论文标题
使用模块化方法渐近formatotic for Signatures $（r，r，p）$
Asymptotic Fermat for signatures $(r,r,p)$ using the modular approach
论文作者
论文摘要
让$ k $成为一个完全真实的领域，$ r \ geq 5 $固定的理性素数。在本文中，我们使用了Freitas和Siksek最近介绍的模块化方法研究非平凡的，原始的解决方案$（x，x，y，z）\ in \ Mathcal {o} _K^3 $ signature $（r，r，r，r，r，r，r，r，r，r，r，r，r，r，r，r，r，p）$ x^r+y^r+y^r+y^r+y^r = z^p $（需要对模块化方法进行改编，我们遵循Freitas的最新工作，该工作在$ K（ζ_R）$的完全真实子场上构建了Frey曲线。当$ k = \ mathbb {q} $时，我们会发现没有非平凡的，原始的整数解决方案$（x，y，z）$，$ 2 | z $ for Signatures $（r，r，r，p）$时37,47,53,59,61,67,71,79,83,83,101,103,107,131,139,149 \} $，$ P $足够大。二次字段的类似结果，例如，当$ k = \ mathbb {q}（\ sqrt {2}）$时，没有非无事的，原始的解决方案$（x，x，y，y，z）\ in \ mathcal {o} _k} _k^_k^3 $带有$ \ sqrt {2} $（5,5，p），（7,7，p）$，$（11,11，p），（13,13，p）$和足够大的$ p $。
Let $K$ be a totally real field, and $r\geq 5$ a fixed rational prime. In this paper, we use the modular method as presented in the recent work of Freitas and Siksek to study non-trivial, primitive solutions $(x,y,z) \in \mathcal{O}_K^3$ of the signature $(r,r,p)$ equation $x^r+y^r=z^p$ (where $p$ is a prime that varies). An adaptation of the modular method is needed, and we follow the recent work of Freitas which constructs Frey curves over totally real subfields of $K(ζ_r)$. When $K=\mathbb{Q}$ we get that there are no non-trivial, primitive integer solutions $(x,y,z)$ with $2|z$ for signatures $(r,r,p)$ when $r \in \{5,7,11,13,19,23, 37,47,53,59,61,67,71,79,83,101,103,107,131,139,149\}$ and $p$ is sufficiently large. Similar results hold for quadratic fields, for example when $K=\mathbb{Q}(\sqrt{2})$ there are no non-trivial, primitive solutions $(x,y,z)\in \mathcal{O}_K^3$ with $\sqrt{2}|z$ for signatures $(5,5,p),(7,7,p)$, $(11,11,p),(13,13,p)$ and sufficiently large $p$.
