Denote by card{} the cardinality of a set.
$\forall a, b (mod\ p), \forall r\in \Bbb N$, If $d := (r, p − 1)$ prove that
$card\{(x, y, z) (mod\ p) : ax^r +by^r ≡ cz^r (mod\ p)\} =card\{(X, Y, Z) (mod\ p) : aX^d +bY^d ≡ cZ^d (mod\ p)\}$
I started by proving that :
If $d := (r, p − 1)$ then $card\{x (mod\ p) : b ≡ ax^r (mod\ p)\}=card\{X (mod\ p) : b ≡ aX^d (mod\ p)\}$ but how can this result be compatible with the sum of equivalence class $mod\ p$?
Thank you for your help.