$5 . x 、 y$ are integers that satisfy the equation $y^{2}+3 x^{2} y^{2}=30 x^{2} +517$, find the value of $3 x^{2} y^{2}$ .
5. The original equation transforms to $\left(y^{2}-10\right)\left(3 x^{2}+1\right)$ $=3 \times 13^{2}>y^{2}-10=1 ， 3 ， 13 ， 39 ， 169$ or $507=>y^{2}=11,13,23,42,179$ or 517. Since $y$ is an integer $>y^{2}=49 \rightarrow y^{2}-10=39 \Rightarrow 3 x^{2}+1=13$ $\therefore 3 x^{2}=12>3 x^{2} y^{2}=12 \times 49=588$.