Find all natural numbers x,y,z

[MM]

Find all natural numbers x, y and z such that [tex]2^{x}+8^{y}=10^{z}[/tex].

[martin123456]

mod3: [tex](-1)^x+(-1)^y \equiv 1[/tex] => x,y are odd
1) [tex]x<3y[/tex]= > [tex]2^{x-z}(1+2^{3y-x})=5^z[/tex]=>[/tex]x=z[/tex]=>[tex]1+2^{3y-x}=5^{x}[/tex].
mod8: [tex]5^{odd} \equiv 5[/tex]=>[tex]3y-x \leq 2[/tex]
mod4; [tex]3y-x \geq 2[/tex]
=> [tex]3y=x+2[/tex]=>[tex]x=1[/tex],[tex]y=1[/tex]
the other cases are similar

[perfectmath]

What is the other cases ??