<p>Let <i>p</i> be an odd prime. For any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$b,c\in {\mathbb {Z}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>, Z.-W. Sun introduced the new-type determinant <Equation ID="Equ8"> <EquationSource Format="TEX">$\begin{aligned}D_p(b,c)=|(i^2+bij+cj^2)^{p-2}|_{1\leqslant i,j\leqslant p-1},\end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>D</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mrow> <mo stretchy="false">|</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>i</mi> <mn>2</mn> </msup> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mi>j</mi> <mo>+</mo> <mi>c</mi> <msup> <mi>j</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>⩽</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>and studied its arithmetic properties. In this paper we mainly prove that <Equation ID="Equ9"> <EquationSource Format="TEX">$\begin{aligned}\left( \frac{D_p(b,1)}{p}\right) =\left( \frac{2b}{p}\right) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfenced close=")" open="("> <mfrac> <mrow> <msub> <mi>D</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> <mi>p</mi> </mfrac> </mfenced> <mo>=</mo> <mfenced close=")" open="("> <mfrac> <mrow> <mn>2</mn> <mi>b</mi> </mrow> <mi>p</mi> </mfrac> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\left( \frac{b^2-4}{p}\right) =-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <mfrac> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>4</mn> </mrow> <mi>p</mi> </mfrac> </mfenced> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p\equiv 1\pmod 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. As an application of our result, we confirm several conjectures of Sun.</p>

Legendre symbols related to $D_p(b,1)$

期刊论文
发表日期：

2025年11月6日

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Xin-Qi Luo,
Wei Xia

摘要

Let p be an odd prime. For any $b,c\in {\mathbb {Z}}$ $b, c \in Z$ , Z.-W. Sun introduced the new-type determinant $\begin{aligned}D_p(b,c)=|(i^2+bij+cj^2)^{p-2}|_{1\leqslant i,j\leqslant p-1},\end{aligned}$ $\begin{matrix} D_{p} (b, c) = {| {(i^{2} + b i j + c j^{2})}^{p - 2} |}_{1 ⩽ i, j ⩽ p - 1}, \end{matrix}$ and studied its arithmetic properties. In this paper we mainly prove that $\begin{aligned}\left( \frac{D_p(b,1)}{p}\right) =\left( \frac{2b}{p}\right) \end{aligned}$ $\begin{matrix} (\frac{D_{p} (b, 1)}{p}) = (\frac{2 b}{p}) \end{matrix}$ when $\left( \frac{b^2-4}{p}\right) =-1$ $(\frac{b^{2} - 4}{p}) = - 1$ and $p\equiv 1\pmod 4$ $p \equiv 1 (mod 4)$ . As an application of our result, we confirm several conjectures of Sun.

Legendre symbols related to \(D_p(b,1)\)

摘要