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<p>In this paper, we find all solutions of the Diophantine equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1075_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">\(F_n^x+F_k^x=F_m^y\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>F</mi> <mi>n</mi> <mi>x</mi> </msubsup> <mo>+</mo> <msubsup> <mi>F</mi> <mi>k</mi> <mi>x</mi> </msubsup> <mo>=</mo> <msubsup> <mi>F</mi> <mi>m</mi> <mi>y</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1075_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{F_m\}_{m\ge 0}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>F</mi> <mi>m</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>m</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is the Fibonacci sequence.</p>

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On the Diophantine equation Fnx + Fkx = Fmy

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年4月4日

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  • Attila Bérczes,
  • Lajos Hajdu,
  • Florian Luca,
  • István Pink

摘要

In this paper, we find all solutions of the Diophantine equation \(F_n^x+F_k^x=F_m^y\) F n x + F k x = F m y , where \(\{F_m\}_{m\ge 0}\) { F m } m ≥ 0 is the Fibonacci sequence.