Let $m,n$ be integers that satisfy $m^2+3m^2n^2=30n^2+517.$ Find $3m^2n^2$

Question

I first tried to use the quadratic equation to get values for $m$ and $n$, but that turned out to bee too long and complicated of a process. Is there any other way i can solve the equation? Completing the square, maybe?

Answer 1

In accordance with comments of Thomas Andrews and Student, $$(m^2-10)(3n^2+1) = 3\cdot13^2.$$ Then, $$\begin{cases} 3\not|\ 3n^2+1,\\ m^2-10 \not= 3,\\ \end{cases}$$ so the only solution with $$3n^2+1 = 13,\quad m^2-10 = 39,$$ or $3n^2 = 12, m^2=49,$ is possible. $4$ and $49$ are squares, so $3m^2n^2 = 12\cdot 49,$ $$\boxed{3m^2n^2 = 588}.$$