Why does the equation x2+y2=3z2x^2 + y^2 = 3z^2x2+y2=3z2 have no positive integer solutions?
The discriminant is negative.
Modulo 4 analysis: x2+y2≡0,1, or 2(mod4)x^2 + y^2 \equiv 0, 1, \text{ or } 2 \pmod{4}x2+y2≡0,1, or 2(mod4), but 3z2≡0 or 3(mod4)3z^2 \equiv 0 \text{ or } 3 \pmod{4}3z2≡0 or 3(mod4).
Fermat's Last Theorem directly applies.
The equation is quadratic in only two variables.