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

Let $K$ be a totally real field, and $r\ge 5$ 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 $2\mathrm{\mid }z$ 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 $\sqrt{2}\mathrm{\mid }z$ for signatures $(5,5,p),(7,7,p)$, $(11,11,p),(13,13,p)$ and sufficiently large $p$.