Asymptotic Fermat equation of signature $(r, r, p)$ over totally real fields

TL;DR

This paper investigates the asymptotic solutions of the Fermat equation of signature $(r, r, p)$ over totally real fields, proving non-existence results for certain cases using the modular method.

Contribution

It establishes the non-existence of asymptotic solutions with even $c$ for a class of totally real fields and analyzes solutions with odd $c$ for the case $r=5$, advancing understanding of Fermat equations over such fields.

Findings

No asymptotic solutions with even $c$ over certain totally real fields.

Existence of asymptotic solutions with odd $c$ for $r=5$ case.

Application of the modular method to Fermat equations over totally real fields.

Abstract

Let $K$ be a totally real number field and $ \mathcal{O}_K$ be the ring of integers of $K$. This manuscript examines the asymptotic solutions of the Fermat equation of signature $(r, r, p)$, specifically $x^r+y^r=dz^p$ over $K$, where $r,p \geq5$ are rational primes and odd $d\in \mathcal{O}_K \setminus \{0\}$. For a certain class of fields $K$, we first prove that the equation $x^r+y^r=dz^p$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2 |c$. Then, we study the asymptotic solutions $(a,b,c) \in \mathcal{O}_K^3$ to the equation $x^5+y^5=dz^p$ with $2 \nmid c$. We use the modular method to prove these results.