Why does the equation x2+3y2=2x^2 + 3y^2 = 2x2+3y2=2 have no integer solutions?
The discriminant is negative.
Modulo 3 analysis: x2+3y2≡x2(mod3)x^2 + 3y^2 \equiv x^2 \pmod{3}x2+3y2≡x2(mod3), which is 0 or 1, but 2≡2(mod3)2 \equiv 2 \pmod{3}2≡2(mod3).
The coefficients don't satisfy the gcd condition.
Because 2 is prime.