Do I make the mistake?

X^4 + Y^4 != Z^4 has been proved by Fermat that if X,Y,Z = integer numbers, the formular is fine. Set x=X^2, y=Y^2, z=Z^2, so x, y, z are (some) integer numbers based on X,Y,Z.

x^4 + y^4 != z^4 //x, y, z are also integer, would be obey to Fermat's Fermat theorem, in which n=4.

X^8 + Y^8 != Z^8 // after instead x,y,z with X,Y,Z.

Do these X,Y,Y,n make Fermat theorem?