<p>The present paper studies the problem of Diophantine equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1755_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$a^{x} + \left( {kb + 1} \right)^{y} = z^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>a</mi> <mi>x</mi> </msup> <mo>+</mo> <msup> <mfenced close=")" open="("> <mrow> <mi>k</mi> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mi>y</mi> </msup> <mo>=</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1755_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$a,b,k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> are positive integers,<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1755_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$x,y,z$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation> are non-negative integers, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1755_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$a, \left( {kb + 1} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mfenced close=")" open="("> <mrow> <mi>k</mi> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> are primes such that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1755_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$a \equiv 1\left( {mod 3} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≡</mo> <mn>1</mn> <mfenced close=")" open="("> <mrow> <mi>m</mi> <mi>o</mi> <mi>d</mi> <mn>3</mn> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1755_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$k$</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> is the multiple of 3. Results depict that there do not exist positive integers <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1755_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$a,b,k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>, and non-negative integers <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1755_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$x,y,z$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation> that satisfy the equation <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40009_2025_1755_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$a^{x} + \left( {kb + 1} \right)^{y} = z^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>a</mi> <mi>x</mi> </msup> <mo>+</mo> <msup> <mfenced close=")" open="("> <mrow> <mi>k</mi> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfenced> <mi>y</mi> </msup> <mo>=</mo> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the Diophantine Equation a^x + (kb + 1)^y = z²

期刊论文
发表日期：

2025年8月15日

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Sudhanshu Aggarwal

摘要

The present paper studies the problem of Diophantine equation $a^{x} + \left( {kb + 1} \right)^{y} = z^{2}$ $a^{x} + {(k b + 1)}^{y} = z^{2}$ , where $a,b,k$ $a, b, k$ are positive integers, $x,y,z$ $x, y, z$ are non-negative integers, $a, \left( {kb + 1} \right)$ $a, (k b + 1)$ are primes such that $a \equiv 1\left( {mod 3} \right)$ $a \equiv 1 (m o d 3)$ , and $k$ $k$ is the multiple of 3. Results depict that there do not exist positive integers $a,b,k$ $a, b, k$ , and non-negative integers $x,y,z$ $x, y, z$ that satisfy the equation $a^{x} + \left( {kb + 1} \right)^{y} = z^{2}$ $a^{x} + {(k b + 1)}^{y} = z^{2}$ .

On the Diophantine Equation ax + (kb + 1)y = z2

摘要

On the Diophantine Equation a^x + (kb + 1)^y = z²