For each of the following equations, determine whether there are no solutions, finitely many solutions, or infinitely many solutions with $x, y$ justify your answers. $$x^2-5y^2=3 \\\ x^2+7y^2=10000 \\\ x^2-6y^2=30$$
I am not quite sure what I should say here. Every one of those equations should have a non trivial solution if I am not mistaken. Perhaps the first one has no solutions, the second has only finitely many and the last infinitely many?