[proposed] NT EXO

Prove that the equation in
$(a,b,c,n)$ $a^2 + b^2 + c^2 = 2^nabc$ has no positive integer solutions.

WLOG
$c$ is even , then mod 4 we deduce that$a^2+b^2 \equiv 0$ hence the three are even
Now we write$a=2a_1$ and similarly for$b_1$ and$c_1$ , we repeat the same argument to get an infinite descent in positive integers, which is impossible ..

@AbdellahElmrini
yes it's just the infinite descent