Asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields

Abstract

In this article, we study the asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields $K$, i.e., $A{x}^p+B{y}^p=C{z}^3$ with prime exponent $p$ and $A,B,C\in {\mathcal{O}}_K\setminus \{0\}$. For certain class of fields $K$, we prove that $A{x}^p+B{y}^p=C{z}^3$ has no asymptotic solutions over $K$ (resp., solutions of certain type over $K$) with restrictions on $A,B,C$ (resp., for all $A,B,C\in {\mathcal{O}}_K\setminus \{0\}$). Finally, we present several local criteria over $K$.

To appear in Journal of Number Theory. arXiv admin note: substantial text overlap with arXiv:2301.09263

References in corpus (1)

• 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

Cited by in corpus (2)

• Asymptotic Fermat equation of signature $(r,r,p)$ over totally real fields
• On the solutions of the generalized Fermat equation over totally real number fields