I make the mistake?

[traevkuq]

X^4 + Y^4 != Z^4 has been proved by Fermat that if X,Y,Z = integer numbers, the formula 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?

[Guest]

Your question seems to involve a variation of Fermat's Last Theorem where n is 4 and you're looking at the equation x^4 + y^4 = z^4. When you set x = X^2, y = Y^2, and z = Z^2, you're essentially looking at the equation (X^2)^4 + (Y^2)^4 = (Z^2)^4, which simplifies to X^8 + Y^8 = Z^8. This is a different equation than the classic Fermat's Last Theorem because the exponent is 8 instead of 4.

Fermat's Last Theorem specifically deals with the case when n is greater than 2. Wiles' proof applies to all such cases. Therefore, for n = 4 (as in your equation X^4 + Y^4 = Z^4), it is not part of Fermat's Last Theorem, and solutions do exist. However, for n = 8 (as in your equation X^8 + Y^8 = Z^8), it is still an open question whether there are integer solutions that violate Fermat's Last Theorem. There had been no proof one way or the other for n = 8, and it remains an unsolved problem.