A survey on the generalized Fermat equation of various signatures over totally real fields

Following the famous proof of Fermat’s Last Theorem by Andrew Wiles using the modularity of elliptic curves over $\mathbb{Q}$, significant developments have been made in the study of Diophantine equations using the modularity method. This article presents a survey of numerous results on the solutions of the generalized Fermat equation of signatures $(p,p,p)$, $(p,p,2)$, $(p,p,3)$, and $(r,r,p)$ over totally real number fields using the modularity method.