On the solutions of the generalized Fermat equation over totally real number fields

Abstract

Let $K$ be a totally real number field and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ over $K$ with prime exponent $p$, where $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even. For certain class of fields $K$, we prove that the equation $Ax^p+By^p+Cz^p=0$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2|abc$. Then, under some assumptions on $A,B,C$, we also prove that $Ax^p+By^p+Cz^p=0$ has no asymptotic solution in $K^3$. Finally, we give several purely local criteria of $K$ such that $Ax^p+By^p+Cz^p=0$ has no asymptotic solutions in $K^3$, and calculate the density of such fields $K$ when $K$ is a real quadratic field.