For an integer m ≥ 2, let ℤ/mℤ be the set of all residue classes mod m. For S ⊆ ℤ/mℤ and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\bar{n}\in\mathbb{Z}/m\mathbb{Z},\,R_{S}(\bar{n})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>/</mo> </mrow> <mi>m</mi> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is defined as the number of solutions to the equation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\bar{n}=\bar{s}+\bar{s^{\prime}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow> <mover> <mi>s</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow> <mover> <msup> <mi>s</mi> <mrow> <mi mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> with an unordered pair <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\bar{s},\bar{s^{\prime}})\in S^{2}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo stretchy="false">(</mo> <mrow> <mover> <mi>s</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow> <mover> <msup> <mi>s</mi> <mrow> <mi mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>S</mi> <mrow> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\bar{s}\ne\bar{s^{\prime}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>s</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>≠</mo> <mrow> <mover> <msup> <mi>s</mi> <mrow> <mi mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>. In this paper, the author determines the structures of sets A and B such that A ⋃ B = ℤ/mℤ, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A\;\cap\;B=\bar{k}\mathbb{Z}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>A</mi> <mspace width="thickmathspace" /> <mo>∩</mo> <mspace width="thickmathspace" /> <mi>B</mi> <mo>=</mo> <mrow> <mover> <mi>k</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$R_{A}(\bar{n})=R_{B}(\bar{n})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>R</mi> <mrow> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>R</mi> <mrow> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\bar{n}\in\mathbb{Z}/m\mathbb{Z}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>n</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>/</mo> </mrow> <mi>m</mi> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>, where k is an integer.

Residue Class Ring with Identical Representation Function

期刊论文
发表日期：

2025年9月17日

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Shiqiang Chen

摘要

For an integer m ≥ 2, let ℤ/mℤ be the set of all residue classes mod m. For S ⊆ ℤ/mℤ and $\bar{n}\in\mathbb{Z}/m\mathbb{Z},\,R_{S}(\bar{n})$ $\bar{n} \in Z / m Z, R_{S} (\bar{n})$ is defined as the number of solutions to the equation $\bar{n}=\bar{s}+\bar{s^{\prime}}$ $\bar{n} = \bar{s} + \bar{s^{'}}$ with an unordered pair $(\bar{s},\bar{s^{\prime}})\in S^{2}$ $(\bar{s}, \bar{s^{'}}) \in S^{2}$ and $\bar{s}\ne\bar{s^{\prime}}$ $\bar{s} \neq \bar{s^{'}}$ . In this paper, the author determines the structures of sets A and B such that A ⋃ B = ℤ/mℤ, $A\;\cap\;B=\bar{k}\mathbb{Z}$ $A \cap B = \bar{k} Z$ and $R_{A}(\bar{n})=R_{B}(\bar{n})$ $R_{A} (\bar{n}) = R_{B} (\bar{n})$ for all $\bar{n}\in\mathbb{Z}/m\mathbb{Z}$ $\bar{n} \in Z / m Z$ , where k is an integer.