The Padovan numbers <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2920_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$(P_n)_{n\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> form a ternary recurrence sequence, associated with the compositional number of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2920_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$n+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> in which each term is 2 or 3. The purpose of this work is to solve the Diophantine equation <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2920_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="116" /> </InlineMediaObject> <EquationSource Format="TEX">$P_n/P_m=P_k/P_\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> <mo stretchy="false">/</mo> <msub> <mi>P</mi> <mi>m</mi> </msub> <mo>=</mo> <msub> <mi>P</mi> <mi>k</mi> </msub> <mo stretchy="false">/</mo> <msub> <mi>P</mi> <mi>ℓ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> in positive integers n, m, k and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2920_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>.

Diophantine Equation Involving Ratios of Padovan Numbers

期刊论文
发表日期：

2025年8月4日

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Carlos Alexis Gómez,
Florian Luca,
Salah Eddine Rihane,
Alain Togbé

摘要

The Padovan numbers $(P_n)_{n\ge 0}$ ${(P_{n})}_{n \geq 0}$ form a ternary recurrence sequence, associated with the compositional number of $n+2$ $n + 2$ in which each term is 2 or 3. The purpose of this work is to solve the Diophantine equation $P_n/P_m=P_k/P_\ell $ $P_{n} / P_{m} = P_{k} / P_{ℓ}$ in positive integers n, m, k and $\ell $ $ℓ$ .