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 $A{x}^p+B{y}^p+C{z}^p=0$ over $K$ with prime exponent $p$, where $A,B,C\in {\mathcal{O}}_K\setminus \{0\}$. For certain class of fields $K$, we prove that the equation $A{x}^p+B{y}^p+C{z}^p=0$ 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 $A{x}^p+B{y}^p+C{z}^p=0$ has no asymptotic solution in $K^3$. Finally, we give several purely local criteria of $K$ such that $A{x}^p+B{y}^p+C{z}^p=0$ has no asymptotic solutions in $K^3$, and calculate the density of such fields $K$ when $K$ is a real quadratic field.

arXiv commentWe have added an omitted nonvanishing hypothesis to Lemma 3.13(1) and Theorem 2.5, with corresponding revisions to the proofs and applications. The unnecessary assumption that ABC is even has been removed

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