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<p>We investigate the upper bound on the size of the set of integral points (<i>x</i>,&#xa0;<i>y</i>,&#xa0;<i>z</i>) for which <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1158_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">\(x+y=z\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation> such that <i>x</i>,&#xa0;<i>y</i>,&#xa0;<i>z</i> are positive squareful numbers and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1158_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(x,y,z\le B\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>≤</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation>, when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1158_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">\(B\rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Our result improves Theorem 1.3 of Browning and Valckenborgh [Experiment. Math. 21 (2012), pp. 204–211].</p>

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The upper bound on sums of three squareful numbers

  • 期刊论文

  • 发表日期：

    2025年7月19日

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  • Xiaodong Zhao

摘要

We investigate the upper bound on the size of the set of integral points (x, y, z) for which \(x+y=z\) x + y = z such that x, y, z are positive squareful numbers and \(x,y,z\le B\) x , y , z ≤ B , when \(B\rightarrow \infty \) B → ∞ . Our result improves Theorem 1.3 of Browning and Valckenborgh [Experiment. Math. 21 (2012), pp. 204–211].