How do you go about systematically solving a Diophantine equation of this form :
$217x^2 + 496y^2 = 15872$ ?
I found that $\gcd(217, 496) = 31$ and reduced that equation to
$7x^2 + 16y^2 = 512$
but then I got stuck there. I want to solve this using the modular arithmetic method, so a solution that takes such an approach will be highly appreciated.
Thanks in advance!