Problem:
Decide if the equation x^2+xy+y^2=2 has integer solutions (x,y) where x e y are both rational numbers.
Solution:
Suppose that the equation x^2+xy+y^2=2 has at least one rational solution \left(\dfrac{x_1}{x_2}, \dfrac{y_1}{y_2} \right), where we can suppose (x_1,x_2), (y_1, y_2) \in \mathbb{Z} \times \mathbb{Z^\ast}, with x_1, x_2 relatively primes and y_1, y_2 relatively primes.
We get \left(\frac{x_1}{x_2}\right)^2+\frac{x_1}{x_2}\frac{y_1}{y_2}+\left(\frac{y_1}{y_2}\right)^2=2
\Longleftrightarrow x^2_1y^2_2 + x_1x_2y_1y_2 + x^2_2y^2_1 = 2x^2_2y^2_2.
Setting a=x_1y_2, b=x_2y_1, c=x_2y_2, we obtain the equation a^2+ab+b^2=2c^2. Suppose that this equation has an integer solution (a, b, c). If this one is a solution, then also (-a, -b, -c) is a solution. Since c \neq 0, we can assume c > 0 and choose \max (a, b, c)>0 as small as possible. Now, it is clear that a and b are both even, otherwise we have 1 \equiv 0 \pmod{2} and also c is even, otherwise we have 0 \equiv 2 \pmod{4}, contradiction. So, a=2a_0, b=2b_0, c=2c_0, a_0,b_0,c_0 \in \mathbb{N}, c_0 > 0, therefore we have 4a^2_0+4a_0b_0+4b^2_0=8c^2_0 \iff a^2_0+a_0b_0+b^2_0=2c^2_0. Then (a_0, b_0, c_0) is a solution of the equation and \max(a_0,b_0,c_0) < \max (a, b, c) which contradicts the minimality of \max (a, b, c).
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