论文标题
在$ x^p+y^p = 2^r z^p $，$ x^p+y^p = z^2 $上的解决方案上
On the solutions of $x^p+y^p=2^r z^p$, $x^p+y^p=z^2$ over totally real fields
论文作者
论文摘要
在本文中，我们研究了对二芬太丁方程的某种类型的非平凡原始解决方案$ x^p+y^p = 2^rz^p $和$ x^p+y^p = z^p = z^2 $ prime odpents $ p $，$ r \ in \ mathbb {n} $，而不是完全真实的字段$ k $。然后，对于$ r = 2,3 $，我们研究了$ \ mathcal {o} _k $的非平凡原始解决方案，用于方程$ x^p+y^p = 2^rz^rz^rz^p $ prime endent $ p $。最后，我们给出了$ k $的几个纯本地标准，以便方程$ x^p+y^p = 2^rz^p $在$ \ natercal {o} _k $上没有非平凡的原始解决方案。
In this article, we study the non-trivial primitive solutions of a certain type for the Diophantine equations $x^p+y^p=2^rz^p$ and $x^p+y^p=z^2$ of prime exponent $p$, $r \in \mathbb{N}$, over a totally real field $K$. Then for $r=2,3$, we study the non-trivial primitive solutions over $\mathcal{O}_K$ for the equation $x^p+y^p=2^rz^p$ of prime exponent $p$. Finally, we give several purely local criteria for $K$ such that the equation $x^p+y^p=2^rz^p$ has no non-trivial primitive solutions over $\mathcal{O}_K$.
