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<p>In this note, we prove a conjecture of Fatehizadeh and Yaqubi regarding the arithmetic mean of the first <i>n</i> Fibonacci numbers. More precisely, we show that there are infinitely many positive integers <i>n</i> such that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2166_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(n \mid \sum _{i=1}^{n} F_i\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∣</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>F</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2166_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">\(n+1 \mid \sum _{i=1}^{n+1} F_i\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>∣</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>F</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On integral consecutive arithmetic means of the first Fibonacci numbers

  • 期刊论文

  • 发表日期：

    2025年8月15日

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  • Florian Luca,
  • Diego Marques

摘要

In this note, we prove a conjecture of Fatehizadeh and Yaqubi regarding the arithmetic mean of the first n Fibonacci numbers. More precisely, we show that there are infinitely many positive integers n such that \(n \mid \sum _{i=1}^{n} F_i\) n ∣ ∑ i = 1 n F i and \(n+1 \mid \sum _{i=1}^{n+1} F_i\) n + 1 ∣ ∑ i = 1 n + 1 F i .