Pell's Equation and Continued Fractions

Question

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?

Answer 1

1. $3$ is not a quadratic residue mod 5, therefore there are no solutions.
2. $(100, 0)$ is a solution, solutions are bounded by $x^2 < 10000, 7y^2 < 10000$ thus there at most finitely many solutions.
3. (6,1) is a solution, there are infinitely many solutions to $x^2-6y^2=1$, thus there are infinitely many solutions.