8. All positive integer solutions of the equation $x^{2}+3 y^{2}=2007$ are $\qquad$ .
8. $x=42, y=9$.

Since $a^{2} \equiv 0,1(\bmod 3), 2007 \equiv 0(\bmod 3)$, therefore, $x \equiv 0(\bmod 3)$.
Let $x=3 x_{1}$, similarly we get $y \equiv 0(\bmod 3)$.
Let $y=3 y_{1}$, then the original equation becomes $x_{1}^{2}+3 y_{1}^{2}=$
223. From $1 \leqslant x_{1}<\sqrt{223}$, we get $1 \leqslant x_{1} \leqslant 14$.
From $223 \equiv 1(\bmod 3)$, we know $x_{1} \equiv \pm 1(\bmod 3)$.
Also, $223 \equiv 3(\bmod 4)$, then $x_{1}$ is even.
Thus, $x_{1} \in\{2,4,8,10,14\}$.
Upon verification, $x_{1}=14, y_{1}=3$. Hence $x=42, y=9$.