In this paper, we will give an upper bound on n satisfying the Diophantine equation <Equation ID="Equ36"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_777_Article_Equ36.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="127" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} F_{n}\pm F_{m}=3^{s}\cdot y^{b} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>F</mi> <mi>n</mi> </msub> <mo>±</mo> <msub> <mi>F</mi> <mi>m</mi> </msub> <mo>=</mo> <msup> <mn>3</mn> <mi>s</mi> </msup> <mo>·</mo> <msup> <mi>y</mi> <mi>b</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in nonnegative integers <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_777_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="219" /> </InlineMediaObject> <EquationSource Format="TEX">$s\ge 0,y\ge 2,b\ge 2,n\ge m>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mi>y</mi> <mo>≥</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>≥</mo> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mi>m</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_777_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( y,3\right) =1.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <mi>y</mi> <mo>,</mo> <mn>3</mn> </mfenced> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Then, we determine all solutions (n, m, s, y) of this equation for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_777_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$b=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_777_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$2\le y\le 10^{4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>.

On the solutions of the Diophantine equation $F_{n}\pm F_{m}=3^{s}\cdot y^{b}$

期刊论文
发表日期：

2025年3月25日

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Zafer Şiar,
İbrahim Erduran

摘要

In this paper, we will give an upper bound on n satisfying the Diophantine equation $\begin{aligned} F_{n}\pm F_{m}=3^{s}\cdot y^{b} \end{aligned}$ $\begin{matrix} F_{n} \pm F_{m} = 3^{s} \cdot y^{b} \end{matrix}$ in nonnegative integers $s\ge 0,y\ge 2,b\ge 2,n\ge m>0$ $s \geq 0, y \geq 2, b \geq 2, n \geq m > 0$ and $\left( y,3\right) =1.$ $(y, 3) = 1 .$ Then, we determine all solutions (n, m, s, y) of this equation for $b=2$ $b = 2$ and $2\le y\le 10^{4}$ $2 \leq y \leq 10^{4}$ .

On the solutions of the Diophantine equation \(F_{n}\pm F_{m}=3^{s}\cdot y^{b}\)

摘要