Let K be a totally real number field and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1168_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathcal {O}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> be the ring of integers of K. This manuscript examines the asymptotic solutions of the Fermat equation of signature (r, r, p), specifically <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1168_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$x^r+y^r=dz^p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mi>r</mi> </msup> <mo>+</mo> <msup> <mi>y</mi> <mi>r</mi> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>z</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> over K, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1168_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$r,p \ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>,</mo> <mi>p</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> are rational primes and odd <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1168_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$d\in \mathcal {O}_K \setminus \{0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>∈</mo> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. For a certain class of fields K, we first prove that the equation <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1168_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$x^r+y^r=dz^p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mi>r</mi> </msup> <mo>+</mo> <msup> <mi>y</mi> <mi>r</mi> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>z</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> has no asymptotic solution <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1168_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$(a,b,c) \in \mathcal {O}_K^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msubsup> <mi mathvariant="script">O</mi> <mi>K</mi> <mn>3</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with 2|c. We also study the asymptotic solutions <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1168_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$(a,b,c) \in \mathcal {O}_K^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msubsup> <mi mathvariant="script">O</mi> <mi>K</mi> <mn>3</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> to the equation <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1168_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$x^5+y^5=dz^p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mn>5</mn> </msup> <mo>+</mo> <msup> <mi>y</mi> <mn>5</mn> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>z</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1168_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$2 \not \mid c$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>∤</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation>. We use the modular method to prove these results.

Asymptotic Fermat equation of signature (r, r, p) over totally real fields

期刊论文
发表日期：

2025年8月4日

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Somnath Jha,
Satyabrat Sahoo

摘要

Let K be a totally real number field and $ \mathcal {O}_K$ $O_{K}$ be the ring of integers of K. This manuscript examines the asymptotic solutions of the Fermat equation of signature (r, r, p), specifically $x^r+y^r=dz^p$ $x^{r} + y^{r} = d z^{p}$ over K, where $r,p \ge 5$ $r, p \geq 5$ are rational primes and odd $d\in \mathcal {O}_K \setminus \{0\}$ $d \in O_{K} \ {0}$ . For a certain class of fields K, we first prove that the equation $x^r+y^r=dz^p$ $x^{r} + y^{r} = d z^{p}$ has no asymptotic solution $(a,b,c) \in \mathcal {O}_K^3$ $(a, b, c) \in O_{K}^{3}$ with 2|c. We also study the asymptotic solutions $(a,b,c) \in \mathcal {O}_K^3$ $(a, b, c) \in O_{K}^{3}$ to the equation $x^5+y^5=dz^p$ $x^{5} + y^{5} = d z^{p}$ when $2 \not \mid c$ $2 ∤ c$ . We use the modular method to prove these results.