<p>In this paper, we prove that when <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1114_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$1< c < \frac{43}{38}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>c</mi> <mo><</mo> <mfrac> <mn>43</mn> <mn>38</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, every sufficiently large positive integer <i>N</i> can be represented in the form <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1114_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="157" /> </InlineMediaObject> <EquationSource Format="TEX">$N=\left[ {p}_{1}^{c}\right] + \left[ {p}_{2}^{c}\right] + \left[ {p}_{3}^{c}\right] ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mfenced close="]" open="["> <msubsup> <mi>p</mi> <mrow> <mn>1</mn> </mrow> <mi>c</mi> </msubsup> </mfenced> <mo>+</mo> <mfenced close="]" open="["> <msubsup> <mi>p</mi> <mrow> <mn>2</mn> </mrow> <mi>c</mi> </msubsup> </mfenced> <mo>+</mo> <mfenced close="]" open="["> <msubsup> <mi>p</mi> <mrow> <mn>3</mn> </mrow> <mi>c</mi> </msubsup> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1114_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$p_1, p_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1114_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$p_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation> are primes and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1114_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\left[ \cdot \right] $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close="]" open="["> <mo>·</mo> </mfenced> </math></EquationSource> </InlineEquation> is the floor function, such that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1114_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="119" /> </InlineMediaObject> <EquationSource Format="TEX">$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>. We improve the previous result.</p>

An equation involving prime numbers and one Linnik prime

期刊论文
发表日期：

2025年5月22日

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Liqun Hu,
Fang Liu,
Siqi Liu

摘要

In this paper, we prove that when $1< c < \frac{43}{38}$ $1 < c < \frac{43}{38}$ , every sufficiently large positive integer N can be represented in the form $N=\left[ {p}_{1}^{c}\right] + \left[ {p}_{2}^{c}\right] + \left[ {p}_{3}^{c}\right] ,$ $N = [p_{1}^{c}] + [p_{2}^{c}] + [p_{3}^{c}],$ where $p_1, p_2$ $p_{1}, p_{2}$ , $p_3$ $p_{3}$ are primes and $\left[ \cdot \right] $ $[\cdot]$ is the floor function, such that $p_1 = x^2 + y^2+1$ $p_{1} = x^{2} + y^{2} + 1$ . We improve the previous result.