann. Math., 8 (1849), 423-429.
• N. P. Romanoff, Über einige Sätze der additiven Zahlentheorie, Math. Ann., 109 (1934), 668-678.
• X.-G. Sun and J.-H. Fang, On the density of integers of the form $(p-1)2^{-n}$ in arithmetic progressions, Bull. Aust. Math. Soc., 78 (2008), 431-436.
• Z.-W. Sun, On integers not of the form $\pm p^a\pm q^b$, Proc. Amer. Math. Soc., 128 (2000), 997-1002.
• Z.-W. Sun and M.-H. Le, Integers not of the form $c(2^a+2^b)+p^{\alpha}$, Acta Arith., 99 (2001), 183-190.
• Z.-W. Sun and S.-M. Yang, A note on integers of the form $2^n+cp$, Proc. Edinb. Math. Soc., 45 (2002), 155-160.
• K.-J. Wu and Z.-W. Sun, Covers of the integers with odd moduli and their applications to the forms $x^m-2^n$ and $x^2-F_{3n}/2$, Math. Comp., 78 (2009), 1853-1866.
• T. Tao, A remark on primality testing and decimal expansions, J. Aust. Math. Soc., 91 (2011), 405-413.
• P.-Z. Yuan, Integers not of the form $c(2^a+2^b)+p^{\alpha}$, Acta Arith., 115 (2004), 23-28.