Asymptotic Fermat for signatures $(r,r,p)$ using the modular approach

TL;DR

This paper extends the modular method to study the equation $x^r + y^r = z^p$ over totally real fields, proving the non-existence of certain primitive solutions for specific signatures and large primes.

Contribution

It adapts the modular approach to totally real fields for signatures $(r,r,p)$, providing new non-existence results for solutions over $Q$ and quadratic fields.

Findings

No primitive solutions with even $z$ for certain signatures over $Q$ when $p$ is large.

No primitive solutions with $Q( oot2)$-integer $z$ for specific signatures when $p$ is large.

Results extend Fermat-type non-existence to totally real fields using the modular method.

Abstract

Let $K$ be a totally real field, and $r\geq 5$ a fixed rational prime. In this paper, we use the modular method as presented in the recent work of Freitas and Siksek to study non-trivial, primitive solutions $(x,y,z) \in \mathcal{O}_K^3$ of the signature $(r,r,p)$ equation $x^r+y^r=z^p$ (where $p$ is a prime that varies). An adaptation of the modular method is needed, and we follow the recent work of Freitas which constructs Frey curves over totally real subfields of $K(\zeta_r)$. When $K=\mathbb{Q}$ we get that there are no non-trivial, primitive integer solutions $(x,y,z)$ with $2|z$ for signatures $(r,r,p)$ when $r \in \{5,7,11,13,19,23, 37,47,53,59,61,67,71,79,83,101,103,107,131,139,149\}$ and $p$ is sufficiently large. Similar results hold for quadratic fields, for example when $K=\mathbb{Q}(\sqrt{2})$ there are no non-trivial, primitive solutions $(x,y,z)\in \mathcal{O}_K^3$ with $\sqrt{2}|z$ for signatures $(5,5,p),(7,7,p)$, $(11,11,p),(13,13,p)$ and sufficiently large $p$.