<p>Fix <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k \geq 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. For any <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$N \geq 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$F_k(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the cardinality of the largest subset of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$\{1,\dots,N\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>N</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> that does not contain <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$k$</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> distinct elements whose product is a square. Erdős, Sárközy, and Sós showed that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="160" /> </InlineMediaObject> <EquationSource Format="TEX">$F_2(N) = (\frac{6}{\pi^2}+o(1)) N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mfrac> <mn>6</mn> <msup> <mi>π</mi> <mn>2</mn> </msup> </mfrac> <mo>+</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="154" /> </InlineMediaObject> <EquationSource Format="TEX">$F_3(N) = (1-o(1))N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq100"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq100.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="135" /> </InlineMediaObject> <EquationSource Format="TEX">$F_k(N) \asymp N/\log N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>≍</mo> <mi>N</mi> <mo stretchy="false">/</mo> <mo>log</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> for even <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k \geq 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$F_k(N) \asymp N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>≍</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> for odd <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k \geq 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. Erdős then asked whether <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="155" /> </InlineMediaObject> <EquationSource Format="TEX">$F_k(N) = (1-o(1)) N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> for odd <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1505_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k \geq 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. Using a probabilistic argument, we answer this question in the negative.</p>

On product representations of squares

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发表日期：

2025年2月7日

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T. Tao

摘要

Fix $k \geq 2$ $k \geq 2$ . For any $N \geq 1$ $N \geq 1$ , let $F_k(N)$ $F_{k} (N)$ denote the cardinality of the largest subset of $\{1,\dots,N\}$ ${1, \dots, N}$ that does not contain $k$ $k$ distinct elements whose product is a square. Erdős, Sárközy, and Sós showed that $F_2(N) = (\frac{6}{\pi^2}+o(1)) N$ $F_{2} (N) = (\frac{6}{π^{2}} + o (1)) N$ , $F_3(N) = (1-o(1))N$ $F_{3} (N) = (1 - o (1)) N$ , $F_k(N) \asymp N/\log N$ $F_{k} (N) ≍ N / log N$ for even $k \geq 4$ $k \geq 4$ , and $F_k(N) \asymp N$ $F_{k} (N) ≍ N$ for odd $k \geq 5$ $k \geq 5$ . Erdős then asked whether $F_k(N) = (1-o(1)) N$ $F_{k} (N) = (1 - o (1)) N$ for odd $k \geq 5$ $k \geq 5$ . Using a probabilistic argument, we answer this question in the negative.