Two results on x^r + y^r = dz^p.

This note proves two theorems regarding Fermat-type equation x^r + y^r = dz^p where r \geq 5 is a prime. Our main result shows that, for infinitely many integers d, the previous equation has no non-trivial primitive solutions such that 2 \mid x+y or r \mid x+y, for a set of exponents p of positive density. We use the modular method with a symplectic argument to prove this result. [ABSTRACT FROM AUTHOR]