<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1007_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$D_{s}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation> be the integer such that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1007_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$5D_{s}=F_{10s+5}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>5</mn> <msub> <mi>D</mi> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow> <mn>10</mn> <mi>s</mi> <mo>+</mo> <mn>5</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1007_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$F_{10s+5}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mrow> <mn>10</mn> <mi>s</mi> <mo>+</mo> <mn>5</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1007_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$10s+5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>10</mn> <mi>s</mi> <mo>+</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>th number in Fibonacci sequence. In this paper, we prove that the class number of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1007_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Q}(\sqrt{-D_{s}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mo>-</mo> <msub> <mi>D</mi> <mi>s</mi> </msub> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is divisible by 5 when <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1007_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$s \not \equiv 0 (\textrm{mod}\text 20)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≢</mo> <mn>0</mn> <mo stretchy="false">(</mo> <mtext>mod2</mtext> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This gives another remaining infinite family of Fibonacci numbers that can be obtained by the Kishi’s method in (J Number Theory 128, 2450–2458, 2008).</p>

A new family of imaginary quadratic fields with class number divisible by 5

期刊论文
发表日期：

2025年1月31日

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Seokho Jin,
Kwang-Seob Kim

摘要

Let $D_{s}$ $D_{s}$ be the integer such that $5D_{s}=F_{10s+5}$ $5 D_{s} = F_{10 s + 5}$ where $F_{10s+5}$ $F_{10 s + 5}$ is the $10s+5$ $10 s + 5$ th number in Fibonacci sequence. In this paper, we prove that the class number of $\mathbb {Q}(\sqrt{-D_{s}})$ $Q (\sqrt{- D_{s}})$ is divisible by 5 when $s \not \equiv 0 (\textrm{mod}\text 20)$ $s ≢ 0 (mod2 0)$ . This gives another remaining infinite family of Fibonacci numbers that can be obtained by the Kishi’s method in (J Number Theory 128, 2450–2458, 2008).