The Fermat-type equations x^5 + y^5 = 2z^p or 3z^p solved through Q-curves

Abstract

We solve the Diophantine equations $x^5+y^5=d{z}^p$ with $d=2,3$ for a set of prime numbers of density 1/4, 1/2, respectively. The method consists in relating a possible solution to another Diophantine equation and solving the later by using Q-curves and a generalized modular technique as in work of Ellenberg and Dieulefait-Jimenez along with some new techniques for eliminating newforms.