Let $A, B, C \in \mathbb{N}$ be such that $\gcd(A,B,C)=1$. Does the equation $A x^n + By^n = C z^n$ have any solutions $x,y,z \in \mathbb{N}$? I know there are no such solutions if $A = B = C =1$, because this is Fermat's last theorem. I was just wondering if something similar was known when there are positive coefficients as well or not. Thank you very much!

Let $A, B, C \in \mathbb{N}$ be such that $\gcd(A,B,C)=1$. Is it known if the equation $A x^n + By^n = C z^n$ has any non-trivial solutions $x,y,z \in \mathbb{N}$? I know there are no such solutions if $A = B = C =1$, because this is Fermat's last theorem. I was just wondering if something similar was known when there are positive coefficients as well or not. Thank you very much!

PS By non-trivial solutions I mean the solutions that do not arise from degenerate cases.