Let p be an odd prime and x be an indeterminate. Recently, Sun conjecture that <Equation ID="Equ13"> <EquationSource Format="TEX">$\begin{aligned} \det \left[ x+\left( \frac{j-i}{p}\right) \right] _{0\le i,j\le \frac{p-1}{2}}={\left\{ \begin{array}{ll} (\frac{2}{p})pb_px-a_p & \text{ if }\ p\equiv 1\ ({\textrm{mod}}\ 4),\\ 1 & \text{ if }\ p\equiv 3\ ({\textrm{mod}}\ 4), \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo movablelimits="true">det</mo> <msub> <mfenced close="]" open="["> <mi>x</mi> <mo>+</mo> <mfenced close=")" open="("> <mfrac> <mrow> <mi>j</mi> <mo>-</mo> <mi>i</mi> </mrow> <mi>p</mi> </mfrac> </mfenced> </mfenced> <mrow> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≤</mo> <mfrac> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </msub> <mo>=</mo> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mrow> <mo stretchy="false">(</mo> <mfrac> <mn>2</mn> <mi>p</mi> </mfrac> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> <msub> <mi>b</mi> <mi>p</mi> </msub> <mi>x</mi> <mo>-</mo> <msub> <mi>a</mi> <mi>p</mi> </msub> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>if</mtext> <mspace width="0.333333em" /> <mspace width="4pt" /> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mspace width="4pt" /> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="4pt" /> <mn>4</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mn>1</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>if</mtext> <mspace width="0.333333em" /> <mspace width="4pt" /> <mi>p</mi> <mo>≡</mo> <mn>3</mn> <mspace width="4pt" /> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="4pt" /> <mn>4</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$a_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$b_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>b</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> are rational numbers related to the fundamental unit and class number of the real quadratic field <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {Q}(\sqrt{p})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mi>p</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we confirm this conjecture based on Vsemirnov’s decomposition of Chapman’s “evil determinant”.

On a generalization of R. Chapman’s “evil determinant”

期刊论文
发表日期：

2026年2月12日

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Li-Yuan Wang,
Hai-Liang Wu,
He-Xia Ni

摘要

Let p be an odd prime and x be an indeterminate. Recently, Sun conjecture that $\begin{aligned} \det \left[ x+\left( \frac{j-i}{p}\right) \right] _{0\le i,j\le \frac{p-1}{2}}={\left\{ \begin{array}{ll} (\frac{2}{p})pb_px-a_p & \text{ if }\ p\equiv 1\ ({\textrm{mod}}\ 4),\\ 1 & \text{ if }\ p\equiv 3\ ({\textrm{mod}}\ 4), \end{array}\right. } \end{aligned}$ $\begin{matrix} det {[x + (\frac{j - i}{p})]}_{0 \leq i, j \leq \frac{p - 1}{2}} = \{\begin{matrix} (\frac{2}{p}) p b_{p} x - a_{p} & if p \equiv 1 (mod 4), \\ 1 & if p \equiv 3 (mod 4), \end{matrix}) \end{matrix}$ where $a_p$ $a_{p}$ and $b_p$ $b_{p}$ are rational numbers related to the fundamental unit and class number of the real quadratic field $\mathbb {Q}(\sqrt{p})$ $Q (\sqrt{p})$ . In this paper, we confirm this conjecture based on Vsemirnov’s decomposition of Chapman’s “evil determinant”.