Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1266_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( F_n\right) _{n \ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close=")" open="("> <msub> <mi>F</mi> <mi>n</mi> </msub> </mfenced> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1266_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( M_n\right) _{n \ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close=")" open="("> <msub> <mi>M</mi> <mi>n</mi> </msub> </mfenced> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> be the Fibonacci and Mulatu sequences. Let b be a positive integer such that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1266_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$b\ge 2.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>≥</mo> <mn>2</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In this paper, we prove that the following Diophantine equation <Equation ID="Equ27"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1266_Article_Equ27.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="221" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} F_n+\epsilon M_m=\rho \left( (b\pm 1)b^{\ell }\pm 1\right) , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>F</mi> <mi>n</mi> </msub> <mo>+</mo> <mi>ϵ</mi> <msub> <mi>M</mi> <mi>m</mi> </msub> <mo>=</mo> <mi>ρ</mi> <mfenced close=")" open="("> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>±</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>b</mi> <mi>ℓ</mi> </msup> <mo>±</mo> <mn>1</mn> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1266_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="240" /> </InlineMediaObject> <EquationSource Format="TEX">$(\epsilon , \rho )\in \{(1, 1), (-1, 1), (-1, -1)\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> have only finitely many solutions in non-negative integers n,  m,  b and positive integer <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1266_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Additionally, we present a method to determine all solutions within the range <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1266_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$2 \le b \le 12.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>b</mi> <mo>≤</mo> <mn>12</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> All this is done using linear forms in logarithms of algebraic numbers and Dujella–Pethő’s reduction method.

On Thabit and Williams numbers base b as sum or difference of Fibonacci and Mulatu numbers and vice versa

期刊论文
发表日期：

2025年1月27日

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Kouèssi Norbert Adédji,
Mohamadou Bachabi,
Alain Togbé

摘要

Let $\left( F_n\right) _{n \ge 0}$ ${(F_{n})}_{n \geq 0}$ and $\left( M_n\right) _{n \ge 0}$ ${(M_{n})}_{n \geq 0}$ be the Fibonacci and Mulatu sequences. Let b be a positive integer such that $b\ge 2.$ $b \geq 2 .$ In this paper, we prove that the following Diophantine equation $\begin{aligned} F_n+\epsilon M_m=\rho \left( (b\pm 1)b^{\ell }\pm 1\right) , \end{aligned}$ $\begin{matrix} F_{n} + ϵ M_{m} = ρ ((b \pm 1) b^{ℓ} \pm 1), \end{matrix}$ where $(\epsilon , \rho )\in \{(1, 1), (-1, 1), (-1, -1)\}$ $(ϵ, ρ) \in {(1, 1), (- 1, 1), (- 1, - 1)}$ have only finitely many solutions in non-negative integers n, m, b and positive integer $\ell .$ $ℓ .$ Additionally, we present a method to determine all solutions within the range $2 \le b \le 12.$ $2 \leq b \leq 12 .$ All this is done using linear forms in logarithms of algebraic numbers and Dujella–Pethő’s reduction method.