Hello,

Consider this equation modulo 4 :

3=-1 mod 4

so we're back to

the only solution to this is

so x=4x', y=4y', z=4z', for some integers x', y', z' different from 0

thus the equation is

divide by 256 and you'll get the initial equation again.

this is the method of infinite ascent (if I'm not mistaking), similar to Fermat's infinite descent, and this means that there is no other solution than (0,0,0)