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.

We correct an error affecting Theorems~2.5 and~2.7 by adding a mild additional condition on the coefficients $A,B,C$. The corresponding proofs and subsequent applications have been revised accordingly. The scope and principal conclusions of these theorems, as well as all other main results, remain unaffected

References in corpus (3)

• On the solutions of $x^2=B{y}^p+C{z}^p$ and $2x^2=B{y}^p+C{z}^p$ over totally real fields
• Asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields
• Asymptotic Fermat equation of signature $(r,r,p)$ over totally real fields