Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1027_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$1<c<\frac{12{,}083}{8652}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>c</mi> <mo><</mo> <mfrac> <mrow> <mn>12</mn> <mo>,</mo> <mn>083</mn> </mrow> <mn>8652</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> be a fixed number, N be a sufficiently large positive number and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1027_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> denote a small positive number. In this paper, we prove that the Diophantine inequality <Equation ID="Equ22"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1027_Article_Equ22.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="235" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \left| p_{1}^{c}+p_{2}^{c}+p_{3}^{c}+p_{4}^{c}+p_{5}^{c}-N\right| <\varepsilon \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfenced close="|" open="|"> <msubsup> <mi>p</mi> <mrow> <mn>1</mn> </mrow> <mi>c</mi> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mrow> <mn>2</mn> </mrow> <mi>c</mi> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mrow> <mn>3</mn> </mrow> <mi>c</mi> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mrow> <mn>4</mn> </mrow> <mi>c</mi> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mrow> <mn>5</mn> </mrow> <mi>c</mi> </msubsup> <mo>-</mo> <mi>N</mi> </mfenced> <mo><</mo> <mi>ε</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>is solvable in prime variables <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1027_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">$p_1,p_2,p_3,p_4,p_5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>4</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>5</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1027_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="119" /> </InlineMediaObject> <EquationSource Format="TEX">$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. This result constitutes a refinement upon that of Dimitrov.

On a quinary Diophantine inequality involving one prime of the special form

期刊论文
发表日期：

2025年1月23日

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Yuhui Liu

摘要

Let $1<c<\frac{12{,}083}{8652}$ $1 < c < \frac{12, 083}{8652}$ be a fixed number, N be a sufficiently large positive number and $\varepsilon $ $ε$ denote a small positive number. In this paper, we prove that the Diophantine inequality $\begin{aligned} \left| p_{1}^{c}+p_{2}^{c}+p_{3}^{c}+p_{4}^{c}+p_{5}^{c}-N\right| <\varepsilon \end{aligned}$ $\begin{matrix} |p_{1}^{c} + p_{2}^{c} + p_{3}^{c} + p_{4}^{c} + p_{5}^{c} - N| < ε \end{matrix}$ is solvable in prime variables $p_1,p_2,p_3,p_4,p_5$ $p_{1}, p_{2}, p_{3}, p_{4}, p_{5}$ such that $p_1=x^2+y^2+1$ $p_{1} = x^{2} + y^{2} + 1$ with integers x and y. This result constitutes a refinement upon that of Dimitrov.