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<p>In this note, we prove that for any given positive integer <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1068_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\kappa \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>, when <i>n</i> is bigger than a constant explicitly depending on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1068_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\kappa \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>, the <i>n</i>-th Fibonacci number has a primitive divisor not less than <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1068_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">\((\kappa +1) n-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On big primitive divisors of Fibonacci numbers

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • Haojie Hong

摘要

In this note, we prove that for any given positive integer \(\kappa \) κ , when n is bigger than a constant explicitly depending on \(\kappa \) κ , the n-th Fibonacci number has a primitive divisor not less than \((\kappa +1) n-1\) ( κ + 1 ) n - 1 .