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Is it decidable when the following system of equation has solutions over the integers?

$$ \begin{matrix} x_1y_2-x_2y_1=a_1,\\ x_1y_3-x_3y_1=a_2,\\ x_2y_3-x_3y_2=a_3. \end{matrix} $$

Here $a_1$, $a_2$ and $a_3$ are integers which $GCD(a_1,a_2,a_3)=1$ and $x_1,x_2,x_3,y_1,y_2,y_3$ are variables.

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Using $x_1=0$, $x_2=-1$, $x_3=-1$, $y_1=a_1$, $y_2=a_2$ and $y_3=a_2-a_3$, you have a solution.

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  • $\begingroup$ This is a trivial solution. Always interesting when the solutions are written as a formula. $\endgroup$ – individ Sep 8 '15 at 11:48

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