Residue Class Ring with Identical Representation Function

Abstract

For an integer m ≥ 2, let ℤ/mℤ be the set of all residue classes mod m. For S ⊆ ℤ/mℤ and

\overline{n}\in \mathbb{Z}/m\mathbb{Z},R_S(\overline{n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\bar{n}\in\mathbb{Z}/m\mathbb{Z},\,R_{S}(\bar{n})$$\end{document}

is defined as the number of solutions to the equation

\overline{n}=\overline{s}+\overline{s^{\mathrm{\prime }}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\bar{n}=\bar{s}+\bar{s^{\prime}}$$\end{document}

with an unordered pair

(\overline{s},\overline{s^{\mathrm{\prime }}})\in S^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\bar{s},\bar{s^{\prime}})\in S^{2}$$\end{document}

and

\overline{s}\ne \overline{s^{\mathrm{\prime }}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\bar{s}\ne\bar{s^{\prime}}$$\end{document}

. In this paper, the author determines the structures of sets A and B such that A ⋃ B = ℤ/mℤ,

A\cap B=\overline{k}\mathbb{Z}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A\;\cap\;B=\bar{k}\mathbb{Z}$$\end{document}

and

R_A(\overline{n})=R_B(\overline{n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{A}(\bar{n})=R_{B}(\bar{n})$$\end{document}

for all

\overline{n}\in \mathbb{Z}/m\mathbb{Z}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\bar{n}\in\mathbb{Z}/m\mathbb{Z}$$\end{document}

, where k is an integer.

Keywords

Residue class / Representation function / 11B13

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