Two results on 𝑥^{𝑟}+𝑦^{𝑟}=𝑑𝑧^{𝑝}

Abstract

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