On Integers of the Form

p+2^{k_1^{r_1}}+\cdots +2^{k_t^{r_t}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p + 2^{k_1^{r_1}} + \dots + 2^{k_t^{r_t}}$$\end{document}

, II

Abstract

Let r₁, …, r_t be given positive integers,

\mathcal{A}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{A}$$\end{document}

the set of all integers of the form

2^{k_1^{r_1}}+\cdots +2^{k_t^{r_t}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2^{k_{1}^{r_1}} + \cdots + 2^{k_{t}^{r_t}}$$\end{document}

, where k₁, …, k_t are positive integers, and

\mathcal{R}(r_1,\dots ,r_t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{R}(r_1, \dots, r_t)$$\end{document}

the set of all positive odd integers that can be represented as p + a, where p is a prime and

a\in \mathcal{A}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \in \cal{A}$$\end{document}

. It is easy to see that if

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_1^{-1} + \dots + r_t^{-1}< 1$$\end{document}

, then the set

\mathcal{R}(r_1,\dots ,r_t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{R}(r_1, \dots, r_t)$$\end{document}

has asymptotic density zero. Chen and Xu [J. Number Theory, 2024, 258: 66–93] proved that if

r_1^{-1}+\cdots +r_t^{-1}\ge 1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_1^{-1} + \dots + r_t^{-1} \geq 1$$\end{document}

, then the set

\mathcal{R}(r_1,\dots ,r_t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{R}(r_1, \dots, r_t)$$\end{document}

has a positive lower asymptotic density. In this paper, it is proved that (i) if

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_1^{-1} + \dots + r_t^{-1}< 1$$\end{document}

, then almost all integers in

\mathcal{R}(r_1,\dots ,r_t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\cal{R}(r_1, \dots, r_t)$$\end{document}

can be represented uniquely as p + a, where p is a prime and

a\in \mathcal{A}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \in \cal{A}$$\end{document}

; (ii) if

r_1^{-1}+\cdots +r_t^{-1}>1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_1^{-1} + \dots + r_t^{-1} > 1$$\end{document}

, then for any positive integer m, the set of positive odd integers that can be represented in at least m ways as p + a, where p is a prime and

a\in \mathcal{A}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \in \cal{A}$$\end{document}

, has a positive lower asymptotic density. Three conjectures are posed for further research.

Keywords

Erdős’ problem / Romanoff type problem / prime / asymptotic density / prime number theorem / 11P32 / 11A67 / 11N36

References

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