2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162577In this paper we are interested in solving the Fermat-type equations x^5+y^5=dz^p where d is a positive integer and p a prime number $\ge 7$. We describe a new method based on modularity theorems which allows us to improve all the results in a previous paper of the first author. We finally discuss the present limitations of the method by looking at the case d=3.major revision: mistakes corrected, several new cases coveredNumber Theorytext