In this paper, we study arithmetic properties of certain determinants involving powers of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1900_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$i^2+cij+dj^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>i</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>c</mi> <mi>i</mi> <mi>j</mi> <mo>+</mo> <mi>d</mi> <msup> <mi>j</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where c and d are integers. For example, for any odd integer <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1900_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1900_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$(\frac{d}{n})=-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mfrac> <mi>d</mi> <mi>n</mi> </mfrac> <mo stretchy="false">)</mo> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> we prove that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1900_Article_IEq4.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="164" /> </InlineMediaObject> <EquationSource Format="TEX">$\det [(\frac{i^2+cij+dj^2}{n})]_{0\leqslant i,j\leqslant n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">det</mo> <msub> <mrow> <mo stretchy="false">[</mo> <mrow> <mo stretchy="false">(</mo> <mfrac> <mrow> <msup> <mi>i</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>c</mi> <mi>i</mi> <mi>j</mi> <mo>+</mo> <mi>d</mi> <msup> <mi>j</mi> <mn>2</mn> </msup> </mrow> <mi>n</mi> </mfrac> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> <mrow> <mn>0</mn> <mo>⩽</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>⩽</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> is divisible by <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1900_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi (n)^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1900_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$(\frac{\cdot }{n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mfrac> <mo>·</mo> <mi>n</mi> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the Jacobi symbol and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1900_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is Euler’s totient function. This confirms a previous conjecture of Sun.

Some Determinants Involving Binary Forms

期刊论文
发表日期：

2025年6月30日

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Yue-Feng She,
Zhi-Wei Sun

摘要

In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$ $i^{2} + c i j + d j^{2}$ , where c and d are integers. For example, for any odd integer $n>1$ $n > 1$ with $(\frac{d}{n})=-1$ $(\frac{d}{n}) = - 1$ we prove that $\det [(\frac{i^2+cij+dj^2}{n})]_{0\leqslant i,j\leqslant n-1}$ $det {[(\frac{i^{2} + c i j + d j^{2}}{n})]}_{0 ⩽ i, j ⩽ n - 1}$ is divisible by $\varphi (n)^2$ $φ {(n)}^{2}$ , where $(\frac{\cdot }{n})$ $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi $ $φ$ is Euler’s totient function. This confirms a previous conjecture of Sun.