  We solve the Diophantine equations $x^5 + y^5 = dz^p$ 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.
