Let N denote a sufficiently large integer satisfying <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1163_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">$N \equiv 4 \ (\mathrm{{mod}}\ 24),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≡</mo> <mn>4</mn> <mspace width="4pt" /> <mo stretchy="false">(</mo> <mi mathvariant="normal">mod</mi> <mspace width="4pt" /> <mn>24</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1163_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$P_r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we proved that every sufficiently large N can be represented by the equation <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1163_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="167" /> </InlineMediaObject> <EquationSource Format="TEX">$N=p^2+x_1^2+x_2^2+x_3^2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mi>p</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mn>3</mn> <mn>2</mn> </msubsup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where p is a prime number, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1163_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$x_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1163_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$P_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-number and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1163_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$x_2x_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1163_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{169}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mn>169</mn> </msub> </math></EquationSource> </InlineEquation>-numbers. This result constitutes a refinement upon that of Tak Wing Ching.

Lagrange’s four squares theorem and Chen’s theorem

期刊论文
发表日期：

2025年7月19日

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Shuangrui Tian

摘要

Let N denote a sufficiently large integer satisfying $N \equiv 4 \ (\mathrm{{mod}}\ 24),$ $N \equiv 4 (\mod 24),$ and $P_r$ $P_{r}$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we proved that every sufficiently large N can be represented by the equation $N=p^2+x_1^2+x_2^2+x_3^2,$ $N = p^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2},$ where p is a prime number, $x_1$ $x_{1}$ is a $P_2$ $P_{2}$ -number and $x_2x_3$ $x_{2} x_{3}$ is a $P_{169}$ $P_{169}$ -numbers. This result constitutes a refinement upon that of Tak Wing Ching.