Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1191_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$[\theta ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>θ</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> denote the integral part of the real number <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1191_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In this paper, we prove that for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1191_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$1<c<\frac{11312}{10029}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>c</mi> <mo><</mo> <mfrac> <mn>11312</mn> <mn>10029</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> and a sufficiently large integer N, the Diophantine equation <Equation ID="Equ37"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1191_Article_Equ37.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="150" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} [p_{1}^{c}]+[p_{2}^{c}]+[p_{3}^{c}]=N \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">[</mo> <msubsup> <mi>p</mi> <mrow> <mn>1</mn> </mrow> <mi>c</mi> </msubsup> <mo stretchy="false">]</mo> </mrow> <mo>+</mo> <mrow> <mo stretchy="false">[</mo> <msubsup> <mi>p</mi> <mrow> <mn>2</mn> </mrow> <mi>c</mi> </msubsup> <mo stretchy="false">]</mo> </mrow> <mo>+</mo> <mrow> <mo stretchy="false">[</mo> <msubsup> <mi>p</mi> <mrow> <mn>3</mn> </mrow> <mi>c</mi> </msubsup> <mo stretchy="false">]</mo> </mrow> <mo>=</mo> <mi>N</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>is solvable in prime variables <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1191_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$p_1,p_2,p_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1191_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="119" /> </InlineMediaObject> <EquationSource Format="TEX">$p_1=x^2+y^2+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with integers x and y. Moreover, we establish the corresponding asymptotic formula.

On a Diophantine equation involving one Linnik prime

期刊论文
发表日期：

2025年9月2日

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Yuhui Liu

摘要

Let $[\theta ]$ $[θ]$ denote the integral part of the real number $\theta .$ $θ .$ In this paper, we prove that for $1<c<\frac{11312}{10029}$ $1 < c < \frac{11312}{10029}$ and a sufficiently large integer N, the Diophantine equation $\begin{aligned} [p_{1}^{c}]+[p_{2}^{c}]+[p_{3}^{c}]=N \end{aligned}$ $\begin{matrix} [p_{1}^{c}] + [p_{2}^{c}] + [p_{3}^{c}] = N \end{matrix}$ is solvable in prime variables $p_1,p_2,p_3$ $p_{1}, p_{2}, p_{3}$ such that $p_1=x^2+y^2+1$ $p_{1} = x^{2} + y^{2} + 1$ with integers x and y. Moreover, we establish the corresponding asymptotic formula.