Residue Class Ring with Identical Representation Function

doi:10.1007/s11401-026-0044-5

Residue Class Ring with Identical Representation Function

2025-09-01

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Shiqiang Chen *

DOI：10.1007/s11401-026-0044-5

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For an integer m >= 2, let & Zopf;/m & Zopf; be the set of all residue classes mod m. For S subset of & Zopf;/m & Zopf; and nis an element of Z/mZ,RS(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 n=s+s '\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 (s,s ')is an element of S2\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 snot equal s '\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 & xcup; B = & Zopf;/m & Zopf;, A boolean AND B=kZ\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 RA(n)=RB(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 nis an element of Z/mZ\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

Residue Class Ring with Identical Representation Function

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