Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2088_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> be the n-th Catalan number. In this note, we prove that the product of two different Catalan numbers cannot be a square of an integer. On the other hand, for each <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2088_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, there are infinitely many k-tuples of pairwise different Catalan numbers with product being squares. We also obtain a characterization of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2088_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$x\in \mathbb {N}_{+}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mo>+</mo> </msub> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2024_2088_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{x}C_{x+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>x</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> is a power-full number and prove that there are infinitely many such x. Moreover we present some numerical results which motivate further problems.

Products of Catalan numbers which are squares

期刊论文
发表日期：

2025年1月23日

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Lajos Hajdu,
Florian Luca,
Szabolcs Tengely,
Maciej Ulas

摘要

Let $C_{n}$ $C_{n}$ be the n-th Catalan number. In this note, we prove that the product of two different Catalan numbers cannot be a square of an integer. On the other hand, for each $k\ge 3$ $k \geq 3$ , there are infinitely many k-tuples of pairwise different Catalan numbers with product being squares. We also obtain a characterization of $x\in \mathbb {N}_{+}$ $x \in N_{+}$ such that $C_{x}C_{x+1}$ $C_{x} C_{x + 1}$ is a power-full number and prove that there are infinitely many such x. Moreover we present some numerical results which motivate further problems.