Let M be a real number greater than 1. We consider continued fractions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_623_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$[0,c_{1},c_{2},\ldots ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_623_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$c_{i}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> are rational numbers greater than or equal to M (denoted by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_623_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$c_{i}\in \mathbb {Q}_{\ge M}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mi>i</mi> </msub> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Q</mi> <mrow> <mo>≥</mo> <mi>M</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>) for any positive integer i. In this paper, we give a sufficient condition for such a continued fraction to converge to an irrational number. Specifically, we determine how many non-integers must exist in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_623_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$c_{1},c_{2},\ldots ,c_{i}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>c</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> in order for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_623_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$[0,c_{1},c_{2},\ldots ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> to be irrational.

A simple condition for generalized continued fractions to converge to an irrational number

期刊论文
发表日期：

2025年2月19日

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Hayato Yoshida

摘要

Let M be a real number greater than 1. We consider continued fractions $[0,c_{1},c_{2},\ldots ]$ $[0, c_{1}, c_{2}, \dots]$ , where $c_{i}$ $c_{i}$ are rational numbers greater than or equal to M (denoted by $c_{i}\in \mathbb {Q}_{\ge M}$ $c_{i} \in Q_{\geq M}$ ) for any positive integer i. In this paper, we give a sufficient condition for such a continued fraction to converge to an irrational number. Specifically, we determine how many non-integers must exist in $c_{1},c_{2},\ldots ,c_{i}$ $c_{1}, c_{2}, \dots, c_{i}$ in order for $[0,c_{1},c_{2},\ldots ]$ $[0, c_{1}, c_{2}, \dots]$ to be irrational.